SIMPLIFICATION

• Natural Numbers: All the counting numbers are called natural

number.
Example: 1, 2, 3, 4, 5, ......
(a) Even Numbers: The numbers which are exactly divisible by 2 are
called even numbers.
Example: 2, 4, 6, 8, ...
(b) Odd Numbers: The numbers which leave a remainder 1 when
divided by 2 are called odd numbers.
Example: 1, 3, 5, 7, ....
(c) Prime Numbers: If a number is not divisible by any other number
except 1 and itself, it is called a prime number.
Example: 2, 3, 5, 7, 11, ....
Co-primes- Two numbers which have no common factor between
them except 1 are said to be co-prime to each other. The two
numbers individually may be prime or composite.
Example: 13 and 29 are co-primes.
(d) Composite Numbers: Numbers which are divisible by other
numbers along with 1 and itself are called composite numbers.
Example: 4, 6, 8, 9, 10, .....
The number 1 is neither prime nor composite.
• Rational and Irrational Numbers: Any number which can be
expressed in the form of p / q, where p and q are integers and q not
equal to 0, is a rational number.
Example: 35, 4, –6, etc.
• Numbers which are represented by non–terminating and non–
recurring decimals are called irrational numbers.
Example: = 1.414....., = 1.732
• Real Numbers: Rational and irrational number taken together are
called real numbers.
• Whole Numbers: Natural numbers along with ‘0’ form the set of
whole numbers.
Example: 0, 1, 2, 3, .....
• Integers: All counting numbers and their negatives along with zero
are called Integers.
Example: ......-4, -3, -2, -1, 0, 1, 2, 3, 4
Tests of divisibility:

Divisibility by 2: A number is divisible by 2 if its unit digit is zero or an

even number.
Example: 248, 130
Divisibility by 3: A number is divisible by 3 if the sum of its digit is
divisible by 3.
Example: 279 --- 2 + 7 + 9 = 18.
18 is divisible by 3, hence 279 is divisible by 3.
Divisibility by 4: A number is divisible by 4 if the number formed by
its last two digits is divisible by 4.
Example: 236784
Here, 84 is divisible by 4, hence 236784 is divisible by 4.
Divisibility by 5: A number is divisible by 5 if the number or its unit
digit is either 5 or 0.
Example: 115, 240, etc
Divisibility by 6: A number is divisible by 6 if it is divisible by both 2
and 3.
Example: 318, 396, etc.
Divisibility by 8: A number is divisible by 8 if the number formed by
its last 3 digit is divisible by 8.
Example: 23816.
Here, 816 is divisible by 8, hence 23816 is divisible by 8
Divisibility by 9: A number is divisible by 9 if the sum of all its digits is
divisible by 9.
Example: 72936 ----- 7 + 2 + 9 + 3 + 6 = 27
27 is divisible by 9, hence 72936 is divisible by 9
Least Common Multiple (LCM)
LCM of two or more numbers is the least number which is divisible by
each of these numbers.
Example: Find the LCM of 36, 84 and 90
Highest Common Factor (HCF)
HCF is the largest factor of two or more given numbers.
HCF is also called Greatest Common Divisor (GCD).
Example: HCF of 144, 336
LCM and HCF of fractions:

LCM of fractions = LCM of Numerators / HCF of Denominators

HCF of fractions = HCF of Numerators / LCM of Denominators

BODMAS Rule.
B -Bracket
O - Of
D - Division
M - Multiplication
A - Addition
S – Subtraction
the order (), { }, []
• (2 + 3 x 6 – 1)
• (2 + 3 ) x (6 – 1 )
• 3 + (2 + (4 + 2 X (4 + 5)))
• 3 + {2 + [4 + 2 x (4 + 5)]}
Approximation
If the given value is 3.009, then the approximate value is 3.
If the given value is 4.45, then the approximate value is 4.50.
Example 1: 2959.85 / 16.001 – 34.99 = ?
(a) 160 (b) 150 (c) 140 (d) 180 (e) 170
(1702 / 68) × 136.05 = ?
(a) 3500 (b) 3550 (c) 3450 (d) 3400 (e) 3525
Multiplication by a number close to 10, 100, 1000, etc

999 = 1000 – 1 ; 101 = 100 + 1

Example: 46 × 98
Multiplication by 5 or powers of 5

2345 × 125
Square of a number which ends with 5.

=
Multiplication of a 2-digit number by a 2-digit number.
Example: 12 × 13
Multiplication of a 3-digit number by a 3-digit number
321 × 132
Finding minimum and maximum values in fractions:
Example: Find maximum value

, ,
69.69 – 51.54 + 73.64 = ? + 32.42
(a) 47.44 (b) 53.88 (c) 58.38
(d) 59.37 (e) None of these

14.28% of 49 = ?
(a) 8 (b) 11 (c) 7
(d) 16 (e) 15
(a) 1
(b) 1
(c) 1
(d) 1
(e) None of these
3/7 of 49/6 of 4/7 = ?
(a) 1 (b) 2 (c) 3
(d) 4 (e) 5

25% of 48 + 50% of 120 = ?% of 1200

(a) 4 (b) 5 (c) 6
(d) 8 (e) 16
(a) 24 (b) (16)2 (c) 25
(d) 16 (e) None of these

4 /6 =?
(a) 3/4 (b) 5/7 (c) 7/11
(d) 5/8 (e) None of these
26.5% of 488 = ?
(a) 205.65 (b) 211.72 (c) 145.67
(d) 129.32 (e) None of these
8000 ÷ 16-200 = ? × 6
(a) 75 (b) 60 (c) 50
(d) 25 (e) None of these

73 × 18 + 486 = ? +
(a) 1485 (b) 1631 (c) 1525
(d) 1225 (e) None of these
 th of th of 11200 = ?
(a) 1100 (b) 1220 (c) 1430
(d) 1200 (e) None of these

(6990 ÷ 15) × (468 ÷ 18) = ?

(a) 12161 (b) 12116 (c) 14000
(d) 13342 (e) None of these
 thof 24% of 500 – 32 = ?
(a) 20 (b) 30 (c) 50
(d) 40 (e) None of these

( -12) x 5 = ?
(a) 175 (b) 180 (c) 225
(d) 195 (e) None of these
(0.88 x 880/8) x 6 = ?
(a) 508.08 (b) 580.80 (c) 408.08
(d) 680.08 (e) None of these

90 x + 73 = ?
(a) 130 (b) 110 (c) 103
(d) 120 (e) None of these
 =?

(a) 15 (b) 25 (c) 35

(d) 20 (e) None of these
(a) 124 (b) 118 (c) 112
(d) 114 (e) None of these

2 × 256 × ? = × ×2
(a) 60 (b) 50 (c) 46
(d) 54 (e) None of these
63 + 371 ÷ 7 = ?
(a) 62 (b) 116 (c) 52
(d) 123 (e) None of these

38% of ? = 3596 – 632

(a) 7800 (b) 7900 (c) 8900
(d) 8700 (e) None of these
922 – 122 = 3535 + ?
(a) 4885 (b) 4785 (c) 5795
(d) 11855 (e) None of these

958 × 21 ÷ 4 = ?
(a) 5029.5 (b) 5039.3
(c) 5049.3 (d) 5019.5 (e) None of these
 of of of ? 3600 = ?
(a) 6000 (b) 7000
(d) 18000 (e) None of these

=?

(a) 70 (b) 60 (c) 20313.6

(d) 50 (e) None of these
 of of 1104 = ?
(a) 70 (b) 60 (c) 20313.6
(d) 50 (e) None of these

=?

(a) 25 (b) 20 (c) 578

(d) 26 (e) None of these
SIMPLIFICATION
PRACTICE
9229.789 – 5021.832 + 1496.989 = ?
(a) 6500 (b) 6000 (c) 6300 (d) 5700

1002 ÷ 49 × 99 – 1299 = ?
(a) 700 (b) 600 (c) 900 (d) 250
29.8% of 260 + 60.01% of 510 – 103.57 = ?
(a) 450 (b) 320 (c) 210 (d) 280

– + =?
(a) 25 (b) 120 (c) 10 (d) 65
√2498 × √626 ÷ √99 =
(a) 110 (b) 90 (c) 160 (d) 125

1599 × 199 ÷ 49 –1398 + 3877 = ?

(a) 9400 (b) 9000 (c) 8700 (d) 8400
4433.764 – 2211.993 – 1133.667 + 3377.442 = ?
(a) 4466 (b) 4377 (c) 363 (d) 4144

𝟐– 𝟐 + 𝟐 – 32.65 = ?

(a) 223 (b) 264 (c) 334 (d) 354

[ – + ]2 = ?
(a) –1800 (b) 1450 (c) –1660 (d) 1680

/ /
?
(a) 18 (b) 22 (c) 26 (d) 30
√4489 - √2601 =
(a) 14 (b) (c) 4 (d) 24

9067 + 2065 – 8400 + 3045 – 1520 = ?

(a) 4257 (b) 4157 (c) 4357 (d) 4047
1/16 of 8432 + 50% of ? = 4429
(a) 6804 (b) 8224 (c) 7884 (d) 7804

250% of ? ÷ 250 – 444 = 200

(a) 634000 (b) 6440 (c) 64400 (d) 64444000
 9= ?
0.01024 × ×
(a) 1 (b) 2 (c) 3 (d) 4

18 × 16 – 3445 ÷ 13 = ? – 344
(a) 369 (b) 367 (c) 368 (d) 467
 {[(3)2]6}5 = 9?
(a) 30 (b) 60 (c) 90 (d) 20

(a) 1 (b) 17 (c) 27 (d) 18

(2√392 – 21) +

(a) 5 (b) 6 (c) 7 (d) 8

(a) 21 (b) 25 (c) 29 (d) 7

Ratio, Proportion & Variation
• Comparisons of two or more Quantities.

• E.g. comparison of the ages, weights, income, savings, volume and

density etc.

• This chapter is very useful in solving the problems of Data

interpretation
Rule of Ratio:

•The comparison of two quantities is meaningless if they are not of

the same kind or in the same units (of length, volume or currency
etc).

•We do not compare 5 litres and 5 toys or 15 metres and 26

centimetres. Therefore, to find the ratio of two quantities (of the
same kind), it is necessary to express them in same units.

• Ratio has no units.

Ratio - The comparison between two quantities in terms of
magnitude is called the ratio, i e , it tells us that the one quantity is
how many times the other quantity.
•For example, Pankaj has 5 pens and Alesha has 3 pens.
•It means the ratio of number of pens between Pankaj and Alesha is
5 is to 3.
•It can be expressed as ‘5 : 3’-ratio is 5 : 3 while 3 : 5 is wrong
So the ratio of any two quantities is expressed as or a : b.

The numerator ‘a’ is called the antecedent and denominator ‘b’ is

called as consequent.
Properties of Ratios

1. The value of a ratio does not change when the numerator

and denominator both are multiplied by same quantities
2. The value of a ratio does not alter (or change) when the
numerator and denominator both are divided by same
quantities
3. The ratio of two fractions can be expressed in ratio of
integers.
4. When two or more than two ratios are multiplied with
each other, then it is called as compounded ratio e g .
5. When the ratio is compounded with itself, it is called as
duplicate, triplicate ratios etc.
Exp. 1) Find the ratio of 25 to 40
Exp. 2) Find the ratio of 90 cm to 1.5 m.
Exp. 3) The number of boys and girls in a school are 576 and 480 respectively.
Express the ratio of the number of boys to that of girls in the simplest form
Exp. 4) Shukla earns 14000rs per month and Mishra earns 18000rs per month.
Find the ratio of Shukla’s salary to Mishra’s salary.
Exp. 5) Out of 144 persons working in an office, 56 are
men and the remaining are women. Find the ratio of number of women to
number of men.
Exp. 6) In a club having 100 members, 20 play carom, 24
play table-tennis and 16 play cricket and the remaining
members do not play any game. No member plays more
than one game. Find the ratio of the number of members
who play.
(a) Carrom to the number of those who play table-tennis.
(b) Cricket to the number of those who play carrom.
(c) Cricket to the number of those who do not play any
game.
(d) Table-tennis to the number of those who do not play
any game.
(e) Some game to the number of those who do not play
any game
Exp. 7) A person earns 1200rs per day and spends 800rs.
Find the ratio of his savings to expenditure
Exp. 8) Simplify the following ratios :
Exp. 9) Divide 14 toffees among Ankita and Anshul in the ratio 5:2.
Exp. 10) Three boys are aged 2 years, 4 years and 8 years. They want to divide
seventy rupees in the ratio of their
ages. How much money would each get?
Exp. 11) An amount of 100 rs is being divided among two persons in the ratio 1
/ 10 : 1 / 15. How much money does each get?
Exp. 12) The lengths of sides of a triangle are in the ratio 2:3:4. If the perimeter
of the triangle is 63 cm, find thelengths of the sides of the triangle.
Exp. 13) Divide 1224 into three parts such that first part be double that of
second part and second part be 1/3 of the third part.
Exp. 14) If A : B = 3 : 4, B : C = 5:2 then find the value of A : B : C
Exp. 15) The ratio of A : B = 1 : 3, B : C = 2 : 5 , C : D = 2 : 3.Find the value
of A : B : C : D
Exp. 16) There are two types of mixtures of milk and water. In the first mixture,
out of 12 litres of mixture, 5 litre is milk only and in the second mixture, 6 litre
is milk and 12 litre is water. Which one mixture is better in terms of milk’s
strength?
Exp. 17) If a / b = 3 : 4 , then find the value of 7a- 4b : 3a + b
(a) 7 : 1 (b) 5 : 13
(c) 12 : 1 (d) none of these
Exp. 18) If a : b = 3 : 2 and b : c = 6 : 5 then a : b : c is equal to
(a) 9: 6 : 5 (b) 9 : 6 : 10 (c) 3 : 3 : 5 (d) 3 : 6 : 5
Exp. 19) The sum of two natural numbers is 64. Which of the following cannot
be the ratio of these two numbers?
(a) 3 : 5 (b) 1 : 3 (c) 7 : 9 (d) 3 : 4
Exp. 20) Monthly incomes of A B and are in the ratio of 4 : 3 and their savings
are in the ratio of 3 : 2. If the expenditure of each will be 600rs, then the
monthly incomes of each are
(a) 1800, 2400 (b) 2400, 1600 (c) 2400, 1800 (d) 1600, 1200
Exp. 21) A ,B and C have 40, x and y and balls with them respectively. If B gives
20 balls to A, he is left with half as many balls as C. If together they had 60
more balls, each of them would have had 100 balls on an average. What is
value of x : y ?
(a) 3 : 2 (b) 4 : 6 (c) 2 : 1 (d) 3 : 4
Exp. 22) The incomes of A , B and C are in the ratio of 12 : 9 : 7 and their
spending's are in the ratio 15 : 9 : 8. If A saves 25% of his income. What is the
ratio of the savings of A ,B and C?
Exp. 23) There are total 100 coins consisting of 20 paise, 50 paise and 1rs in the
ratio of 7 : 8 : 5. What is the no. Of coins of 50 paise if the difference
between the amount
yielded by 20 paise and 1 rs coin is 18?
Proportion
•An equality of two ratios is called a proportion and we say
that four numbers are in proportion.

•That is, if a / b = c / d or a : b = c : d , then we say that a, b, c and d

are in proportions and we write them as a : b: : c : d, where the
symbol ‘::’ indicates proportion and it is read as ‘a is to b as c is to d’.

•Here a d and are called extremes (or extreme terms) and b c and are
called as means (or middle terms). Thus four numbers are said to be in
proportion, if the ratio of the first to the second number is equal to
the ratio of the third to the fourth number. For Example, 2 3 4 6
•Proportionality Test

•If four numbers (quantities) are in proportion, then product of the

extremes is equal to the product of the means and if these are not in
proportion, then product of extremes is not equal to the product of
the means. That is if a : b :: c : d, then a x d = b x c

•Thus it is clear that if three out of four terms of a proportion are

given, we can find the fourth term by using this proportionality test.
•Proportionality Theorems.

(a) Invertendo: If = =

(b) Alternando: If = =

(c) Componendo: If = =

( ) ( )
(d) Dividendo If = =

( ) ( )
(e) Componendo and Dividendo: If = =
( ) ( )
•Continued Proportion

•If a, b and c and are three numbers such that a : b = b : c,

then these numbers a b , and c are said to be in continued
proportion or simply in proportion.

•That is, if a : b = b : c then =axc

•Here b is said to be the mean proportional to a, c and c is

said to be the third proportional to a and b For Example,
3 : 9 : : 9 : 27.
Exp. 1) The first, second and fourth terms of a proportion are 5, 15 and 90
respectively. Find the third term.
Exp. 2) The ratio of length to width of a rectangular sheet of paper is 5 : 3. If the
width of the sheet is 18 cm, find its
length.
Exp. 3) If 81 , x , x , 256 are in proportion, find x.
Exp. 4) The ratio between the number of men and women in an office is 5 : 7 If
the number of women working in the office is 56, find the number of men
working in the office.
Exp. 5) The age of Chandi and Radhika are in the ratio 5 : 3 If Chandi’s age is 20
years, find the age of Radhika.
Exp. 6)The ratio of the number of boys to that of girls in a school is 9 : 11. If the
number of girls in the school is 2035, find :
(a) number of boys in school
(b) number of students in school.
Exp. 7) What is the least possible number which must be
subtracted from 16, 19 and 23 so that the resulting
numbers are in continued proportion?
Exp. 8) If( a + b ) : ( a - b ) = 15 : 1 then the value of - is:
Exp. 9) The mean proportional between 8 and 98 is :
Exp. 10) The students in three classes are in the ratio of
2 : 3 : 4. If 40 students are added in each class, the ratio
becomes 4 : 5 : 6. Find the total number of students in all
the three classes is :
Exp. 11) The dimensions of a photograph are 4 and
1.8 cms. If the breadth of the enlarged photo is 4.5 cm and
it was enlarged proportionally then what is the new
length of new photograph?
Exp. 12) Two equal containers are filled with the mixture
of milk and water. The concentration of milk in each of the containers is 20%
and 25% respectively. What is the ratio of water in both the containers
respectively?
1. If A : B = 4 : 5, B : C = 3 : 4,C : D = 7 : 11, then A : D is
(a) 3 : 4 (b) 21 : 55 (c) 21 : 44 (d) 7 : 5
2. Mean proportional between 17 and 68 is :
(a) 51 (b) 24 (c) 4 (d) 34
3. Third proportional between 16 and 36 is :
(a) 64 (b) 144 (c) 81 (d) 49
4. If a : b = 2 : 3, then (5a + b ) : ( 3a + b )is :
(a) 13 : 12 (b) 15 : 17 (c) 12 : 13 (d) 13 : 11
 5. a = 2b = 3c = 4d, then a : b : c : dis
(a) 12 : 3 : 6 : 4 (b) 3 : 4 : 6 : 12
(c) 6 : 12 : 4 : 3 (d) 12 : 6 : 4 : 3
6. The fourth proportional to 4, 7 and 20 is :
(a) 28 (b) 21 (c) 18 (d) 35
7. If = = = then =?
(a) 2 (b) 3 (c) 4 (d) 5
9. If (a + b ) : ( a - b ) = 3 : 2, the ( - ):( ) equals :
(a) 5 : 13 (b) 12 : 13
(c) 9 : 4 (d) none of these
10. Two whole numbers, whose sum is 64, cannot be in
the ratio :
(a) 1 : 7 (b) 3 : 5 (c) 5 : 11 (d) 1 : 2
11. Two numbers are in the ratio 3 : 4. The difference between their squares is
28. Find the greater number
(a) 12 (b) 8 (c) 24 (d) 16
12. If A B and shared 1300rs in the ratio 1 : 12 , how much
did A get?
(a) 120 (b) 1200 (c) 100 (d) 1000
13. 3960rs are divided among A, B and C so that half of A’s
part, one-third of B’s part and one-sixth of C’s part are
equal. Then C’s part is :
(a) 720 (b) 2160 (c) 1080 (d) 810
14. A sum of ` 21000 is divided among A, B and C such that
shares of A and B are in the ratio of 2 : 3 and those of
B and C and are in the ratio 4 : 5. The amount received by A
is :
(a) ` 6000 (b) ` 4500 (c) ` 4800 (d) ` 8400
15. A certain amount was divided between A and B in the ratio7 : 9. If B’s share
was ` 7200, the total amount was :
(a) ` 1280 (b) ` 6300 (c) ` 5600 (d) ` 12800
16. ` 11250 are divided among A, B and C so that A may receive one-half as
much as B and C together receive and B receives one-fourth of what A and C
together receive. The share of A is more than that of B by :
(a) ` 2500 (b) ` 1500 (c) ` 1800 (d) ` 650
17. A girl 1.2 metre tall casts a shadow 1.1 m at the time when a building casts a
shadow 6.6 m long. The height of the building is :
(a) 2.7 m (b) 7.2 m (c) 6.0 m (d) 5.5 m
18. The prices of Bajaj Scooter and Bajaj Pulser are in the ratio of 4 : 9. If the
Bajaj Pulser costs ` 30000 more than a Bajaj Scooter, the price of Bajaj Pulser is :
(a) ` 63000 (b) ` 45000
(c) ` 54000 (d) ` 60000
19. What is the ratio whose terms differ by 40 and the
measure of which is 2/7?
(a) 16 : 56 (b) 14 : 49
(c) 15 : 36 (d) 16 : 72
20. Two numbers are in the ratio 3 : 5. If 9 be subtracted from each, then they
are in the ratio of 12 : 23. The
second number is :
(a) 53 (b) 54
(c) 55 (d) 52
21. In a mixture of 120 litres, the ratio of milk and water is 2 : 1. If the ratio of
milk and water is 1 : 2 , then the amount of water (in litres) is required to be
added is :
(a) 20 (b) 40 (c) 80 (d) 120
Unitary Method
At first we find the value of one unit and, then we find the value of
required number of units by multiplying the value of one unit with
the required number of units.
For example, if the price of 10 bananas is ` 50, find the price of 25
bananas. Then, as per unitary method, first we find the price of one
unit of bananas, which is ` 5. Thus, we can find the price of 25
bananas by multiplying the price of one unit
with the number of desired units, which is ` 125.
Direct Proportion

Two quantities are said to be directly proportional if the

increase (or decrease) in one quantity causes the increase
(or decrease) in the other quantity by same proportion. e.g.,

(i) The cost of articles varies directly with the number of

articles. More articles more cost, less articles less cost.

(ii) The work done varies directly with the number of men
(work force) at work. More men at work, more work done in
the same time. Less men, less work done in the same time.
Inverse Proportion

Two quantities are said to vary inversely if the increase (or

decrease) in one quantity causes the decrease (or increase) in
the other quantity by same proportion. e.g.,

The time taken to finish a work varies inversely to the number

of men at work.

More men at work, less time taken to finish the same work.
Less men at work, more time taken to finish the same work.
Exp. 1) If 6 note books cost ` 45, how much would
8 notebooks cost?
Exp. 2) If 45 students can consume a stock of food in
2 months, find for how many days the same stock of food
will last for 27 students?
Exp. 3) A man working 8 hours a day takes 5 days to
complete a project. How many hours a day must he work
to complete it in 4 days?
2. If 20 persons can do a piece of work in 7 days, then the
number of persons required to complete the work in
28 days :
(a) 4 (b) 5
(c) 14 (d) 10
3. If 20 men can reap a field in 38 days, in how many
days will 19 men reap the field?
(a) 21 days (b) 19 days
(c) 76 days (d) 40 days
4. 56 workers can reap a field in 8 days. If the work is to
be completed in 7 days, the extra workers needed are :
(a) 7 (b) 8 (c) 14 (d) 16
Proportion
•An equality of two ratios is called a proportion and we say
that four numbers are in proportion.

•That is, if a / b = c / d or a : b = c : d , then we say that a, b, c and d

are in proportions and we write them as a : b: : c : d, where the
symbol ‘::’ indicates proportion and it is read as ‘a is to b as c is to d’.

•Here a d and are called extremes (or extreme terms) and b c and are
called as means (or middle terms). Thus four numbers are said to be in
proportion, if the ratio of the first to the second number is equal to
the ratio of the third to the fourth number. For Example, 2 3 4 6
•Proportionality Test

•If four numbers (quantities) are in proportion, then product of the

extremes is equal to the product of the means and if these are not in
proportion, then product of extremes is not equal to the product of
the means. That is if a : b :: c : d, then a x d = b x c

•Thus it is clear that if three out of four terms of a proportion are

given, we can find the fourth term by using this proportionality test.
•Proportionality Theorems.

(a) Invertendo: If = =

(b) Alternando: If = =

(c) Componendo: If = =

( ) ( )
(d) Dividendo If = =

( ) ( )
(e) Componendo and Dividendo: If = =
( ) ( )
•Continued Proportion

•If a, b and c and are three numbers such that a : b = b : c,

then these numbers a b , and c are said to be in continued
proportion or simply in proportion.

•That is, if a : b = b : c then =axc

•Here b is said to be the mean proportional to a, c and c is

said to be the third proportional to a and b For Example,
3 : 9 : : 9 : 27.
Exp. 1) The first, second and fourth terms of a proportion are 5, 15 and 90
respectively. Find the third term.
Exp. 2) The ratio of length to width of a rectangular sheet of paper is 5 : 3. If
the width of the sheet is 18 cm, find its
length.
Exp. 3) If 81 , x , x , 256 are in proportion, find x.
Exp. 4) The ratio between the number of men and women in an office is 5 : 7 If
the number of women working in the office is 56, find the number of men
working in the office.
Exp. 5) The age of Chandi and Radhika are in the ratio 5 : 3 If Chandi’s age is 20
years, find the age of Radhika.
Exp. 6)The ratio of the number of boys to that of girls in a school is 9 : 11. If the
number of girls in the school is 2035, find :
(a) number of boys in school
(b) number of students in school.
Exp. 7) What is the least possible number which must be
subtracted from 16, 19 and 23 so that the resulting
numbers are in continued proportion?
Exp. 8) If( a + b ) : ( a - b ) = 15 : 1 then the value of - is:
Exp. 9) The mean proportional between 8 and 98 is :
Exp. 10) The students in three classes are in the ratio of
2 : 3 : 4. If 40 students are added in each class, the ratio
becomes 4 : 5 : 6. Find the total number of students in all
the three classes is :
Exp. 11) The dimensions of a photograph are 4 and
1.8 cms. If the breadth of the enlarged photo is 4.5 cm and
it was enlarged proportionally then what is the new
length of new photograph?
Exp. 12) Two equal containers are filled with the mixture
of milk and water. The concentration of milk in each of the containers is 20%
and 25% respectively. What is the ratio of water in both the containers
respectively?
1. If A : B = 4 : 5, B : C = 3 : 4,C : D = 7 : 11, then A : D is
(a) 3 : 4 (b) 21 : 55 (c) 21 : 44 (d) 7 : 5
2. Mean proportional between 17 and 68 is :
(a) 51 (b) 24 (c) 4 (d) 34
3. Third proportional between 16 and 36 is :
(a) 64 (b) 144 (c) 81 (d) 49
4. If a : b = 2 : 3, then (5a + b ) : ( 3a + b )is :
(a) 13 : 12 (b) 15 : 17 (c) 12 : 13 (d) 13 : 11
 5. a = 2b = 3c = 4d, then a : b : c : dis
(a) 12 : 3 : 6 : 4 (b) 3 : 4 : 6 : 12
(c) 6 : 12 : 4 : 3 (d) 12 : 6 : 4 : 3
6. The fourth proportional to 4, 7 and 20 is :
(a) 28 (b) 21 (c) 18 (d) 35
7. If = = = then =?
(a) 2 (b) 3 (c) 4 (d) 5
9. If (a + b ) : ( a - b ) = 3 : 2, the ( - ):( ) equals :
(a) 5 : 13 (b) 12 : 13
(c) 9 : 4 (d) none of these
10. Two whole numbers, whose sum is 64, cannot be in
the ratio :
(a) 1 : 7 (b) 3 : 5 (c) 5 : 11 (d) 1 : 2
11. Two numbers are in the ratio 3 : 4. The difference between their squares is
28. Find the greater number
(a) 12 (b) 8 (c) 24 (d) 16
12. If A B and shared 1300rs in the ratio 1 : 12 , how much
did A get?
(a) 120 (b) 1200 (c) 100 (d) 1000
13. 3960rs are divided among A, B and C so that half of A’s
part, one-third of B’s part and one-sixth of C’s part are
equal. Then C’s part is :
(a) 720 (b) 2160 (c) 1080 (d) 810
14. A sum of ` 21000 is divided among A, B and C such that
shares of A and B are in the ratio of 2 : 3 and those of
B and C and are in the ratio 4 : 5. The amount received by A
is :
(a) ` 6000 (b) ` 4500 (c) ` 4800 (d) ` 8400
15. A certain amount was divided between A and B in the ratio7 : 9. If B’s share
was ` 7200, the total amount was :
(a) ` 1280 (b) ` 6300 (c) ` 5600 (d) ` 12800
16. ` 11250 are divided among A, B and C so that A may receive one-half as
much as B and C together receive and B receives one-fourth of what A and C
together receive. The share of A is more than that of B by :
(a) ` 2500 (b) ` 1500 (c) ` 1800 (d) ` 650
17. A girl 1.2 metre tall casts a shadow 1.1 m at the time when a building casts a
shadow 6.6 m long. The height of the building is :
(a) 2.7 m (b) 7.2 m (c) 6.0 m (d) 5.5 m
18. The prices of Bajaj Scooter and Bajaj Pulser are in the ratio of 4 : 9. If the
Bajaj Pulser costs ` 30000 more than a Bajaj Scooter, the price of Bajaj Pulser is :
(a) ` 63000 (b) ` 45000
(c) ` 54000 (d) ` 60000
19. What is the ratio whose terms differ by 40 and the
measure of which is 2/7?
(a) 16 : 56 (b) 14 : 49
(c) 15 : 36 (d) 16 : 72
20. Two numbers are in the ratio 3 : 5. If 9 be subtracted from each, then they
are in the ratio of 12 : 23. The
second number is :
(a) 53 (b) 54
(c) 55 (d) 52
21. In a mixture of 120 litres, the ratio of milk and water is 2 : 1. If the ratio of
milk and water is 1 : 2 , then the amount of water (in litres) is required to be
added is :
(a) 20 (b) 40 (c) 80 (d) 120
Unitary Method
At first we find the value of one unit and, then we find the value of
required number of units by multiplying the value of one unit with the
required number of units.
For example, if the price of 10 bananas is ` 50, find the price of 25
bananas. Then, as per unitary method, first we find the price of one
unit of bananas, which is ` 5. Thus, we can find the price of 25
bananas by multiplying the price of one unit
with the number of desired units, which is ` 125.
Direct Proportion

Two quantities are said to be directly proportional if the

increase (or decrease) in one quantity causes the increase (or
decrease) in the other quantity by same proportion. e.g.,

(i) The cost of articles varies directly with the number of

articles. More articles more cost, less articles less cost.

(ii) The work done varies directly with the number of men
(work force) at work. More men at work, more work done in
the same time. Less men, less work done in the same time.
Inverse Proportion

Two quantities are said to vary inversely if the increase (or

decrease) in one quantity causes the decrease (or increase)
in the other quantity by same proportion. e.g.,

The time taken to finish a work varies inversely to the

number of men at work.

More men at work, less time taken to finish the same work.
Less men at work, more time taken to finish the same work.
Exp. 1) If 6 note books cost ` 45, how much would
8 notebooks cost?
Exp. 2) If 45 students can consume a stock of food in
2 months, find for how many days the same stock of food
will last for 27 students?
Exp. 3) A man working 8 hours a day takes 5 days to
complete a project. How many hours a day must he work
to complete it in 4 days?
2. If 20 persons can do a piece of work in 7 days, then the
number of persons required to complete the work in
28 days :
(a) 4 (b) 5
(c) 14 (d) 10
3. If 20 men can reap a field in 38 days, in how many
days will 19 men reap the field?
(a) 21 days (b) 19 days
(c) 76 days (d) 40 days
4. 56 workers can reap a field in 8 days. If the work is to
be completed in 7 days, the extra workers needed are :
(a) 7 (b) 8 (c) 14 (d) 16
Ratio & Proportion
PRACTICE
If two-third of A is four-fifth of B, then A : B = ?
(a) 5 : 6 (b) 6 : 5 (c) 10 : 9 (d) 9 : 10
Three numbers are in the ratio of 3 : 4 : 5. The sum of the largest and the
smallest equals the sum of the second and 52. The smallest number is :
(a) 20 (b) 27 (c) 39 (d) 52
If x : y = 2 : 1, then (x2 – y2) : (x2 + y2) is :
(a) 3 : 5 (b) 5 : 3 (c) 4 : 5 (d) 5 : 6
If a : b : c = 3 : 4 : 7, then the ratio (a + b + c) : c is equal to -
(a) 2 : 1 (b) 14 : 3 (c) 7 : 2 (d) 1 : 2
Three numbers are in the ratio The difference between the greatest
and the smallest numbers is 36. The numbers are :
(a) 72, 84, 108 (b) 60, 72, 96 (c) 72, 84, 96 (d) 72, 96, 108
If 78 is divided into three parts which are in the ratio of 1 : :
The middle part is :
(a) 9 (b) 13 (c) 17 (d) 18
The sum of two numbers is equal to 25 and their difference is 20. The
ratio of the two numbers is:
(a) 9 : 1 (b) 7 : 9 (c) 3 : 5 (d) 2 : 7
The difference between a two-digit number and the number obtained by
interchanging the digits is 36. What is the difference between the sum and the
difference of the digits of the number if the ratio between the digits of the
number is 1 : 2 ?
(a) 4 (b) 8 (c) 16 (d) 20
Seats for Mathematics, Physics and Biology in a school are in the ratio 5 :
7 : 8. There is a proposal to increase these seats by 40%, 50% and 75%
respectively. What will be the ratio of increased seats?
(a) 2 : 3 : 4 (b) 6 : 7 : 8 (c) 6 : 8 : 9 (d) 4 : 8 : 9
In a mixture 60 liters, the ratio of milk and water is 2 : 1. What quantity
of water should be added so that the ratio becomes 1 : 3?
(a) 80 L (b) 100 L (c) 120 L (d) 60 L
The ratio of the numbers of boys and girls of a school with 504 students
is 13 : 11. What will be the new ratio if 12 more girls are admitted?
(a) 91 : 81 (b) 81 : 91 (c) 9 : 10 (d) 10 : 9
The ratio of the number of boys and girls in a college is 7 : 8. If the percentage
increase in the number of boys and girls be 20% and 10% respectively, what
will be the new ratio?
(a) 8 : 9 (b) 17 : 18 (c) 21 : 22 (d) 22 : 21
The salaries of A, B and C are in the ratio 2 : 3 : 5. If the increments of 15%, 10%
and 20% are allowed respectively in their salaries, then what will be new ratio
of their salaries?
(a) 3 : 3 : 10 (b) 10 : 11 : 20 (c) 23 : 33 : 60 (d) 3 : 4 : 5
Two numbers are such that the ratio between them is 4 : 7. If each is increased
by 4, the ratio becomes 3 : 5. The larger number is :
(a) 36 (b) 48 (c) 56 (d) 64
Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased
by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit’s salary?
(a) Rs. 17,000 (b) Rs. 20,000 (c) Rs. 25,500 (d) Rs. 34,000
A sum of money is divided among C, A and B in ratio of 4 : 5 : 6 and another
sum of money is divided between M and N equally if B got 2000 more than M
then how much C get?
(a) Rs. 1000 (b) Rs. 500 (c) Rs. 4000 (d) Can't be determined
Three number A, B and C are in ratio of 12 : 15 : 25. If sum of these numbers be
364 find the ratio between difference of B and A and the difference of C and B?
(a) 3 : 2 (b) 3 : 10 (c) 3 : 5 (d) 4 : 2
In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 2 : 3 : 4. If there is
Rs. 50 in all, how many 5 p coins are there?
(a) 50 (b) 100 (c) 50 (d) 200
In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 1 : 2 : 3. If there are
Rs. 30 in all, how many 5p coins are there?
(a) 500 (b) 50 (c) 150 (d) 200
A box has 210 coins of denominations Rs. 1 and fifty paise only. The ratio of
their respective values is 13 : 11. The number of Rs. 1 coins is:
(a) 65 (b) 66 (c) 77 (d) 78
A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is
added to the mixture, the ratio becomes 4 : 5. Find the quantity of alcohol in
the given mixture.
(a) 5 Ltr. (b) 7.5 Ltr. (c) 10 Ltr. (d) 12 Ltr.
15 litres of mixture contains 20% alcohol and the rest water. If 3 litres of water
be mixed with it, the percentage of alcohol in the new mixture would be :
(a) 15% (b) 16 % (c) 17% (d) 18 %
85 L of a mixture contains milk and water in the ratio 27 : 7. How much more
water is to be added to get a new mixture containing milk and water in the ratio
3:1?
(a) 5 L (b) 6.5 L (c) 7.25 L (d) 8 L
The ratio of syrup and water in a bottle is 3 : 1. What part of the mixture is
taken out and same amount of water is added to it so that the ratio of syrup
and water in the mixture become 1 : 1?
(a) (b) (c) (d)
A mixture of milk and water contains 75% milk. If 8 liter of mixture is taken out
and 7 liters of milk is added, then ratio of milk to water becomes 7 : 2. Find
the quantity of mixture initially.
a) 56 b) 64 c) 72 d) 54
A mixture of 30 litres contains alcohol and water in ratio 3 : 7. How much
alcohol must be added to this mixture so that the ratio of alcohol and water
becomes 2 : 3 ?
a) 5ltr. b) 6 ltr. c) 7 ltr. d) 4 ltr.
600 g of sugar solution has 40% sugar in it. How much sugar should be added to
make it 50% in the solution?
a) 160 g b) 120 g c) 130 g d) 140 g
20 litres of a mixture contains milk and water in the ratio of 5 : 3. If four litres of
this mixture is replaced by four litres of milk, then the ratio of the milk to that
of the water in the new mixture will be
a) 3:7 b) 7:3 c) 7:9 d) 9:7
A vessel contains a mixture of milk and water in ratio 4:3. If 14 liters of mixture
is taken out and replaced with water, the ratio of milk and water becomes 3:4.
Find the initial quantity of mixture?
a) 42 b) 24 c) 56 d) 65
The ratio between the present ages of P and Q is 5 : 7 respectively. If the
difference between Q’s present age and P’s age after 6 years is 2, what is the
total of P’s and Q’s present ages?
(a) 48 years (b) 52 years (c) 56 years (d) 58 years
Radha and Rani are sisters. Five years back, the age of Radha was three times
that of Rani, but one year back the age of Radha was two times that of Rani.
What is the age difference between them?
(a) 8 (b) 9 (c) 10 (d) 11
5 years ago, the ratio of the age of A to that of B was 4:5. Five years hence, the
ratio of the age of A to that of B will be 6:7. If at present, C is 10 years younger
than B, then what will be the ratio of the present age of A to that of C?
(a) 5:4 (b) 3:2 c) 4:5 d) 6:7
On year ago, the ratio of the age (in years) of A to that of B was 4 : 3. The ratio
of their respective ages, 3 years from now, will be 6 : 5. What will be the ratio of
respective ages of A and B, 9 years from now?
(a) 8 : 7 (b) 10 : 9 (c) 9 : 8 (d) 7 : 6
Ten years before, the ages of a mother and her daughter were in the ratio
3 : 1. In another 10 yr. from now, the ratio of their ages will be 13 : 7. What are
their present ages?
(a) 39 yr, 21 yr (b) 55 yr, 25 yr (c) 75 yr, 25 yr (d) 49 yr, 31 yr
The sum of ages of a father, a mother, a son Sonu and daughters Savita and
Sonia is 96 yr. Sonu is the youngest member of the family. The year Sonu was
born, the sum of the ages of all the members of the family was 66 yr. If the
father’s age now is 6 times that of Sonu’s present age, then 12 yr. Hence, the
father’s age will be
(a) 44 yr (b) 45 yr (c) 46 yr (d) 48 yr
The ratio of the father’s age to his son’s age is 7 : 3. The product of their ages is
756. The ratio of their ages after 6 years will be :
(a) 5 : 2 (b) 2 : 1 (c) 11 : 7 (d) 13 : 9
The present ages of three persons are in proportion 4 : 7 : 9. Eight years ago,
the sum of their ages was 56. Find their present ages (in years).
(a) 8, 20, 28 (b) 16, 28, 36 (c) 20, 35, 4 (d) 13, 16, 25
The ratio of the present ages of a man and his wife is 4 : 3. After 4 years, this
ratio will be 9 : 7. If at the time of marriage, the ratio was 5 : 3, then how many
years ago were they married?
(a) 8 years (b) 10 years (c) 12 years (d) 15 years
10 yr ago, the ratio of ages of A and B was 13 : 17. After 17 yr from now, the
ratio of their ages will be 10 : 11. The present age of B is
(a) 23 yr (b) 40 yr (c) 27 yr (d) 44 yr
The difference between the ages of Meena and Seema is 3 yr and the ratio
between their ages is 7 : 8. What is the sum of their ages?
(a) 43 yr (b) 41 yr (c) 45 yr (d) 48 yr
Four years ago, the father’s age was three times the age of his son. The total of
the ages of the father and the son after four years, will be 64 years. What is the
father’s age at present?
(a) 32 years (b) 36 years (c) 44 years (d) 40 years
Gold is 19 times as heavy as water and copper is 9 times as heavy as
water. In what ratio should these be mixed to get an alloy 15 times as
heavy as water ?
(a) 1 : 1 (b) 2 : 3 (c) 1 : 2 (d) 3 : 2
A and B are two alloys of gold and copper prepared by mixing metals in
the ratio 7 : 2 and 7 : 11 respectively. If equal quantities of the alloys are
melted to form a third alloy C, the ratio of gold and copper in C will be :
(a) 5 : 7 (b) 5 : 9 (c) 7 : 5 (d) 9 : 5
An alloy of copper and zinc is taken in the ratio 1 : 2, and another alloy of
the same metal is taken in the ratio of 2 : 3. How many parts of the two
alloys must be taken to obtain a new alloy consisting of copper and zinc
that are in the ratio 3 : 5?
(a) 7 and 9 (b) 5 and 7 (c) 3 and 5 (d) 5 and 3
In two alloys, gold and silver are in the ratios of 4 : 1 and 1 : 3. After
alloying together 10 kg. of the first alloy, 16 kg. of the second and several
kilograms of pure gold, an alloy was obtained in which the ratio of gold to
silver was 3 : 2. Find the weight of the new alloy.
(a) 12 kg. (b) 19 kg. (c) 24 kg. (d) 35 kg.
Zinc and copper are in the ratio 2 : 1 in 75 kg. of an alloy. The amount of
copper to be further added to the alloy so as to make the ratio of zinc and
copper 1 : 2.
a) 72kg b) 57 kg c) 21 kg d) 75 kg
In an alloy, the ratio of Copper and Zinc is 5:2. if 1.250 kg of Zinc is mixed
in 17.5 kg alloy, then the ratio of copper and zinc will be…
a) 2:1 b) 2:3 c) 3:2 d) 1:2
What is the fourth proportional to 189, 273 and 153?
(a) 117 (b) 299 (c) 221 (d) 187
What is the third proportional to 10 and 20?
(a) 30 (b) 25 (c) 50 (d) 40
The monthly incomes of A and B are in the ratio 4 : 3. Each saves Rs.600. If
their expenditures are in the ratio 3 : 2, then what is the monthly income
of A?
(a) Rs.1800 (b) Rs.2000 (c) Rs.2400 (d) Rs.3600
The ratio of the income of A to that of B is 5 : 7. A and B save Rs.4,000 and
Rs.5,000 respectively. If the expenditure of A is equal to 66 % of the
expenditure of B, then the total income of A and B is:
(a) Rs.28,800 (b) Rs.26,400 c) Rs.25,200 (d) Rs.24,000
A sum of ₹x is divided among A, B and C such that the ratio of the shares
of A and B is 6:7 and that of B and C is 3:2. If the difference the shares of A
and C is ₹540, then the value of x is:
(a) 7155 (b) 7290 (c) 7020 (d) 7425
A sum is divided among A, B, C and D such that the ratio of the shares of A
and B is 2:3, that of B and C is 1:2 and that of C and D is 3:4. If the
difference between the shares of A and D is Rs 648, then the sum of their
shares is:
(a) Rs 1944 (b) Rs 2484 (c) Rs 2052 (d) Rs 2160
A sum of Rs.15525 is divided among Sunil, Anil and Jamil such that if Rs.22, Rs.35
and Rs.48 be diminished from their shares respectively, their remaining sums
shall be in the ratio 7 : 10 : 13. What would have been the ratio of their sums if
Rs.16, Rs.77, Rs.37 respectively were added to their original shares?
(a) 9 : 13 : 17 (b) 18 : 26 : 25 (c) 36 : 52 : 35 (d) None of these
A’s income is Rs.140 more than B’s income and C’s income is Rs.80 more than
D’s. If the ratio of A’s and C’s income is 2 : 3 and the ratio of B’s and D’s income is
1 : 2, then the incomes of A, B, C and D are respectively.
(a) Rs.260, Rs.120, Rs.320 and Rs.240 (b) Rs.300, Rs.160, Rs.600 and Rs.520
(c) Rs.400, Rs.260, Rs.600 and Rs.520 (d) Rs.320, Rs.180, Rs.480 and Rs.360
A and B have their monthly incomes in the ratio 8 : 5 while their monthly
expenditures are in the ratio 5 : 3. If they have saves Rs.12,000 and Rs.10,000
monthly respectively, then the difference in their monthly incomes is
(a) Rs.52,000 (b) Rs.42,000 (c) Rs.44,000 (d) Rs.46,000
A and B have their monthly incomes in the ratio 8 : 5 while their monthly
expenditures are in the ratio 5 : 3. If they have saves Rs.12,000 and Rs.10,000
monthly respectively, then the difference in their monthly expenditure is
(a) Rs.52,000 (b) Rs.40,000 (c) Rs.44,000 (d) Rs.46,000
A man divides his property so that his son’s share to his wife’s and wife’s
share to his daughter’s are both as in the ratio 3 : 1. If the daughter gets
Rs.10,000 less than son, the value (in rupees) of the whole property is
(a) Rs.16,250 (b) Rs.16,000 (c) Rs.18,250 (d) Rs.17,000
Rs.120 is distributed among A, B and C so that A’s share is Rs.20 more than
B’s and Rs.20 less than C’s. What is B’s share?
(a) Rs.10 (b) Rs.15 (c) Rs.20 (d) Rs.25
Mixtures and Alligations
• The concept of Alligations is simply an extension of Averages.

• Used to find out the percentage of impurity or reduction in the original

quantity where the repeated dilution or depreciation takes place.

• We can use it in the chapters such as Simple and Compound interest, Profit
and Loss and Ratio and Proportion .
How much Pepsi at Rs 6 a litre is added to 15 litre of ‘dew’ at Rs 10 a litre so
that the price of the mixture be ` 9 a litre?
(a) 5 (b) 8
(c) 10 (d) none of these
In a municipal parking there are some two wheelers and rest are 4 wheelers. If
wheels are counted, there are total
520 wheels but the incharge of the parking told me that there are only 175
vehicles. If no vehicle has a stepney,
then the no. of two wheelers is :
(a) 75 (b) 100 (c) 90 (d) 85
In my pocket there are ` 25 consisting of only the denominations of 20 paise and
50 paise. Thus there are total 80 coins in my pocket. The no. of coins of the
denomination of 50 paise is :
(a) 30 (b) 70 (c) 50 (d) 25
 In the 75 litres of mixture of milk and water, the ratio of milk and water is
4 : 1. The quantity of water required to
make the ratio of milk and water 3 : 1 is
(a) 1 litre (b) 3 litres (c) 4 litres (d) 5 litres
In my office the average age of all the female employees is 21 years and that
of male employees is 32 years, where the average age of all the (male and
female) employees is 28 years. The total no. of employees in my office could
be :
(a) 35 (b) 78 (c) 231 (d) 90
A car agency has 108 cars. He sold some cars at 9% profit
and rest at 36% profit. Thus he gains 17% on the sale of all
his cars. The no. of cars sold at 36% profit is :
(a) 25 (b) 32 (c) 35 (d) 75
A car agency has 108 cars. He sold some cars at 9% profit and rest at 36%
profit. Thus he gains 17% on the sale of all his cars. The no. of cars sold at 36%
profit is :
(a) 25 (b) 32 (c) 35 (d) 75
Rs 69 were divided among 115 students so that each girl gets 50 paise less than
a boy. Thus each boy received twice the paise as each girl received. The no. of
girls in the class is :
(a) 92 (b) 42 (c) 33 (d) 23
A butler stole wine from a butt of sherry containing 50% of spirit, then he
replenished it by different wine containing 20% spirit. Thus there was only 30%
strength (spirit) in the
new mixture. How much of the original wine did he steal?
(a) 1/3 (b) 2/3 (c) 1/2 (d) 1/4
Mr. Mittal purchased two steel factories, one in India and other one in Malaysia for total ` 72
crores. Later on he sold the Indian factory at 16% profit and Malasian factory at 24% profit. Thus
he gained a total profit of 19%. The selling price of Indian factory is :
(a) 45 crore (b) 52.2 crore
(c) 8.55 crore (d) can not be determined
In a 25 litre mixture of milk and water, the water is only 20%. How many
litres of water is required to increase the
percentage of water to 90%?
(a) 45 litre (b) 70 litre
(c) 115 litre (d) 175 litre
A milkman sells the milk at the cost price but he mixes the
water (freely available) in it and thus he gains 9.09%. The
quantity of water in the mixture of 1 litre is :
(a) 83.33 mL (b) 90.90 mL
(c) 99.09 mL (d) can’t be determined
The price of petrol is Rs 60 per litre and the price of spirit is Rs 40 per litre. In what ratio the
petrol and spirit be mixed such that the profit after selling the mixture at Rs 75 per litre be
25%?
(a) 1 : 1
(b) 3 : 2
(c) 5 : 1
(d) such a mixture is not possible
A trader sells total 315 TV sets. He sells black and white TV sets at a loss of 6% and colour TV
sets at a profit of 15%.
Thus he gains 9% on the whole. The no. of B/W TV sets, which he has sold, is :
(a) 126 (b) 216 (c) 135 (d) 90
In a class of 30 students, the average weight of boys is 20 kg and the average
weight of the girls is 25 kg. The fraction of boys out of the total students of the
class is :
(a) 4 : 5
(b) 5 : 6
(c) 3 : 4
(d) data insufficient
Baniya sells two types of tea viz. Desi Chai and Videshi Chai. He sells Desi Chai
at ` 18 per kg and incurs a loss of
10% whereas on selling the Videshi Chai at ` 30 per kg, he gains 20%. In what
proportion should the Desi Chai and Videshi Chai be mixed such that he can
gain a profit of 25%
by selling the mixture at ` 27.5 per kg?
(a) 3 : 2 (b) 2 : 3 (c) 2 : 5 (d) 3 : 5
 The average age of boys in a class is 16.66, while the
average age of girls is 18.75. Thus the average age of all the
40 students of the class is 17.5. If the difference between the
no. of boys and girls is 8, then the no. of girls in the class is :
(a) 12 (b) 16
(c) 18 (d) Data insufficient
The ratio of water and alcohol in two different containers is 2 3: and 4 5: . In
what ratio we are required to mix the
mixtures of two containers in order to get the new mixture in which the ratio
of alcohol and water be 7 5: ?
(a) 7 3: (b) 5 3: (c) 8 5: (d) 2 7
The average marks of the students in four sections A, B, C and D together is 60%. The average
marks of the students of A, B, C and D individually are 45%, 50%, 72% and 80%
respectively. If the average marks of the students of sections
A and B together is 48% and that of the students of B and C
together is 60%. What is the ratio of number of students in
sections A and D?
(a) 2 3: (b) 4 3: (c) 5 3: (d) 3 5
The diluted wine contains only 8 litres of wine and the rest is water. A new
mixture whose concentration is 30%, is to be formed by replacing wine. How
many litres of mixture shall be replaced with pure wine if there was initially
32 litres of water in the mixture ?
(a) 4 (b) 5
(c) 8 (d) none of these
The average weight of boys in a class is 30 kg and the
average weight of girls in the same class is 20 kg. If the
average weight of the whole class is 23.25 kg, what could be
the possible strength of boys and girls respectively in the
same class ?
(a) 14 and 26 (b) 13 and 27
(c) 17 and 27 (d) none of these
The shopkeeper mixed 40 kg refined oil with vegetable oil worth ` 60 per kg.
Thus he gains ` 10 after selling the
mixture of the two oils. The price of the first oil is :
(a) 20 (b) 25
(c) 45 (d) can’t be determined
In a mixture of milk and water, there is only 26% water.
After replacing the mixture with 7 litres of pure milk, the
percentage of milk in the mixture become 76%. The
quantity of mixture is :
(a) 65 litre (b) 91 litre
(c) 38 litre (d) none of these
The ratio of expenditure and savings is 3 2: . If the income
increases by 15% and the savings increases by 6%, then by
how much per cent should his expenditure increases?
(a) 25 (b) 21
(c) 12 (d) 24
4 kg of a metal contains 1 : 5
copper and rest is zinc. Another
5 kg of metal contains 1 : 6
copper and rest is zinc. The ratio
of copper and zinc into the mixture of these two metals :
(a) 49 : 221 (b) 39 : 231
(c) 94 : 181 (d) none of these
450 litres of a mixture of milk and water contain the milk and water in the ratio
9 : 1. How much water should be
added to get a new mixture containing milk and water in the ratio 3 : 1 ?
(a) 54 (b) 90
(c) 45 (d) 63
The ratio of petrol and kerosene in the container is 3 : 2 when 10 litres of the
mixture is taken out and is replaced
by the kerosene, the ratio becomes 2 : 3. The total quantity
of the mixture in the container is :
(a) 25 (b) 30
(c) 45 (d) cannot be determined
From a container, 6 litres milk was drawn out and was replaced by water. Again
6 litres of mixture was drawn out
and was replaced by the water. Thus the quantity of milk and water in the
container after these two operations is
9 : 16. The quantity of mixture is :
(a) 15 (b) 16
(c) 25 (d) 31
A milkman brings 100 litres pure milk from a dairy farmer and he sells 10 litres of it to the first customer, then he
refills his vessel by adding 10 litres water. After this, he proceeds to the next house and sells 10 litres of it to the
second customer and then he refills his vessel again by adding 10 litres of water. Thus, every time he sells 10 litres
of milk - pure or impure - he keeps on replacing it with 10 litres of pure water. Maximum how many customers can
get at least 50% milk in the mixture that they purchase from this milkman?
(a) 5 (b) 6
(c) 7 (d) None of these
Mixture and Alligation
Practice
Several students have taken an exam. There was an error in the answer key which
affected the marks of 48 students, and their average marks reduced from 78 to
66. The average of remaining students increased by 3.5 marks. This resulted the
reduction of the average of all students by 4.5 marks. The number of students
that attended the exam is:
(a) 96 (b) 84 (c) 93 (d) 100
A class consists of 20 boys and 30 girls. In the mid-semester examination, the
average score of the girls was 5 higher than that of the boys. In the final exam,
however, the average score of the girls dropped by 3 while the average score of
the entire class increased by 2. The increase in the average score of the boys is
(a) 9.5 (b) 10 (c) 7.5 (d) 8
In a company with 600 employees, the average age of the male employees is 42
years and that of the female employees is 41 years. If the average age of all the
employees in the company is 41 years 9 months, then the number of female
employees is:
(a) 150 (b) 250 (c) 450 (d) 350
An alloy is prepared by mixing three metals A, B and C in the proportion 3 : 4 : 7
by volume. Weights of the same volume of the metals A,B and C are in the ratio
5 : 2 : 6 . In 130 kg of the alloy , the weight , in kg of the metal C is
(A) 96 (B) 84 (C) 70 (D) 48
If a beer of 750ml bottle has 9% Alcohol and a wine bottle of 500 ml has 14%
alcohol in it. If 2 bottles of beer and 3 bottles of wine are mixed together, then
find the percentage of alcohol in the solution?
(a) 12.5% (b) 12% (c) 11.5% (d) 10.5%
A vessel is full of milk. 23L of milk is taken out & replaced by water this process is
repeated two more times. Find the initial amount of milk in the vessel. If at the
end the ratio of milk and water becomes 1331:397.
a)230L b)253L c)460L d)276L
In an examination, a student scores 6 marks for every correct answer and loses 4
marks for every wrong answer. If he attempted 80 questions and obtained 310
marks, how many questions did he attempt correctly?
(a) 59 (b) 65 (c) 63 (d) 67
A sample of 50 litres of glycerin is found to be adulterated to the extent of 28%.
How much pure glycerin should be added to it so as to bring down the percentage
of impurity to 10%?
(a) 75 ltr (b) 90ltr (c) 80 ltr (d) 120 ltr
A bottle full of whisky contains 50% alcohol. A part of this whisky is replaced by
another containing 18% alcohol and the percentage of alcohol was found to be
26%. The quantity of whisky replaced is:
(a) (b) (c) (d)
Sushma bought 15 tables and 11 chairs for Rs.10,800. She sold the tables at a
profit of 8% and the chairs at a loss of 10%. If her total gain was Rs.270, then the
total cost of the 4 tables was:
a) Rs.2,000 b) Rs.1800 c) Rs.2,400 d) Rs.2,500
A and B are solutions of acid and water. The ratios of water and acid in A and B are
4:5 and 1:2, respectively. If x liters of A is mixed with y liters of B, then the ratio of
water and acid in the mixture becomes 8: 13. What is x:y?
(a) 5:6 (b) 2:5 (c) 3:4 (d) 2:3
In a 56 litres mixture of milk and water, the ratio of milk to water is 5 : 2. In order
to make the ratio of milk to water 7 : 2, some quality of milk is to be added to the
mixture. The quantity of the milk present in the new mixture will be:
(a) 40 liters (b) 16 liters (c) 48 liters (d) 56 liters
A container contains 20 L mixture in which there is 10% sulphuric acid. Find the
quantity of sulphuric acid to be added in it to make the solution to contain 25%
sulphuric acid.
(a) 3 L (b) 5 L (c) 4 L (d) 2 L
Profit and Loss
• Cost Price (CP): The money paid by the shopkeeper to the manufacturer or
whole -seller to buy goods is called the cost price (cp) of the goods purchased
by the shopkeeper.

• Selling Price (SP): The price at which the shopkeeper sells the goods is called
selling price (s.p) of the goods sold by the shopkeeper to the customer.

•Profit: If the selling price of an article is more than its cost price, then the
dealer (or shopkeeper) makes a profit (or gain) i.e., Profit = SP – CP; SP > CP

•Loss: If the selling price of an article is less than its cost price, the dealer
suffers a loss i.e., Loss = CP – SP; CP > SP
A man purchases an item for Rs. 120 and he sells it at a 20 percent profit, find
his selling price
Find the cost price of an article which is sold for Rs. 200 at a loss of 20%
Concept 1:
MARK UP AND DISCOUNT

• Marked Price: To avoid loss due to bargaining by the customer and to

get profit over the cost price, the trader increases the cost price. This increase
is known as markup and the increased price (i.e., cp+markup) is called the
marked price or printed price or list price of the goods.
Marked Price = CP + markup

• Discount: Discount means reduction of marked price to sell at a lower rate

or literally discount means concession.
Basically, it is calculated on the basis of marked price.
Selling price = Marked price – Discount

Selling price = MP – [ (% Discount ) x MP ] / 100

If the cost price of an articale is Rs. 300 and the percent markup is 30%. What
is the marked price?
Concept 2:
Dishonest Dealer Case: If a trader professes to sell his goods at cost
price, but uses false weights, then

A shopkeeper sold an article at cost price but use the weight of 960 gm in
place of 1 kg weight. Find his profit%?
Concept 3:
Where two articles are sold at same price but one of them at a profit and
another at a loss and the percentage profit is the same as the percentage
loss, In this case there is always a loss.

Each of two car is sold for Rs. 1000. The first one is sold at 25% profit and the
other one at 25% loss. What is the percentage loss or gain in the deal?
Concept 4:
When two successive discounts on an article are x% and y% resp. Then net discount:
[x + y – xy/100]%

A shopkeeper given two sucessive discount of 50% and 50% find the real (equivalent)
discount?
A man buys an article for Rs. 27.50 and sells
it for Rs. 28.60. Find the gain percent?
(a)4%
(b) 3%
(c) 5%
(d) 10%
If a radio is purchased for Rs. 490 and sold
for Rs. 465.50. Find the loss%?
(a)6%
(b) 5%
(c) 4%
(d) 3%
Find SP when CP = Rs. 56.25 and Gain =
20%?
(a)Rs. 72
(b) Rs. 67.5
(c) Rs. 50
(d) Rs. 75
Find SP when CP = Rs. 80.40, loss = 5%?
(a)Rs. 81
(b) Rs. 84.72
(c) Rs. 76.38
(d) Rs. 82.9
Find CP when SP = Rs. 40.60, gain = 16%?
(a)Rs. 35
(b) Rs. 50
(c) Rs. 75
(d) Rs. 89
Find CP when SP = Rs. 51.70, loss = 12%?
(a)Rs. 58.75
(b) Rs. 62.25
(d) Rs. 69.27
(c) Rs. 65
A person incurs 5% loss by selling a watch for
Rs. 1140. At what price should the watch be
sold to earn 5% profit?
(a)Rs. 1380
(d) Rs. 1400 (b) Rs. 1160
(e) None of these(c) Rs. 1260
If the cost price is 96% of the selling price,
then what is the profit percent?
(a)5.72%
(b) 3.72%
(c) 8.92%
(d) 2.8%
A discount dealer professes to sell his goods
at cost price but uses a weight of 960 gms
instead of a Kg weight. Find his gain%?
(a) 27 / 4 %
(b) 8 / 3 %
(c) 25 / 6 %
(d) 21/ 4 %
A man sold two cows at Rs. 1995 each. On
one he lost 10% and on the other he gained
10%. What his gain or loss percent?
(a)4%
(b) 2%
(c) 0.5%
(d) 1%
Two discounts of 40% and 20% equal to a
single discount of?
(a)48%
(b) 53%
(c) 52%
(d) 60%
Amit buys 5 watches for Rs. 9450 and later
sells them for Rs. 9700. How much profit
does Amit make per watch?
(a)Rs. 75
(b) Rs. 80
(c) Rs. 60
(d) Rs. 95
The price of 12 chair and 8 table is Rs. 676.
What is the price of 21 chair and 14 table?
(a)Rs. 1183
(b) Rs. 4732
(c) Rs. 1180
(d) Cannot be determine
Aditya sold TV to Sanjay at 12% more than
the CP. If Sanjay paid Rs. 17696 for that TV
then what was the original price of the TV?
(a)Rs. 15,500
(b) Rs. 15,820
(c) Rs. 15,520
(d) Rs. 15,800
Amit purchased 13 chair of Rs. 115 each and
sold all at Rs. 1220. Then find the profit or
Loss on the transaction
(a)Rs. 280 Loss
(b) Rs. 275 Loss
(c) Rs. 325 Profit
(d) Rs. 350 Profit
 Aditya purchase a book with a 20% discount
on the marked price. How much did he pay if
the book marked was Rs. 500?
(a)Rs. 400
(b) Rs. 300
(c) Rs. 200
(d) Rs. 500
By selling a book for Rs. 360, 20% profit was
earned. What is the CP of the book?
(a)Rs. 300 (b) Rs. 200 (c) Rs. 250
(d) Cannot be determined (e)None of these
Profit earned by selling an article of Rs. 1630
is same as the loss incurred by selling the
article for Rs. 1320. What is the CP?
(a)Rs. 1475 (b) Rs. 1300 (c) Rs. 1350
(d) Rs. 1275 (e) None of these
 If the CP of 50 items is equal SP of 40 items
then what is the profit or
loss%?
(a)20% (b) 15% (c) 25%
(d) 35% (e) None of these
Sonal buys mangoes at the rate of 3 kgs for
Rs. 21 and sells them at 5 kgs for Rs. 50. To
earn a profit of Rs. 102, he must sell how
many mangoes?
(a)34 kgs
(b) 52 kgs
(c) 26 kgs
(d) 32 kgs
If a banana cost is Rs. 1.25 and apple cost is
Rs. 1.75 the what will be the
cost of 2 Dozen of Banana and 3 Dozen of
apple?
(a)Rs. 93 (b) Rs. 83 (c) Rs. 85
(d) Rs. 70 (e) Rs. 40
Nutan bought a watch with 24% discount. If
she pays Rs. 779 for that
watch then what is the marked price of
watch?
(a)Rs. 950 (b) Rs. 975 (c) Rs. 1000
(d) Rs. 1025 (e) None of these
A man sold his two horses for Rs. 770 each,
on one he gained 10% & on the other he lost
10%. The average gain or loss percentage is
(a)100% (b) 0.96% (c) 4%
(d) 1% (e) None of these
Profit and Loss
PRACTICE
Two third of consignment was sold at a profit of 5% and the remainder at a loss of
2%. If the total profit was ₹ 400, the value of the consignment was?
(a) ₹ 12000 (b) ₹ 14000 (c) ₹ 15000 (d) ₹ 13000
If the CP of 13 bats is ₹ 390. What is the price when it is sold at 10% loss?
(a) ₹ 200 (b) ₹ 300 (c) ₹ 350 (d) ₹ 400
If an item is sold for ₹ 924 then there is a profit of 10% then what is the cost
price?
(a) ₹ 840 (b) ₹ 860 (c) ₹ 880 (d) ₹ 900
Cost price of 15 articles is equal to selling price of 10 articles. The profit percent
is…
(a) 30% (b) 40% (c) 45% (d) 50%
A single discount, equivalent to a successive discount of 40% and 30% is?
(a)55% (b) 56% (c) 57% (d) 58%
The profit earned by selling an article for Rs. 900 is double the loss incurred when
the same article is sold for Rs. 450. At what price should the article be sold to
make 25% profit?
(a)Rs. 400 (b) Rs. 500 (c) Rs. 700 (d) Rs. 750
A shopkeeper sold some article at the rate of Rs. 35 per article and earned profit
of 40%. At what price each article should have been sold so that profit of 60% was
earned?
(a)Rs. 45 (b) Rs. 42 (c) Rs. 39 (d) Rs. 40
Due to a 20% rise in price of sugar, a bachelor is able to buy 1.5 kg less for Rs. 135.
What is the increased price of sugar per kg?
(a)Rs. 15 (b) Rs. 21 (c) Rs. 18 (d) Rs. 24
A trader mixes 26 kg of rice at Rs. 20 per kg with 30 kg of rice of other variety at
Rs. 36 per kg and sells the mixture at Rs. 30 per kg. His profit percent is:
(a) No profit, no loss (b) 5% (c)8% (d)10%
Sanjay purchased a chair marked at ₹ 800 at 2 successive discount of 10% and
15% respectively. He spent ₹ 28 on transportation and sold the chair for ₹ 800.
How much is his gain percentage?
(a) 14% (b) 30% (c) 25% (d) 40%
Aditya purchased 14 shirt & 25 pants at ₹ 45 and ₹ 55 respectively what should be
the approximate overall average selling price of shirt and pant so that 40% profit
is earned?
(a) ₹ 72.5 (b) ₹ 71 (c) ₹ 72 (d) ₹ 70
In what ratio must a grocer mix two varieties of tea worth ₹ 60 per kg and ₹ 65
per kg so that by selling the mixture at ₹ 68.20 per kg he may gain 10%?
(a) 3 : 2 (b) 3 : 4 (c) 3 : 5 (d) 4 : 5
How many kilogram of sugar costing ₹ 9 per kg must be mixed with 27 kg of sugar
costing ₹ 7 per kg so that there may be a gain of 10% by selling the mixture at
₹ 9.24 per kg?
(a) 36 kg (b) 42 kg (c) 54 kg (d) 63 kg
A person sells 36 apple per rupee and suffers a loss of 4%. Find how many apple
per rupees to be sold to have a gain of 8%.
(a) 32 (b) 16 (c) 4 (d) 15
A person sold a book at 20% profit. If he had bought it at 10% less cost and sold for
₹ 90 more then he would have gained 40% profit. Find cost price of book.
(a) ₹ 800 (b) ₹ 1600 (c) ₹ 1500 (d) ₹ 1200
P sold a car to Q at a profit of 15% and Q sold the car to S at a profit of 10%. If S
bought the car in Rs.25,30,000, then what is the cost price of car for Q? (in Rs.)
(a) 22,00,000 (b) 23,00,000 (c) 20,00,000 (d) 22,50,000
Gaurav sell a shirt to Mukesh at 20% profit, Mukesh sell the shirt to Deepak at a
profit of 25%, Deepak sell this shirt to Abhishek at Rs3600 and received a loss of
10%. At what price Mukesh purchase the shirt?
(a) Rs 3200 (b) Rs 4000 (c) Rs 3800 (d) Rs 3000
Cost price of B is 200 more than cost price of A. B is sold at 10% profit and A is sold
at 40% loss and selling price of A and B are in the ratio 4 : 11. If A is sold at 20%
loss then what will be selling price of A.
(a) 320 (b) 400 (c) 240 (d) 160
A shopkeeper marked his article 50% above the cost price and gives a discount of
20% on it. If he had marked his article 75% above the cost price and gives a
discount of 20% on it then find the earlier profit is what percent of the profit
earned latter?
(a) 50% (b) 60% (c) 33% (d) 40%
When a book is sold at its Marked Price it gives a profit of 40%. What will happen
if it is sold at half the marked Price?
(a) 30% profit (b) 25% loss (c) 30% loss (d) 40% profit
Find the total profit on selling 18 articles whose cost price is 25% less than the
mark price and a 15% discount is given on each article while selling price of each
article is 34.
(a) 90 (b) 72 (c) 54 (d) 108
A sold an article for Rs.8000 and incurred a loss. Had he sold the article for
Rs.9800, his gain would have been twice the amount of loss. At what price should
the article be sold to earn 20% profit?
(a) Rs.10840 (b) Rs.9820 (c) Rs.10320 (d) Rs.9840
In a medical transaction, 17 times the cost price is equal to 8 times the sum of the
cost price and the selling price. What is the gain or loss percentage?
(a) Loss 15% (b) Gain 17.5% (c) Gain 12.5% (d) Loss 30%
A shopkeeper earns a profit of 21% after selling a book at 21% discount on the
printed price. The ratio of the cost price and selling price of the book is:
(a) 100 : 79 (b) 100 : 121 (c) 79 : 100 (d) 121 : 100
A shopkeeper earns a profit of 21% after selling a book at 21% discount on the
printed price. The ratio of the cost price and marked price of the book is:
(a) 100 : 79 (b) 100 : 121 (c) 79 : 121 (d) 121 : 100
A shopkeeper sold two articles for ₹ 10591 each. On one he gained 19% and on
the other he lost 11%. What was his overall gain or loss percent (correct to one
decimal place)?
(a) Loss 2.7% (b) Loss 10% (c) Profit 5% (d) Profit 1.8%
A dealer gains 20% by selling an article at 25% discount on its marked price. If the
cost price of the article is decreased by 15%, how much discount percentage
should he now give on the same marked price so as to earn the same percentage
of profit as before?
(a) 32.50% (b) 35% (c) 36.25% (d)37.75%
A trader sells an article at 16% below its cost price. Had he sold it for ₹ 192.20
more, he would have gained 15%. The cost price (in Rs) of the article is:
(a) 720 (b) 620 (c) 520 (d) 420
A dealer offers a cash discount of 20% and still makes a profit of 20%. If he further
sells 8 articles at a rate of 6 articles, then how much percentage above the cost
price does he mark on each article?
(a)77.5% (b)100% (c)112.5% (d)87.5%
A fruit merchant bought some bananas. One fifth of them got rotten and were
thrown away. He sold two fifth of the bananas with him at 15% profit at the
remaining bananas at 10% profit. Find his overall loss or profit percent?
(a)Profit 9.6% (b) Loss 10.4% (c)Loss 9.6% (d) Profit 10.4%
Time Distance and Speed
•The basic concept of time and distance is used in solving the question based
on motion in a straight line.

•The applications of time & distance are used to solve the problems related to
trains and races.

•The relation between time, distance and speed is Distance = Time × Speed
A car covers 200 km in 4 hours, then find the speed of the car.
Conversion of Units:

(i) When we convert km/h into m/s, we multiply the speed by 5/18.
1km/h = 5 /18 m/s.

(ii) When we convert m/s into km/h, we multiply the speed by 18 / 5

1 m/s = 18 / 5 km/h
Conversion of Units:

(i) When we convert km/h into m/min, we multiply the speed by 50/3.
1km/h = 50 /3 m/min.

(ii) When we convert m/min into km/h, we multiply the speed by 3/ 50

1 m/min = 3 / 50 km/h
Convert 72 km/h into m/s.
Concept 1:
Average speed: A certain distance is covered at ‘x’ km/h and the same
distance is covered at ‘y’ km/h then the average speed during the whole
journey.

Average speed = km /h
Rohit covers a certain distance by car driving at speed of 40 km/h and he
returns to the starting point riding on a scooter with a speed of 10 km/hr. Find
the average speed of the whole journey?
Concept 2: A person covers a distance in T hours and the first half at S1 km/h
and the second half at S2 km/h, then the total distance covered by the
person.
A car covers a distance in 10 hrs, the first half at 40 km/h and the second half
at 20 km/h. Find the distance travelled by car?
Concept 3: If two persons P and Q start at the same time in opposite
directions from two points and after passing each they complete their
journeys in 'a' and 'b' hrs respectively then
Shivam sets out to cycle from Delhi to Ghaziabad and at the same time
Hemant starts from Ghaziabad to Delhi, After passing each other they
complete their journeys in 4 and 16 hours respectively. At what rate does
Hemant cycle if Shivam cycle at 18 km per hour?
Concept 4: If a man travelled a certain distance by bus at a rate of x km/h and
walked back at the rate of ‘y’ km/h. If the whole journey took ‘t’ hours, then
the distance he travelled is
A man travelled a certain distance by train at a rate of 15 km/h and walked
back at the rate of 12 km/h. The whole journey took 9 hours. Find the
distance he travelled?
Walking 4 / 5 of his usual speed, a man is 16 minutes late. Find the usual time
taken by him to cover that distance?
Concept 6: (i) If speed is constant, then distance is directly proportional to the
time; D = TK
(ii) If time is constant, then distance is directly proportional to the speed;
D=k
(iii) If Distance is constant, then speed is inversely proportional to the time;
S=K/T
A person covers a certain distance with a speed of 54 km/ h in 15 min. If he
wants to cover the same distance in 30 min, what should be his speed?
Concept 7:
(i) When a train passes a pole or any other object, the distance covered by
train is equal to the length of the train.
(ii) If a train passes a bridge, platform etc, then distance travel by train is
equal to the sum of the length of train and the stationary object through
which the train is passing.
A 100 m long train passes a platform of 200 m long. Find the distance covered
by the train in passing the platform?
Concept 8:
(i) When two trains are moving is opposite directions, then their relative
speed is equal to the sum of the speed of both trains.
(ii) When two trains are moving is same directions, then their relative speed is
equal to the difference of the speed of both trains.
Two trains are moving in the same direction with speed of 40 km/h and 50
km/h respectively. Find the relative speed?
Concept 9: Two trains start at the same time from P and Q and proceed
towards each other at the rate of x km/h and y km/h respectively. When they
meet it is found that one train has travelled D km more than the other. The
Distance between P and Q is ( x + y / x – y ) D
Two trains start at the same time from Kanpur and Delhi and proceed towards
each other at the rate of 73 km/h and 47 km/h respectively. When they meet
it is found that the train has travelled 13 km more than the other. Find the
distance b/w Kanpur and Delhi?
Concept 10: When the speed of two trains are in the ratio x : y. They are
moving in opposite directions on parallel tracks. The first train crosses a
telegraph pole in ‘t1’seconds where as the second train crosses the pole in
‘t2’ seconds. Time taken by the trains to cross each other completely is given
by Time taken
The speed of two trains are in the ratio 4 : 5. They are moving in opposite
directions along the parallel tracks. If each takes 3 seconds to cross a pole.
Find the time taken by the train to cross each other completely?
A man travels first 50 km at 25 kmph, next 40 km at 20 kmph and then 90 km at 15
kmph. His average speed for the whole journey (in kmph) is :
(a) 25 (b) 20 (c) 18
(d) 40 (e) None of these
A man walks at the rate of 5 km/hr for 6 hours and at 4 km/hr for 12 hours. The
average speed of the man (in km/hr) is :
If a person travels 10 1/5 km in 3 hours, then the distance covered by him in 5 hours
will be :
If a train 110m long passes a telegraph pole in 3 seconds, then the time taken (in
seconds) by it to cross a railway platform 165 m long, is :
A train 700 m long is running at the speed of 72 km/hr. If it crosses a tunnel in 1
minute, then the length of the tunnel (in metres) is:
(a) 700 (b) 600 (c) 550
(d) 500 (e) None of these
If a 200 m long train crosses a platform of the same length as that of the train in 20 seconds,
then the speed of the train is:
(a) 50 km/hr (b) 60 km/hr (c) 72 km/hr
(d) 80 km/hr (e) None of these
Two trains, each of length 125 metre, are running in parallel tracks in opposite
directions. One train is running at a speed 65 km/hour and they cross each other in 6
seconds. The speed of the other train is:
(a) 75 km /hour (b) 85 km/ hour (c) 95 km/ hour
(d) 105 km/ hour (e) None of these
A man with 3 / 5 of his usual speed reaches the destination 2 ½ hours
late. Find his usual time to reach the destination?
(a) 4 hours (b) 3 hours (c) 3 3 /4 hours (d) 4 1/ 2 hours (e) None
of these
A train running at 7 / 11 of its normal speed reached a place in 22 hours. How much
time could be saved if the train would have run at its normal speed?
(a) 14 hours (b) 7 hours (c) 8 hours
(d) 16 hours (e) None of these
Walking at three-fourth of his usual speed, a man covers certain distance in 2 hours more
than the time he takes to cover the distance at his usual speed. The time taken by him to
cover the distance with his usual speed is:
(a) 4.5 hours (b) 5.5 hours (c) 6 hours
(d) 5 hours (e) None of these
A man goes from a place A to B at a speed of 12 km/hr and returns from B to A at a
speed of 18 km/hr. The average speed for the whole journey is:
(a) 14 2/5 km/hr (b) 15 km/hr (c) 15 1/2 km/hr
(d) 16 km/hr (e) None of these
Two trains started at the same time, one from A to B and the other from B to A. If they
arrived at B and A respectively 4 hours and 9 hours after they passed each other, the
ratio of the speeds of the two trains was:
(a) 2 : 1 (b) 3 : 2 (c) 4 : 3
(d) 5 : 4 (e) None of these
A starts from a place P to go to a place Q. At the same time B starts from Q to P. If
after meeting each other A and B took 16 and 25 hours more respectively to reach
their destinations, the ratio of their speeds is:
(a) 3 : 2 (b) 5 : 4 (c) 9 : 4
(d) 9 : 13 (e) None of these
A train of 320 m cross a platform in 24 seconds at the speed of 120 km/ h. while a
man cross same platform in 4 minute. What is the speed of man in m/s?
(a) 2.4 (b) 1.5 (c) 1.6
(d) 2.0 (e) None of these
A car travel first 39 km distance in 45 minute while next 25 km distance in 35 minutes.
What is its average speed?
(a) 45 Km/h (b) 35 Km/h (c) 48 Km/h
(d) 90 Km/h (e) None of these
A truck cover 224 km in 4 hours, the average speed of a bike is 1 / 4 th the average
speed of the truck how much distance will the bike cover in seven hour?
(a) 96 km (b) 98 km (c) 95 km
(d) 92 km (e) None of these
 If a person walks at 14 km/h instead of 10 km/h he would have walked 20 km more.
The actual distance travelled by him is:
(a) 85 Km (b) 50 Km (c) 80 Km
(d) 70 Km (e) None of these
Train fare between Patna to Munger for one adult is three times the train fair of one
child. If adult fair is 102 then. What will be the fare of 2 adult and 3 children together
for same distance?
(a) 306 (b) 212 (c) 206
(d) 214 (e) None of these
A car travels a distance of 75 km at the speed of 25 km/h. It covers the next 25 km of its
journey at the speed of 5 km/h and the last 50 km of its journey at the speed of 25 km/h.
What is the average speed of car?
(a) 15 Km/h (b) 12.5 Km/h (c) 40 km/h
(d) 25 km/h (e) None of these
If the length of the train is 700 m and length of platform is 500 m. Find the time taken by the
train moving at 54 km/h to cross platform.
(a) 75 sec (b) 80 sec (c) 85 sec
(d) 90 sec (e) None of these
 Amit start to go to Delhi from patna at speed of 50 km/h. Distance between Delhi and Patna
is 1000 km. He takes rest of 20 minutes every 3 hours of journey. How much time will he take
to arrive Delhi?
(a) 20 hours (b) 21 hours (c) 22 hours
(d) 23 hours (e) None of these
 A car cover a distance of 330 km in a certain amount of time at speed of 55 km/h. What is
the average speed of bike that cover distance of 15 km less than car in 1 hour less than time
taken by car?
(a) 50 Km/h (b) 60 Km/h (c) 63 Km/h
(d) 65 Km/h (e) None of these
A car traveling at a speed of 40 km/hour can complete a journey in 9 hours. How long will it
take to travel the same distance at 60 km/ hour?
(a) 6 hours (b) 3 hours (c) 4 hours
(d) 4.25 hours (e) None of these
A 75 meter long train is moving at 20 kmph. It will cross a man standing on the platform in
(a) 12 seconds (b) 14 seconds (c) 13.5 seconds
(d) 15.5 seconds (e) None of these
In what time will a train 100 meter long cross an electric pole, if itsspeed be
144 km/hour?
(a) 2.5 seconds
(d) 3 seconds (b) 5 seconds
(e) None of these(c) 12.5 seconds
A train running at a speed of 60 kmph crosses a platform double its
length in 32.4 seconds. What is the length of the platform?
(a) 180 m
(d) 90 m (b) 240 m
(e) None of these(c) 360 m
A train running at the speed of 66 kmph crosses a signal pole in 18seconds.
What is the length of the train?
(a) 330 m
(d) 320 m (b) 300 m
(e) None of these(c) 360 m
TIME DISTANCE SPEED
PRACTICE
An athlete runs an 800 m race in 96 seconds. His speed (in km/h) is:
(a) 40 km/h (b) 20 km/h (c) 25 km/h (d) 30 km/h
A train covers a distance in 30 minutes if it runs at a speed of 54 Km/h on an
average. The speed at which the train must run to reduce the time of the
journey to 20 min is:
A) 18Km/h B) 60 Km/h C) 81 Km/h D) 75 Km/h
A man driving at 3/4th of his original speed reaches his destination 20
minutes later than the usual time. Then the usual time is
A) 45 min B) 60 min C) 75 min D) 120 min
Given that the lengths of the paths of a ball thrown with different speeds by two
boys are the same, if they take 0.6 seconds and 1 second respectively to cover the
said length, what is the average speed of travel for the first throw, if the same for
the second is 96km/h?
(a)100 km/h (b)150 km/h (c)160 km/h (d)200 km/h
Speed of A is 50% more than the speed of B and the speed of B is 25 % less
than the speed of C. If B takes 45 minutes more than C to cover a distance,
Find the actual time taken by A?
A) 90 min B) 120 min C) 150 min D) 60 min
The diameter of a wheel is 70 cm. It completes 600 revolutions in 1 minute.
The speed, in km/h, of the vehicle is: (Taken = )
(a) 78.4 (b) 79.2 (c) 77.8 (d) 78.2
A person travelled from station A to station B at 40km/hr and from B to A at
30km/hr. The entire journey took 6.3 hours. What is the distance (in km)
between A and B?
A) 117 B) 108 C) 91 D) 99
Ranjeet drives his car at an average speed of 50 km/hr and reaches his destination
in 8 hours. Rehman covers the same distance in 5 hours. If Ranjeet increases his
speed by 10 km/hr and Rehman increases his speed by 20 km/hr, then what will be
the difference between the times taken by them to reach the destination?
A) 2 hr 40 min B) 3 hr 40 min C) 3 hr 20 min D) 2 hr 30 min
An hour-long test has 60 problems. If a student completes 30 problem in 25
minutes, then the required seconds he has taken on average for computing each of
the remaining problems is
A) 70s B) 30s C) 50s D) 40s
A person has to cover a distance of 160 km in 15 hours. If he covers of the
distance in of the time, then what should be his speed (in km/hr) to cover the
remaining distance in the remaining time?
A) 6 B) 6.4 C) 8 D) 6.5
Travelling at 60 km/h, a person reaches his destination in a certain time. He covers
60% of his journey in th of the time. At what speed (in km/h) should he travel to
cover the remaining journey so that he reaches the destination right on time?
(a) 36 (b) 42 (c) 48 (d) 40
The distance between places A and B is 999 km. An express train leaves place A at 6
am and runs at a speed of 55.5 km/hr. The train stops on the way for 1 hours 20
minutes. It reaches B at
(a) 1:20 am (b) 12 pm (c) 6 pm (d) 11 pm
A man walks at a speed of 8 km/h. After every kilometre, he takes a rest for 4
minutes. How much time will he take to cover a distance of 6 km?
(a) 60 min (b) 65 min (c) 70 min (d) 69 min
A man is walking at a speed of 20 Km/h. After every 3 kms, he takes a rest for 5
minutes. How much time will he take to cover a distance of 40 km?
(a) 2 hrs (b) 3 hrs 5 minutes (c) 2 hrs 40 minutes (d) 3 hrs 12 minutes
Excluding stoppages, the speed of a bus is 60 kmph and including stoppages, it is
45 kmh. For how many minutes does the bus stop per hour?
(a) 12 (b) 9 (c) 15 (d) 10
A man travelled a distance of 42 km in 5 hours. He travelled partly on foot at the
rate of 6 km/h and partly on bicycle at the rate of 10 km/h. The distance travelled
on foot is:
(a) 10 km (b) 12 km (c) 18 km (d) 8 Km
A man travelled a distance of 35 Km in 5 hours. He travelled partly on foot at the
rate of 4Km/h and the rest on bicycle at the rate of 9 Km/h. The distance travelled
on foot is.
A) 10 Km B) 12 Km C) 8 Km D) 15 Km
Amita travels from her house at 3 km/h and reaches her school 6 minutes late.
The next day she travels at 4 km/h and reaches her school 10 minutes early.
What is the distance between her house and the school?
(a) 4.8 km (b) 5.4 km (c) 4.2 km (d) 5.6 km
Walking at 3 km per hour, Pintu reaches his school 5 minutes late. If he walks at
4 km per hour, he will be 2 minutes late. The distance of Pintu’s school from his
house is
(a) 600 m (b) 1000 m (c) 2500 m (d) 500 m
The speed of train A is 16 km/h less than the speed of train B. To cover a distance
of 384 km, B takes 4 hours less time than A. What is the speed (in km/h) of train
B?
A) 50 B) 32 C) 45 D) 48
Due to inclement weather, an air plane reduced its speed by 300 km/hr and
reached the destination of 1200 km late by 2 hrs. Then the schedule duration of the
flight was
A) 1 hr B) 2 hr C) 1.5 hr D) 2.5 hr
Train A takes 1 hour more than train B to travel a distance of 720 km. Due to engine
trouble speed of train B falls by one third, so it takes 3 hours more than Train A to
complete the same journey. What is the speed of Train A (in km/hr)?
(a) 80 (b) 90 (c) 60 (d) 70
A person travels 75 km at a speed of 25 km/h, next 60 km at a speed of 20 km/h
and the last 90 km at a speed of 15 km/h. His average speed is :
(a) 25.5 km/h (b) 18.75 km/ (c) 20.25 km/h (d) 15 km/h
A train travels the distance between stations P and Q at a speed of 126 Km/h,
while in the opposite direction it comes back at 90 Km/h. Another train travels the
same distance at the average speed of the first train. The time take by the second
train to travel 525 Km is:
A) 4 hours B) 5 hours 20 min C) 5 hours D) 4 hours 20 min
A person covers 40% of a distance with a speed of 60 km/h and the remaining
with a speed of 40 km/h. What is his average speed for the whole journey in
km/h?
(a) (b) (c) (d)
The speed of two railway engines is in the ratio 5:4. If they move on parallel tracks
in the same direction and if the slower engine is ahead of the faster engine by 8
km when the latter starts, then how far will the faster engine have to travel before
it overtakes the slower one?
(a) 32 (b) 48 (c) 40 (d) 3
Buses departs from a bus terminal with the speed of 80 km/h at an interval of
20 minutes. Find out speed of man who is going away from the bus terminal in
the same direction, if he gets buses an interval of 25 minutes.
A) 12 km/h B) 25 km/h C) 16 km/h D) 15 km/h
A 180 m long train running at 20 m/s will take what time ( in seconds ) to cross a
child walking at 10 m/s in the same direction ?
(a) 12 (b) 36 (c) 15 (d) 18
A 360 m long train running at a uniform speed, crosses a platform in 55 seconds
and a man standing on the platform in 24 seconds. What is the length (in metre)
of the platform?
(a) 480 (b) 445 (c) 410 (d) 465
Time and Work
Concept 1:
• If a person can complete a work in 'D' days, then the work done by him in 1
day is
• Efficiency is inversely proportional to the time taken (T) when the work
done is constant. Eα
Example: Ram can do a work in 40 days. Hari is 4 times more efficient
than Ram. In how many days Hari can finish the work?
Concept 2:
If M1 persons can do W1 work in D1 days working H1 hours and M2 person
can do W2 work in D2 days working H2 hours, then relation between them
is
24 men working 8 hours a day make a road in 15 days. In how many days 48
men working 6 hours a day will make the three times long road?
Concept 3:
If A does a work in 'a' days and B in 'b' days then both can complete the
work in
Example: A complete the work in 10 days and B in 15 days. In how many
days A + B can complete the work?
Concept 4:
If A and B can complete a work in x days and A alone can finish that work
in y days, then number of days B takes to complete the work is days
Example: A and B can complete a work in 20 days and A alone can finish
that work in 30 days. In how many days B can complete the work?
Concept 5:
A, B, C can do a work in x, y and z days respectively. They will finish the
work in days
Example: A, B and C can do a work in 10, 12 & 15 days respectively. In how
many days all of them together will finish the work?
Concept 6:
If A and B can do a piece of work in x days, B and C can do the same work
in y days and A and C can do it in z days, then, working together they can
complete that work in days
Example: A and B can complete a work in 20 days. B and C can complete
the same work in 30 days. C and A can complete the same work in 40 days. In
how many days they working together to complete the work?
Concept 7:
If A takes 'a' days more to complete a work than the time taken by (A+B)
to do some work and B takes 'b' days more than the time taken by (A+B)
to do same work. Then (A + B) do the work in ab days .
Example: A takes 4 days more to complete a work than the time taken by
(A + B) to do the same work and B takes 9 days more than the time taken by
(A + B) to do the same work. In how many days A + B complete the work?
Concept 8:
A can do a certain piece of work in d1 days and B in d2 days. Then, the
ratio of wages of A and B are:
A's share : B's share = : = d2 : d1
A, B and C can do a piece of work in d1, d2 and d3 days.
Then the ratio of wages of A, B and C are
A's share : B's Share : C's share = : :
Concept 9:
If A, B and C can do a piece of work in x, y and z days respectively. The
contract for the work is Rs. r and all of them work together.
Then,

Share of A = Rs. ( ryz / xy + yz + zx ) ,

Share of B = Rs. (rzx / xy + yz + zx ),
Share of C = Rs. ( rxy / xy + yz + zx )
Example: A, B and C can do a work in 20 days, 25 days and 30 days
respectively. They finished together that work and gained Rs.
3700 as wage. Find the wages of A, B and C respectively
Concept 10:
A can do a piece of work in x days. With the help of B, A can do the same
work in y days. If they get Rs. a for that work
Then,
Share of A = Rs. (ay / x )
, And Share of B = Rs. a(x – y) / x
Example: A can do a piece of work in 20 days. With the help of B, A can
do the same work in 15 days. If A + B gets Rs. 1500 for the work,
find the share of A and B respectively?
18 boys can do a piece of work in 24 days. In how many days can 27
boys do the same work?
(a) 16 (b) 32 (c) 23
(d) 48 (e) None of these
How many days will 1648 persons take to construct a dam, if 721
persons can build the same in 48 days?
(a) 21 days (b) 20 days (c) 23 days
(d) 24 days (e) None of these
If 10 persons can do a job in 20 days, then 20 persons with twice the
efficiency can do the same job in :
(a) 5 days (b) 10 days (c) 20 days
(d) 40 days (e) None of these
A and B can separately do a piece of work in 6 days and 12 days
respectively. How long will they together take to do the work?
(a) 9 days (b) 18 days (c) 6 days
(d) 4 days (e) None of these
A job can be completed by 12 men in 12 days. How many extra days
will be needed to complete the job if, 6 men leave after working for 6
days?
(a) 3 (b) 6 (c) 12
(d) 24 (e) None of these
A and B can do a piece of work in 12 days and 15 days, respectively.
They began to work together but A left after 4 days. In how many
more days would B alone complete the remaining work?
(a) 20/3
(b) 25/3
(c) 6
(d) 5 (e) None of these
Working efficiencies of A and B for completing a piece of work are in
the ratio 3 : 4. The number of days to be taken by them to complete the
work will be in the ratio
(a) 3 : 2 (b) 2 : 3 (c) 3 : 4
(d) 4 : 3 (e) None of these
5 men can prepare 10 toys in 6 days working 6 hours a day. Then in
how many days can 12 men prepare 16 toys working 8 hrs a day?
(a) 5 days (b) 3 days (c) 4 days
(d) 6 days (e) None of these
If A and B together can complete a work in 18 days, A and C together
in 12 days, and B and C together in 9 days, then B alone can do the
work in
(a) 18 days (b) 24 days (c) 30 days
(d) 40 days (e) None of these
P, Q and R contract a work for Rs. 550. Together, P and Q are supposed to do
7/11 of the work. How much does R get?
(a) Rs. 200 (b) Rs. 300 (c) Rs. 150
(d) Rs. 250 (e) None of these
A and B can complete a piece of work in 15 days and 10 days respectively.
They contracted to complete the work for Rs. 30000. The share of A in the
contracted money will be
(a) Rs. 18000 (b) Rs. 16500 (c) Rs. 12500
(d) Rs. 12000 (e) None of these
A daily-wage labourer was engaged for a certain number of days for
Rs 5,750 but being absent on some days he was paid only Rs 5,000.
What was his maximum possible daily wage?
(a) Rs 125 (b) Rs 250 (c) Rs 375
(d) Rs 500 (e) None of these
A can finish a work in 24 days, B in 9 days and C in 12 days. B and C
start the work but are forced to leave after 3 days. The remaining
work was done by A in:
(a) 5 days (b) 6 days (c) 10 days
(d) 10 1 / 2 days (e) None of these
If 3 men or 6 women can do a piece of wok in 16 days, in how many
days can 12 men and 8 women do the same piece of work?
(a) 4 days (b) 5 days (c) 3 days
(d) 2 days (e) None of these
A can do a work in 15 days and B in 20 days. If they together work on
it for 4 days, then the fraction of the work that is left is:
(a) 8/15 (b) 15/7 (c) 1/4 (d) 1/10 (e) None of these
If a job is to be completed in 10 days, it requires 270 persons. If 180
persons take up the same job, they will finish it in
(a) 27 days (b) 12 days (c) 15 days
(d) 18 days (e) None of these
A and B can do a job in 6 and 12 days, respectively. They began the
work together but A leaves after 3 days. Then, the total number of
days needed for the completion of the work is
(a) 4 (b) 5 (c) 6
(d) 9 (e) None of these
How many men will be required to plough 100 acres of land in 10 days
if 10 men require 8 days to plough 20 acres of land?
(a) 30 (b) 40 (c) 60
(d) 50 (e) None of these
A and B can do a piece of work in 20 days and 12 days, respectively. A
started the work alone and then after 4 days B joined him till the
completion of the work. How long did the work last?
(a) 10 days (b) 20 days (c) 15 days
(d) 6 days (e) None of these
18 women can complete a work in 12 days and 12 men can complete
the same work in 9 days. In how many days will 8 men and 8 women
complete that work?
(a) 9 days (b) 6 days (c) 12 days
(d) 8 days (e) None of these
A group of men decided to do a work in 10 days, but five of them
absented themselves. If the rest of the group finished the work in 12
days, find the original number of men?
(a) 20 men (b) 30 men (c) 40 men
(d) 50 men (e) None of these
To complete a work, A takes 50% more time than B. If together they
take 18 days to complete the work, how much time shall B take to do
it?
(a) 30 days (b) 35 days (c) 40 days
(d) 45 days (e) None of these
TIME & WORK
PRACTICE
Some staff promised to do a job in 18 days, but 6 of them went on leave.
So, the remaining men took 20 days to complete the job. How many men
were there originally?
(a) 55 (b) 62 (c) 56 (d) 60
A track of 100 m can be built by 7 men or 10 women in 10 days. How many
days will 14 men and 20 women take to build a track of 600 m?
(a) 15 (b) 20 (c) 25 (d) 30
If 12 men or 18 women can make a wall in 14 days, then working at the
same rate, 8 men and 16 women can make the same wall in:
(a) 9 days (b) 5 days (c) 7 days (d) 8 days
8 children and 12 men complete a certain piece of work in 9 days.
If each child takes twice the time taken by a man to finish the work, in how
many days will 12 men finish the same work?
(a) 8 days (b) 10 days (c) 11 days (d) 12 days
A man can do a work in 10 days. With the help of a boy he can do the same
work in 6 days. If they get Rs. 50 for that work, what is the share of that boy?
(a) Rs. 20 (b) Rs. 40 (c) Rs. 50 (d) Rs. 60
5 women or 7 men can earn Rs. 5,250 per day, how much would 7 women
and 13 men earn per day?
(a) Rs. 11,600 (b) Rs. 11,700 (c) Rs. 16,100 (d) Rs. 17,100
A man, a woman and a boy can together complete a piece of work in 3 days.
If a man alone can do it in 6 days and a boy alone in 18 days, how long will a
woman alone take to complete the work?
(a) 9 days (b) 21 days (c) 24 days (d) 27 days
If 16 men or 20 women can do a piece of work in 25 days, in what time will
28 men and 15 women do it?
(a) 14 days (b) 33 days (c) 18 days (d) 10 days
6 women or 12 men can do a piece of work in 20 days. In how many
days can 8 women and 16 men do twice as big as this work?
(a) 2 (b) 5 (c) 15 (d) 10
There is sufficient food for 400 men for 31 days. After 28 days 280 men leave
the place. For how many days will the rest of the food last for the rest of the
men?
(a) 5 days (b) 10 days (c) 12 days (d) 15 days
A and B together can do a piece of work in 12 days. A alone can do it in 18
days. In how many days B alone can do the work?
(a) 32 days (b) 30 days (c) 36 days (d) 24 days
A and B can do a work together in 18 days. A is three times as efficient as B.
In how many days can B alone complete the work?
(a) 60 days (b) 54 days (c) 72 days (d) 64 days
A and B together can complete a work in 3 days. They start together but
after 2 days, B left the work. If the work is completed after two more days, B
alone would do the work in
(a) 5 days (b) 6 days (c) 9 days (d) 10 days
A can do a piece of work in 6 days and B in 9 days. How many days will both
take together to complete the work ?

a) 7.5 b) 5.4 c) 3.6 d) 3

A can do a work in 20 days and B in 40 days. If they work on it together
for 5 days. Then fraction of the work that is left, is:
(a) 5/8 (b) 5/15 (c) 7/15 (d)1/10
P can complete ¼ of a work in 10 days, Q can complete 40% of the same work
in 15 days, R can complete 1/3 of the work in 13 days and S can complete 1/6
of the work in 7 days, Who will be able to complete the work first?
(a) P (b) Q (c) R (d) S
A is 50% as efficient as B. C does half of the work done by A and B together in
same time in same time. If C alone does the work in 20 days, then A, B and C
together can do work in:
(a) 5 days (b) 6 (c) 6 days (d) 7 days
A, B and C together earn Rs. 2700 in 18 days. A and C together earn Rs. 940 in
10 days. B and C together earn Rs. 1520 in 20 days. Find the daily earning of C?
(a) Rs. 20 (b) Rs. 40 (c) Rs. 10 (d) Rs. 15
A, B and C together earn Rs. 2700 in 18 days. A and C together earn Rs. 940 in
10 days. B and C together earn Rs. 1520 in 20 days. Find the daily earning of C?
(a) Rs. 20 (b) Rs. 40 (c) Rs. 10 (d) Rs. 15
P, Q and R contracted to do a work for Rs. 4200. P can do the work in 6 days, Q
in 10 days and R in 12 days. If they work together to do the work, what is the
share of R?
(a) Rs. 2000 (b) Rs. 1200 (c) Rs. 1000 (d) Rs. 1500
A can complete a work in 10 days, B in 12 days and C in 15 days. All of them began the
work together; but A had to leave the work after 2 days of the start and B also left 3
days before the completion of the work. How long did the work last?

(a) 7 days (b) 8 days (c) 10 days (d) 12 days

A, B and C can complete a piece of work in 15, 30 and 40 days respectively. They
started the work together but A left 2 days before the completion of the work and B
left 4 days before the completion of the work. In how many days was the work
completed?
(a) 7 days (b) 10 days (c) 10 days (d) 10 days
A , B and C can do a piece of work individually in 8, 12 and 15 days, respectively. A and
B start working but A quits after working for 2 days. After this, C joins B till the
completion of work. In how many days will the work be completed?
(a) 5 (b) 4 (c) 6 (d) 3
A and B can complete a piece of work in 45 and 40 days respectively. Both started to
work together, but after some days A left and B alone completed the rest work in 23
days. For how many days did A work?
(a) 12 days (b) 10 days (c) 8 days (d) 9 days
A 10 hectare field is reaped by 2 men, 3 women and 4 children together in 10 days. If
working capabilities of a man, a woman and a child are in the ratio 5 : 4 : 2, then a 16
hectare field will be reaped by 6 men, 4 women and 7 children in
(a) 5 days (b) 6 days (c) 7 days (d) 8 days
Pipes and Cistern
• Pipes and Cistern problems generally consist of a cistern (tank) to which
one or more pipes fill the cistern or empty the cistern.

• These problems of pipes and cisterns can be solved by using the same
method used in time and work.
• Important Points:

1. If a pipe can fill a tank in x hours, then the part filled in 1 hour =
1/x
2. If a pipe can empty a tank in y hours, then the part of the full tank
emptied in 1 hour = 1/y
If a pipe fills a tank in 6 h, then what part of the tank will the pipe fill in 1
h?
An inlet pipe fills 1/8 part of a tank in 1 h. How much time will the
pipe take to fill the empty tank?
(a) 4h (b) 2h (b) 6h
(d) 8h (e) None of these
An outlet pipe can empty a cistern in 3 h. In what time will the pipe
empty two-third part of the cistern?
(a) 4h (b) 2h (b) 3h
(d) 5h (e) None of these
There are two taps A and B to fill up a water tank. The tank can be
filled in 40 min, if both taps are on. The same tank can be filled in 60
min, if tap A alone is on. How much time will tap B alone take, to fill
up the same tank?
(a) 64 min (b) 80 min (b) 96 min
(d) 120 min (e) None of these
A pipe can fill a tank in 10 h, while an another pipe can empty it in
6 h. Find the time taken to empty the tank, when both the pipes are
opened simultaneously?
(a) 11h (b) 15h (b) 18h
(d) 16h (e) None of these
Three taps are fitted in a cistern. The empty cistern is filled by the
first and the second taps in 3 and 4h, respectively. The full cistern is
emptied by the third tap in 5 h. If all three taps are opened
simultaneously, the empty cistern will be filled up in?
(a) 1 14/23h (b) 2 14/23 h (b) 2 h 40 min
(d) 1 h 56 min (e) None of these
Pipe A can fill a tank in 30 min, while pipe B can fill the same tank in
10 min and pipe C can empty the full tank in 40 min. If all the pipes
are opened together, how much time will be needed to make the
tank full?
(a) 9 3/13 h (b)9 4/13 h (c) 9 7/13 h (d) 9 9/13 h (e) None of these
Three taps A, B and C together can fill an empty cistern in 10 min.
The tap A alone can fill it in 30 min and the tap B alone can fill it in
40 min. How long will the tap C alone take to fill it?
(a) 16 min (b) 24 min (b) 32 min
(d) 40 min (e) None of these
A, B and C are three pipes connected to a tank. A and B together fill
the tank in 6 h, B and C together fill the tank in 10 h and A and C
together fill the tank in 12 h. In how much time A, B and C fill up the
tank together?
(a) 9 h (b) 5 3/7 h (b) 5 7/2 h
(d) 5 5/7 h (e) None of these
Inlet A is four times faster than inlet B to fill a tank. If A alone can fill
it in 15 min, how long will it take if both the pipes are opened
together?
(a) 10 min (b) 12 min (b) 15 min
(d) 14 min (e) None of these
There are two inlets A and B connected to a tank. A and B can fill the
tank in 16 h and 10 h, respectively. If both the pipes are opened
alternately for 1 h, starting from A, then how much time will the
tank take to be filled?
(a) 13 1/4 h (b) 11 6/8 h (b) 12 2/5 h
(d) 12 1/4 h (e) None of these
A pipe can empty a cistern in 27 hours. Find the time in which 23 part
of the cistern will be emptied?
(a) 9 hours (b) 12 hours (b) 15 hours
(d) 18 hours (e) None of these
A water tank is 2/3rd full. Pipe A can fill the tank in 10 minutes and
the pipe B can empty it in 6 minutes. If both the pipes are open, how
long will it take to empty or fill the tank completely?
(a) 6 minutes to empty (b) 6 minutes to fill
(c) 9 minutes to empty (d) 9 minutes to fill
(e) None of these
12 pumps working 6 hours a day can empty a completely filled
reservoir in 15 days. How many such pumps working 9 hours a day
will empty the same reservoir in 12 days?
(a) 15 (b) 9 (c) 10
(d) 12 (e) None of these
PIPES & CISTERN
PRACTICE
Two pipes A and B can fill a tank in 60 hours and 40 hours respectively and
pipe C can empty the tank in 15 hours. If pipes A and B are opened for 12
hours, then pipe C is also opened. After how many hours, the tank will
be emptied?
A. 15 hours B. 20 hours C. 12.5 hours D. 10 hours
Two pipes A and B can fill a tank in 80 minutes and 60 minutes respectively. There
is also an outlet C. If all the three pipes are opened together, the tank takes 40
minutes to fill completely. How much time will C take to empty the full tank?

A. 2.5 Hr B. 2.4 Hr C. 2 Hr D. 4 Hr
Two pipes A and B can fill a cistern in 15 hours and 10 hours respectively. A tap C
can empty the full cistern in 30 hours. All the three taps were open for 3 hours,
when it was remembered that the emptying tap had been left open. It was then
closed. How many hours more would it take for the cistern to be filled?
A. 3 hr 30 m B. 3 hr 12m B. 3 hr 36 m D. 4hr 35m
Pipe A, B and C together fill the tank in 8 hours, Pipe B and C together will fill the
tank 10 hours and Pipe A and C together can fill the tank in 12 hours. In how many
hours pipe A and B together can fill the tank?
A. 15 hours B. 18 hours C. 20 hours D. 24 hours
Two pipe can fill a tank in 6 hour and 8 hour and another pipe can empty it in
12 hours. If these pipes are opened at 7 am, 8 am and 9 am, so at what time
tank will be filled?
A. 11:24am B. 11:30am C. 11:36am D. 11:40am
A tank has 3 inlets A, B, C. A & B together fill half of tank in same time that C
alone takes to fill 1/4 of tank. If 3 of them together can fill tank completely in 8
hrs. What was the time (in hours) taken by C alone to fill empty tank?
A. 24 B. 36 C. 48 D. 64 E. 42
Pipe A can fill a tank in 40 hours and the ratio of the efficiency of A to B is 3: 2.
If Pipe B and pipe C together can fill the tank in 40 hours, in how many hours
pipe A and pipe C together can fill the tank completely?
A. 30 hours B. 40 hours C. 20 hours D. 25 hours
The ratio of time taken by pipe F to empty the cistern and pipe E to fill the
cistern is 5:3, if both pipes are opened together for 14 minutes, cistern gets
filled by 4/15th of its capacity, then find the time taken by pipe E to fill the
cistern.
A. 42 minutes B. 21 minutes C. 10.5 minutes D. 31.5 minutes
An inlet pipe B alone can fill a tank in 10 hours and efficiencies of inlet pipes A
and B are respectively 120% and 60% of that of an outlet pipe C. If pipes A and B
started together to fill the tank and after 2 hours pipe C is also started, then in
what time the tank will be filled?
A. 8 hours B. 5 hours C. 6 hours D. 3 hours
Pipe A can fill a tank in 24 minutes while pipe B can empty it in 36 minutes. If
both the pipes were opened on alternate minutes, then how much time will they
take to fill the tank completely.
a) 138 minutes b) 139 minutes c) 140 minutes d) 141 minutes
A tank has tapes A, B and C. A and B can fill the tank in 10, 12 hours respectively
whereas C can empty it 15 hours, if tap A, B and C are open alternatively for an
hour, then tank will be filled in what time?
a) 21hours b) 22hours c) 23hours d) 24hours
A pipe can fill a tank in 20 minute. And another waste pipe empty tank 5 lit./sec.
If both pipe are working together can fill the tank in 100 min.
Find the capacity of the tank?
a) 6000 liters b) 6500 liters c) 7000 liters d) 7500 liters
Three pipes A, B and C can fill cistern in 6 hrs. After working together for 2 hours,
C is closed and A and B fill the cistern in 8 hrs. Then find the time in which the
cistern can be filled by pipe C?
(a) 6 hrs (b) 12 hrs (c) 14 hrs (d) 20 hrs
There are 4 filling pipes and 3 emptying pipes each capable of filling and
emptying in 12 minutes and 15 minutes respectively. If all the pipes are opened
together and as a result they fill 10 litres of water per minute. Find the capacity
of the tank.
A) 65 Lt B) 70 Lt. C) 75 Lt. D) 80 Lt.
Pump A and Pump B can fill a tank in 72 hours and 48 hours, respectively. Pump C
can empty 25% of the tank in 18 hours. If Pump A and B opened simultaneously
and Pipe C opened when the tank is half filled, then find the time taken to fill the
tank?
A. 36.4 hours B. 37.4 hours C. 38.4 hours D. 39.4 hours
Pipe P and Q can fill the tank alone in 16 min and 20 min respectively. Both the
pipes opened together but P left 8 min before filling the tank.
Find the total time taken by both of them to fill the tank?
A. 13 min 20 secs B. 12 min 24 secs C. 14 min 36 secs D. 11 min 42 secs
Average
Average: Average is defined as the sum of different data (terms) divided by
total number of terms.

Average = Sum of given terms(S)/Total number of terms(N)

Find the average of given terms 2, 3, 4, 5, 6

Some Basic Formulae
1. Average of ‘n’ natural number = (n + 1)/2
2. Average of ‘n’ even numbers = (n + 1)
3. Average of ‘n’ odd numbers = n
4. Average of ‘n’ consecutive natural numbers = First number + Last number / 2
5. Average of sum of squares of first ‘n’ natural numbers = (n + 1)(2n + 1)/6
Some Important Points:
1. If the value of each number is increased by the same value ‘a’, then
the average of all numbers will also increase by ‘a’.
2. If the value of each number is decreased by the same value ‘a’,
then the average of all numbers will also decrease by ‘a’.
3. If the value of each number is multiplied by the same value ‘a’,
then the average of all numbers will also get multiplied by ‘a’.
4. If the value of each numbers is divided by the same value ‘a’, then
the average of all numbers will also get divided by ‘a’.
1. If the average of ‘x’ numbers is a and that of ‘y’ numbers is b, then
the average of (x + y) numbers = xa + yb / x + y

The average of 10 numbers is 15 and that of 15 numbers

is 20. Find the average of all 25 numbers?
2. If the average of ‘n’ quantities is equal to ‘x’ when a quantity is
removed the average becomes ‘y’. Then the value of the removed
quantity is = [n (x – y) + y]

The average age of 24 men and 1 woman is equal to 35 years. If 1 woman left, the
average becomes 34 years. Find the age of woman who left the class?
A motorist covered the first 20 km of his journey in 30 minutes and
the remaining 29 km in 40 minutes. His average speed in km/hr is:
The mean of 19 observations is 4. If one more observation of 24 is
added to the data, the new mean will be:
(a) 4 (b) 5 (c) 6
(d) 7 (e) None of these
A student bought 4 books for Rs. 120 from one book shop and 6
books for Rs. 150 from another. The average price (in rupees), he
paid per book was :
(a) 27 (b) 27.50 (c) 135
(d) 138 (e) None of these
The average of the price per kg. of rice at 10 different places was Rs.
4.85. After a week, the price per kg. was increased by 20 paise at 3
places and decreased by 10 paise at one place. The new average of
price per kg is:
(a) Rs. 4.88 (b) Rs. 4.86 (c) Rs. 4.90
(d) Rs. 5.35 (e) None of these
The average weight of a group of 20 boys was calculated to be 89.4
kg and it was later discovered that one weight was misread as 78 kg
instead of the correct one of 87 kg. The correct average weight is :
(a) 88.95 kg (b) 89.25 kg (c) 89.55 kg
(d) 89.85 kg (e) None of these
 A class has 20 boys and 30 girls. The average age of boys is 11 years
and that of girls is 12 years. What is the average age of the whole
class
(a) 11.6 years (b) 12 years (c) 10 years
(d) 11.2 years (e) None of these
 Find the average of the following set of scores 253, 124, 255, 534, 836,
375, 101, 443, 760
(a) 427 (b) 413 (c) 141
(d) 490 (e) None of these
Find average of all prime numbers between 60 and 90.
(a) 72 (b) 74.7 (c) 74
(d) 73.6 (e) None of these
Average age of 5 boys is 16 yr, of which that of 4 boys is 16 yr 3
months. The age of the 5th boy is
(a) 15 yr (b) 15 yr 6 months (c) 15 yr 4 months
(d) 15 yr 2 months (e) None of these
The average age of 30 girls is 13 yr. The average of first 18 girls is 15
yr. Find out the average age of remaining 12 girls?
(a) 12 yr (b) 10 yr (c) 16 yr
(d) 10.5 yr (e) None of these
The average of 13 results is 60. If the average of first 7 results is 59
and that of last 7 results is 61, what will be the seventh result?
(a) 90
(d) 60 (b) 50
(e) None of the above(c) 75
The average of nine numbers is 50. The average of the first five
numbers is 54 and that of the last three numbers is 52. Then, the sixth
number is?
(a) 34 (b) 24 (c) 44
(d) 30 (e) None of these
The average age of 30 women decreases by 3 months if a new person
Neha is included in place of a 25 yr old woman. Calculate the age of
Neha?
(a) 17.5 yr (b) 20 yr (c) 30 yr
(d) 22 yr (e) None of the above
A cricketer scored some runs in his continuous 9 innings. He scored
100 runs in his 10th innings and this increased his average by 8 runs.
What was the average of his runs at the end of 10th innings?
(a) 20 (b) 24 (c) 28
(d) 32 (e) None of the above
The average age of three boys is 15 yr. If the ratio of their ages is 3 :
5 : 7, what is the age of the oldest boy?
(a) 7 yr (b) 14 yr (c) 20 yr
(d) 21 yr (e) None of these
Nine friends have a dinner in a hotel. Eight of them spent Rs. 12 each
on their meals and the ninth spent Rs. 16 more than the average
expenditure of all the nine. Find out the total money spent by them?
(a) Rs. 126 (b) Rs. 135 (c) Rs. 111
(d) Rs. 141 (e) None of these
The average of certain first consecutive even number is 101. Find
their sum?
(a) 25,000 (b) 33,600 (c) 10100
(d) 24,960 (e) None of these
The average of certain first consecutive natural numbers is 20.5. Find
their sum.
(a) 282
(d) 820 (b) 450
(e) None of these(c) 518
A, B, C, D & E are five consecutive even numbers. Average of A and
E is 46. What is the largest number?
(a) 52 (b) 42 (c) 50
(d) 48 (e) None of these
The average salary of 20 workers in an office is Rs. 1900 per month.
If the manager’s salary is added, the average becomes Rs. 2000 per
month. The manager’s annual salary (in Rs) is:
(a) Rs. 24000 (b) Rs. 25200 (c) Rs. 45600
(d) Rs. 48000 (e) None of these
AVERAGE
PRACTICE
The average weight of 62 students of a school is 54 kg. If the weight of the teacher
be included, the average rises by 3 kg. Find the weight of the teacher?

a) 186 b) 189 c) 243 d) 234

The mean weight of 34 students of a school is 42 kg. If the weight of the teacher be
included, the mean rises by 400 grams. Find the weight of the teacher?
a) 28 kg b) 56kg c) 76 kg d) 18 kg
The average age of 6 workers in a company is 37 year. If a new worker is employed,
the average is decreased by 2 years. Find the age of new worker?

a) 23 kg b) 56kg c) 76 kg d) 18 kg
The average weight of 10 members in a family increases by 2.5 kg when a new member
comes in place of one of them weighing 70 kg. Find the weight of new member.

a) 45 kg b) 75kg c) 72.5 kg d) 95 kg
The captain of a cricket team of 11 members is 26 years old and the wicket keeper is
3 years older. If the ages of these two are excluded. The average age of the remaining
players is one year less then the average age of the whole team. Find out the average
age of the team?
a) 23 b) 56 c) 76 d) 18
The average age of 30 students of a class is 15 years. A student of 20 years left the
class and two more students, whose age difference was 5 years, joined the class in
place of him. If the average of the class was still 15 years then find the age of the
younger new student?
a) 43 b) 56 c) 15 d) 18
A batsman played 40 innings with average score of 50 runs. His highest score exceeds his
lowest score by 172 runs. If these two innings are excluded, then average score of his
remaining innings is 48 runs. What is the highest score of this batsman?
a) 132 b) 156 c) 174 d) 182
Average weight of ‘m’ boys is 43 kg if the weight of their teacher who weighs 63 kg is
also included then average becomes 45 kg. Find the value of ‘m.’

a) 6 b) 8 c) 9 d)10
A batsman in his 12th inning makes a score of 63 runs and thereby increase his average
score by 2. What is his average after 12th inning.
a) 39 b) 41 c) 36 d) 44
The average of 12 numbers is 20. The average of first 5 numbers is 18 and that of last 6
numbers is 22. Find the 6th number.
a) 22 b) 16 c) 26 d) 18
Average temperature of a place in 25 days is recorded as 15°C. Later it was found that
average temperature of four days which was 14°C by mistake taken as 16.5°C. Find the
original average temperature of that place within that duration.
a) 14.8°C b) 14°C c) 14.4°C d) 14.6°C e) 14.2°C
The average marks of 14 students was calculated as 71. but later it is found that there
was an error in noting the marks of 2 students as 42 instead of 56 and 74 instead of
32. What will be the correct average of the students?
a) 53 b) 66 c) 69 d) 59
The average marks of 21 students in a class is 42. If 15 students with an average mark
of 78 join them, then find the average marks of all the students.
a) 53 b) 56 c) 57 d) 58
The mean weight of 180 students in school is 50kg. The mean weight of boys is 60kg
and that of girls is 45kg. Find the numbers of boys and girls in the school?

a) 100 & 80 b) 60 & 120 c) 70 & 110 d) 150 & 30

Average weight of a class is 60kg and average weight of boys in the class is 80kg. Ratio
of boys to girls in the class is 5: 4. If there are 72 students in the class, then find the
average weight of girls in the class.
a) 54 kg b) 42kg c) 35 kg d) 45kg e) 38kg
The average salary of all the workers in a workshop is Rs. 8000. The average salary
of 7 technicians is Rs. 12000 and the average salary of the rest is Rs. 6000. The total
number of workers in the workshop is:
a) 14 b) 28 c) 21 d) 30
Find the average of first 20 multiple of "3".

a) 30.5 b) 31.5 c) 32.5 d) 33.5

Average of 27 consecutive even number is 82. Then find the largest and smallest
numbers.

a) 110 b) 112 c) 114 d) 108

Average of 13 consecutive multiples of 11 is 14641. Find the second largest number
of the series.
a) 14652 b) 16496 c) 14696 d) 14608
The sum of the 5 consecutive odd numbers is 215. What is the sum of a different set
of 5 consecutive numbers whose second lowest number is 37 less than double of
the lowest number of the first set ?

a) 210 b) 230 c) 250 d) 240

LCM and HFC
Highest common factors (HCF) and least common multiple (LCM).

Common Factors
When any factor which is the factor of two or more given
numbers then it is said that this particular factor is common.

For example 6 = 2 x 3
15 = 3 x 5
We see that 3 is a common factor in both 6 and 15

6, 8, 30
Highest Common Factor (HCF) or Greatest Common Divisor (GCD)

HCF of two or more than two numbers is the greatest possible number that
can divide all these numbers exactly, without leaving any remainder. For
example, find the HCF of 84 and 126.
Q 84 = 2 x 3 x 7 x 2 = 42 x 2
126 = 2 x 3 x 7 x 3 = 42 x 3
So, the HCF of 84 and 126 2 x 3 x 7 = 42
There are two methods of finding the HCF.
(i) Factor Method
(ii) Division Method

(i) Factor Method

In this method first we break (or resolve) the numbers into prime factors then
take the product of all the common factors. This resultant product is known as
the HCF of the given numbers.
Exp. 1) Find the HCF of 1680 and 3600.
(ii) HCF by Division Method

Exp. 1) Find the HCF of 120 and 180

Exp. 2) Find the HCF of 420 and 1782
Exp. 3) Find the HCF of 210, 495 and 980.
HCF with Remainders
Case 1. Find the greatest possible number with which when
we divide 37 and 58, it leaves the respective remainder
of 2 and 3..
Case 2. Find the largest possible number with which when
60 and 98 are divided it leaves the remainders 3 in each
case
Case 3. Find the largest possible number with which when
38, 66 and 80 are divided the remainder remains the same
Least Common Multiple (LCM).

There are two methods to find the LCM.

(i) Factor Method
(ii) Division Method
(i) Factor Method.

Exp. 1) Find the LCM of 48, 72, 140.

Exp. 2) Find the LCM of 42, 63 and 231.
Division Method.

Exp. 3) Find the LCM of 108, 135 and 162

Exp. 5) Find the least possible number which can be
divided by 32, 36 and 40.
Exp. 6) What is the least possible number of 5 digits
which is divisible by all the numbers 32, 36 and 40.
Exp. 7) Find the largest possible number of 4 digits
which is exactly divisible by 32, 36 and 40.
Exp. 8) Find the number of numbers lying between 1
and 1000 which are divisible by each of 6, 7 and 15.
Exp. 9) Find the least possible perfect square number
which is exactly divisible by 6, 40, 49 and 75.
Exp. 10) Three bells in the Kalaram temple toll at the
interval of 48, 72 and 108 second individually. If they have
tolled all together at 6 : 00 AM then at what time will they
toll together after 6 : 00 AM?
LCM with Remainders.

Case 1. When the remainders are same for all the divisors.
Case 2. When the remainders are different for different divisors, but the
respective difference between the divisors and the remainders remains
constant.
Case 3. When neither the divisors are same nor the respective differences
between divisors and the remainders remain constant.
Case 1
Exp. 13) What is the least possible number which when divided by 24, 32
or 42 in each case it leaves the remainder 5?
Exp. 14) What is the least possible number which when divided by 21, 25,
27 and 35 it leaves the remainder 2 in each case?
Case 2
Exp. 15) What is the least possible number which when divided by 18, 35
or 42 it leaves. 2, 19, 26 as the remainders, respectively?
Exp. 16) What is the least possible number which when divided by 2, 3, 4,
5, 6 it leaves the remainders 1, 2, 3, 4, 5 respectively?
Exp. 17) What is the least possible number which when divided by 13 it
leaves the remainder 3 and when it is divided by 5 it leaves the remainder 2
Exp. 18) How many numbers lie between 11 and 1111 which when divided
by 9 leave a remainder of 6 and when divide by 21 leave a remainder of 12?
HCF and LCM of Fractions.

HCF of fractions = HCF of Numerator/LCM of denominator

, , ,
LCM of fractions : The least possible number of fraction which is exactly
divisible by all the given fractions is called the LCM of the fractions.

LCM of fractions = LCM of Numerator/HCF of Denominator

, , ,
LCM-HCF
PRACTICE
HCF of 1007 and 1273 is :
(a) 1 (b) 17 (c) 23 (d) 19
The GCD of two whole numbers is 5 and their LCM is 60. If one of the numbers is 20,
then other number would be :
(a) 25 (b) 13 (c) 16 (d) 15
The number of possible pairs of numbers, whose product is 5400 and HCF is 30 :
(a) 1 (b) 2 (c) 3 (d) 4
A merchant has 140 litres, 260 litres and 320 litres of three kinds of oil. He wants to
sell the oil by filling the three kinds of oil separately in tins of equal volume. The
volume of such a tin is :
(a) 20 litres (b) 13 litres (c) 16 litres (d) 70 litres
There are three drums with 1653 litre, 2261 litre and 2527 litre of petrol. The
greatest possible size of the measuring vessel with which we can measure up the
petrol of any drum, while every the vessel must be completely filled:
(a) 31 (b) 27 (c) 19 (d) 41
The largest possible length of a tape which can measure 525 cm, 1050 cm and 1155
cm length of cloths in a minimum number of attempts without measuring the
length of a cloth in a fraction of the tape’s length
(a) 25 (b) 105 (c) 75 (d) 115
The three numbers are in the ratio 1 : 2 : 3 and their HCF is 12. These numbers are :
(a) 4, 8, 12 (b) 5, 10, 15 (c) 24, 48, 72 (d) 12, 24, 36
Mr. Akshay wants to plant 36 mango trees, 144 orange trees and 234 apple trees in
his garden. If he wants to plant the equal no. of trees in every row, but the rows of
mango, orange and apple trees will be separate, then the minimum number of
rows in his garden is :
(a) 18 (b) 23 (c) 36 (d) 42
Minimum how many similar tiles of square shape are required to furnish the floor
of a room with the length of 462 cm and breadth of 360 cm?
(a) 4420 (b) 4220 (c) 4120 (d) 4620
Two pencils are of 24 cm and 42 cm. If we want to make them of equal size then
minimum no. of similar pencils is
(a) 6 (b) 11 (c) 12 (d) 22
LCM-HCF
PRACTICE
HCF and LCM of two numbers is 8 and 48 respectively. If the ratio of two numbers
is 2:3, then the larger of the two numbers is?
(a) 16 (b) 48 (c) 18 (d) 24
The HCF of two numbers is 21 and their LCM is 221 time the HCF. If one of the
numbers lies between 200 and 300, then the sum of the digits of the other
number is:
(a) 17 (b) 18 (c) 14 (d) 15
The LCM of two numbers x and y is 204 times their HCF. If their HCF is 12 and
the difference between the numbers is 60, then x + y = ?
(a) 660 (b) 426 (c) 348 (d) 852
The least common multiple of two number 1728 and k is 5184 then how many
values of k are possible?
(a) 6 (b) 4 (c) 3 (d) 7
The ratio of two numbers 9:14 and their LCM is 1008. The numbers are:
a) 63,98 b) 72,112 c) 154,105 d) 81,126
The sum of LCM and HCF of two numbers is 4956. Those numbers have ratio of
11:16. What's the difference between LCM and HCF of two numbers?
a) 4480 b) 4620 c) 4900 d) 4780
Sum of two numbers is 128 and their HCF and LCM are 8 and 504 respectively. The
sum of the reciprocal of those numbers will be ……
a)2/63 b)8/27 c)1/28 d)16/63
If HCF and LCM of two numbers are 5 and 495. If sum of both numbers is 100. Find
difference of them.
a) 4480 b) 4620 c) 4900 d) 4780
LCM and HCF of two four digits number are 19261 and 103 find the sum of these
two number?
a)2884 b)4296 c)1996 d)2464
Data Interpretation
What is data interpretation:
• When data is organized into tables and charts it is done with the purpose
of making it meaningful.
• The objective of data interpretation is to assess whether a student can
understand bars and charts and Answer some questions based on them.
• This act of organizing and interpreting data to get meaningful information
under a given set of conditions is Data interpretation.

About data interpretation:

• This is the calculation intensive portion, it consists of a myrid of graph.
charts and tables and analyze data.
• The key to crack this area is to quickly Identify the key pieces of
information that you will require to work on.
Basic key that will help you to solve this topic:
Calculation
• Square
• Cube
• Table
• BODMAS
• Percentage
• Profit and loss
• Ratio and proportion
• Average
Types of Data Interpretation:
• Data table
• Line graph
• Pie charts
• Bar graph
• Mixed graph
Approach for data interpretation:
• First you look carefully at the table or graph and the direction. Note the
years to which, the data refers to and the units.
• Sometimes the figures may be given in thousands. While the Answer may
be millions Resulting in mistakes.

• The level of approximation that can be done is assessed from the choices.
If the answer is wide, time should not be wasted in working out exact
figures.
• If the choice ‘none of the above exists, a close approximation may be
required’.
Q1} Read the question carefully, it will give an indication as to which row and
column should be seen. A carefull reading of the question will reveal exactly
what is to be done and the units in which the answer is required.

• There may be one or two very large question requiring calculations.

Attempt these at the last.
• Revise bar charts, table and line graphs before attempting D.I. question
remember that the D.I. section is a scoring one and also time saving.
 Maths Chemistry Physics Geography History Computer
Students
(150) (130) (120) (100) (60) Science (40)

Golu 90 50 90 60 70 80
Mithi 100 80 80 40 80 70
Suraj 90 60 70 70 90 70
Gapplu 80 65 80 80 60 60
Mahi 80 65 85 95 50 90
Khushi 70 75 65 85 40 60

Sheetal 65 35 50 77 80 80
1. What are the average marks obtained by all the seven students in
physics?
Maths Chemistry Physics History Computer
(a)77.26 (b) 89.14 (c) 91.37 Students
(150) (130)
Geography
(120) (60)
(100) Science (40)

(d) 96.11 (e) None of these

Golu 90 50 90 60 70 80
Mithi 100 80 80 40 80 70
Suraj 90 60 70 70 90 70
Gappl
u
80 65 80 80 60 60

Mahi 80 65 85 95 50 90
Khus
hi
70 75 65 85 40 60

Sheet
al
65 35 50 77 80 80
2. The number of students who obtained 60% or above marks in all
subjects is
Maths Chemistry Physics History Computer
(a)1 (b) 2 (c) 3 Students
(150) (130)
Geography
(120)
(100) (60) Science (40)

(d) 4 (e) None of these

Golu 90 50 90 60 70 80
Mithi 100 80 80 40 80 70
Suraj 90 60 70 70 90 70
Gappl
u
80 65 80 80 60 60

Mahi 80 65 85 95 50 90
Khus
hi
70 75 65 85 40 60

Sheet
al
65 35 50 77 80 80
3. What was the aggregate of marks obtained by suraj in all the six
subjects ?
(a)409 (b) 419 (c) 429 Students
Maths
(150)
Chemistry
(130)
Physics
Geography
(120)
(100)
History
(60)
Computer
Science (40)

(d) 449 (e) None of these

Golu 90 50 90 60 70 80
Mithi 100 80 80 40 80 70
Suraj 90 60 70 70 90 70

Gapplu 80 65 80 80 60 60

Mahi 80 65 85 95 50 90

Khushi 70 75 65 85 40 60

Sheetal 65 35 50 77 80 80
4. In which subject is the overall percentage the best?
(a)Maths (b) Chemistry (c) Physics
(d) History (e) None of these Students
Maths
(150)
Chemistry
(130)
Physics
(120)
Geography
(100)
History
(60)
Computer
Science (40)

Golu 90 50 90 60 70 80
Mithi 100 80 80 40 80 70
Suraj 90 60 70 70 90 70

Gapplu 80 65 80 80 60 60

Mahi 80 65 85 95 50 90

Khushi 70 75 65 85 40 60

Sheetal 65 35 50 77 80 80
 Line Graph:
Answer the question based on the given line graph Following line graph
shows the ratio of export to import of company A and company B over the
year
Line Graph
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
2005 2006 2007 2008 2009 2010
Company A Company B
1. In how many of the given years were the exports more than the
imports for company A?
(a) 2 (b) 3 (c) 4 Line Graph
(d) 5 (e) None of these 2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
2005 2006 2007 2008 2009 2010
Company A Company B
2. If the imports of company A in 2007 were increased by 40%. What
would be the ratio of exports to the increased imports?
(a) 1.20 (b) 1.25 (c) 1.30
(d) 1.35 (e) None of these
Line Graph
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
2005 2006 2007 2008 2009 2010
Company A Company B
3. If the exports of company B in 2008 was Rs 237 Crore, what was the
amount of imports in that years ?
(a) 189.6 crore (b) 243 crore (c) 281 crore
(d) 316 crore (e) None of these
Line Graph
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
2005 2006 2007 2008 2009 2010
Company A Company B
4. In which year were the exports of company A minimum proportionate to its
imports.
(a)2008 and 2009 (b) 2010 (c) 2008 and 2010
(d) 2005 and 2007 (e) None of these
Line Graph
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
2005 2006 2007 2008 2009 2010
Company A Company B
Pie Charts or Circle Graphs:
1. What percentage of candidates passed the exam from institute T out
of the total no. of candidate enrollled from the same institute?
(a)50% (b) 62.5% (c) 75%
(d) 80% (e) None of these
2. The no. of candidates passed from institute S and P together exceeds
the no. of candidates enrolled from institutes T and R together by?
(a)228 (b) 279 (c) 399
(d) 407 (e) None of these
3. What is % of candidates passed to the candidate enrolled for institutes
Q and R together ?
(a)68% (b) 80% (c) 74%
(d) 65% (e) None of these
4. What is the ratio of candidates passed to the candidates enrolled
from institute P ?
(a)9 : 11 (b) 14 : 17 (c) 6 : 11
(d) 9 : 17 (e) None of these
 Chart Title
70 65
60
60 55 55
50 50 50 50
50 45 45
40
40 35
30
20
10
0
2006 2007 2008 2009 2010 2011
Company X Company Y
1. The income of two company X and Y in 2010 were in the rato of 3 : 4.
respectively. What is the respective ratio of their expenditures in 2010?
(a)7 : 22 (b) 14 : 19 (c) 15 : 22
(d) 27 : 35 (e) None of these
2. If the expenditure of company Y in 2007 was Rs 220 crore, what was
its income in 2007?
(a)Rs. 312 crore (b) Rs. 297 crore (c) Rs. 283 crore
(d) Rs. 275 crore (e) None of these
3. If the expenditures of company X and Y in 2006 were equal and the
total income of the two companies in 2006 was Rs 342 crore, what
was the total profit of the two company together in 2006?
(a)Rs. 240 crore (b) Rs. 171 crore (c) Rs. 120 crore
(d) Rs. 102 crore (e) None of these
4. The expanditure of company X in the year 2008 was Rs 200 crore and
the income of company X in 2008 was the same as its expenditure in
2011. The income of company X in 2011 was?
(a)Rs. 465 crore (b) Rs. 385 crore (c) Rs. 335 crore
(d) Rs. 295 crore (e) None of these
5. If the income of two company were equal in 2009, then what was the
ratio of expenditure of company X to that of company Y in 2009?
(a)6 : 5 (b) 5 : 6 (c) 11 : 6
(d) 16 : 15 (e) None of these
Mixed Graph: Pie Chart breakup shows that number of employees in
different department of an organization
Table shows the percentage of men in each department (Rest one woman
Total number of employess = 1200

Percentage
IT Departments
of Men
Accounts 20%
IT 35
15%
PRODUCTION 87
Distribution Prodution HR 25
Accounts 25% MARKETING 75
12%
DISTRIBUTION 50
Marketing
23% HR ACCOUNTS 65
5%
1. What is the Respective Ratio of the number of men from the marketing
department to those from the accounts department
(a)23 : 13 (b) 13 : 9 (c) 27 : 19
(d) 17 : 11 (e) None of these
2. The number of women from the IT department are what percent of
the number of men from the same department (Rounded off to two
deigits after decimal)
(a)159.38% (b) 190.07% (c) 185.71%
(d) 168.23% (e) None of these
3. The total number of men from all departments together forms what
percent of the total no. of employees in the organization?
(a)67% (b) 63% (c) 55%
(d) 58% (e) None of these
4. What is the total number of women from the production department
and the HR department together ?
(a)78 (b) 84 (c) 92
(d) 64 (e) None of these
5. Which department has the highest number of women employees
(a)IT (b) marketing (c) Accounts
(d) Distribution (e) None of these
Data Interpretation
PRACTICE
Given below the table shows total number of room booked in five different hotels
on five days of a week. Read the table carefully and answer the questions:
Q1. Total rooms booked in ‘Oberai’ on
Tuesday & Thursday together is what percent
less than total rooms booked in ‘Grand’ on
Monday & Thursday?

A. 25% B. 20% C. 16% D. 34%

Q2. Find difference between total number of
rooms booked in ‘Oberai’, ‘Lodhi’ & ‘Taj’ on
Monday together and total number of rooms
booked in ‘Taj’ , ‘Grand’& ‘Eros’ on Thursday
together?

A. 140 B. 210 C. 70 D. 110

Q3. Find ratio between total rooms booked in
‘Eros’ on Wednesday & Thursday together to
total rooms booked in ‘Lodhi’ on Thursday &
Friday together?

A. 27:26 B. 19:17 C. 29:32 D. 53:49

Q4. Find sum of average numbers of room
booked in ‘Eros’ on Monday, Wednesday &
Friday and average number of rooms booked
in ‘Grand’ on Monday & Friday?

A. 580 B. 380 C. 495 D. 460

Q5. Find percentage increase in rooms
booked on Friday in ‘Oberai’ over total rooms
booked on Monday in same Hotel?

𝟐 𝟏 𝟏
A. 46% B. 66 C. 37 D. 33
𝟑 𝟐 𝟑
Given bar graph shows the number of student passed in 𝑿𝒕𝒉 class from 6 different school.

Q1. Pass percentage of school S is equal to that

of school Q. Find total strength of school Q is
what % more than that of school S.

A. 20% B. 40% C. 50% D. 25%

Q2. If fail percentage of school P is 65% then,
find number of student failed from school P is
what percentage of number of students passed
from school T.

A. 30% B. 100% C. 120% D. 130%

Q3. If ratio between total student who passed to
who failed from all school is 7:3, then find the
total number of failed student from all schools
together.

A. 225 B. 125 C. 250 D. 275

Q4. Student passed from school P, Q, U and T
together is how much more than that of school
R and S together.

A. 200 B. 250 C. 190 D. 235

Q5. Failed student of school U is 15 more than
that of school R. If ratio between total strength of
school U to school R is 3 : 2, then find the total
number of failed student from both schools
together.

A.57 B.23 C.45 D.63

Pie-chart shown below shows distribution of biscuits sold by six sellers. Study the chart
carefully and answer the following questions.
TOTAL NO. OF BISCUITS SOLD = 2500 Q1. C sold 30% biscuit to males, 45% to females
and remaining to transgender. Then biscuit
bought by males and females together is how
F much more than the biscuit bought by
12% 13%
A transgender.
14% B
16%
C A. 220 B. 225 C. 250 D. 265
D
18% E
27%
TOTAL NO. OF BISCUITS SOLD = 2500 Q2. Average number of biscuit sold by B, C and E
all together is how much percent more than
biscuit sold by A?

12% 13%
F A. 12.25% B. 16.5% C. 18.75% D. 20%
A

14% B
16%
C
D
18% E
27%
TOTAL NO. OF BISCUITS SOLD = 2500 Q3. What is difference between the number of
biscuit sold by E, F and D together to the number
of biscuit sold by A and C together?

12% 13%
F A. 150 B. 175 C. 100 D. 80
A

14% B
16%
C
D
18% E
27%
TOTAL NO. OF BISCUITS SOLD = 2500 Q4. Biscuit sold by F and D together is how much
percentage more than the biscuit sold by E?

A. 225% B. 125% C. 75% D. 175%

F
12% 13%
A

14% B
16%
C
D
18% E
27%
TOTAL NO. OF BISCUITS SOLD = 2500 Q5. B sold three types of biscuit i.e. X, Y and Z in
the ratio 2 : 3 : 4. Find the difference between Z
type biscuit and X type biscuit sold by B?

12% 13%
F A. 120 B. 130 C. 140 D. 150
A

14% B
16%
C
D
18% E
27%
Simple Interest
Principal = Rs. ‘P’
Time = 'T' years
Rate of interest = ‘R%’
Simple Interest = ‘S.I.’
Amount = ‘A’
 P T R S.I. A
100 1st yr 20% 20 120
2nd yr
3rd yr
4th yr
5th yr
 𝑷×𝑹×𝑻
SI =
𝟏𝟎𝟎

A = P + SI
𝑷×𝑹×𝑻
A=P+
𝟏𝟎𝟎

𝑹𝑻
A = P [1 + ]
𝟏𝟎𝟎
 P T R S.I. A
100 1 yr 20% 20 120
2nd yr
3rd yr
4th yr
5th yr
A sum of money becomes four times in 20 years at SI. Find the rate of interest.
What is amount of Rs. 800 on 5% per annum for 3 years?

P R T SI A

𝑹𝑻
A = P [1 + ]
𝟏𝟎𝟎
What is principal when amount received is of Rs. 920 on 5% per annum for 3
years?

P R T SI A

𝑹𝑻
A = P [1 + ]
𝟏𝟎𝟎
A sum fetched a total simple interest of Rs. 3200 at the rate of 6.25 % per
annum in 4 years. What is the sum ?
a) Rs. 13800 b) Rs. 11800 c) Rs. 12800 d) Rs. 14800
P R T SI A
Saksham borrowed certain sum of money at simple interest at the rate of 5% p.a. for
the first three years, 9% p.a. for the next five years and 15% p.a. for the period
beyond 8 years. If the total interest paid by him at the end of 12 years is Rs.4800,
how much money did he borrow?
a) Rs.4000 b) Rs.2000 c) Rs.4500 d) Rs.5600

P R T S.I.
Two equal sums of money were invested, one at 4% and the other at 4 ½ %. At the
end of 7 years, the simple interest received from the latter exceeded that received
from the former by Rs. 31.50. Each sum was:
a) Rs. 1000 b) Rs. 500 c) Rs. 750 d) Rs. 900

P R T SI A
In how many years simple interest obtained on Rs. 8000 at 3% per annum will be
equal simple interest on Rs. 6000 at 4% for 5 years?
a) 4 years b) 5 years c) 6 years d) 8 years

P R T SI A
A person lent certain sum of money at 5% per annum simple interest and in 15 years
the interest amounted to Rs 250 less than the sum lent. What was the sum lent ?
a) Rs. 1000 b) Rs. 1500 c) Rs. 2400 d) Rs. 3000

P R T SI A
A person lent certain sum of money at 9.5% per annum simple interest in 40 years the
interest amounted to Rs. 22,400 more than the sum lent. What was the sum lent?
a) Rs. 8100 b) Rs. 8500 c) Rs. 8000 d) Rs. 8700

P R T SI A
At a rate of SI the interest on a sum of money for 10 years will be ⅗th part of the
amount. Then rate of SI per annum is (in %)
a) 30 % b) 15 % c) 10 % d) 7.5 %

P R T SI A
A certain sum of money triples itself in 5 years at simple interest. In how many years it
will be five times?
a) 5 years b) 18 years c) 10 years d) 15 years

P R T SI
When a person invests some money under simple interest then at the end of some
years the amount become 13 times of the principal and the numerical value of the rate
of interest per annum is thrice of the time. At the end of 10 years, the amount will
become how many times of the principal at the same rate of interest?
a) 6 times b) 8 times c) 9 times d) 7 times

P R T SI
A money lender claims to lend money at the rate of 12.5% per annum simple interest.
However, he takes the interest in advance when he lends a sum for one year. At what
interest rate does he lend the money actually?
a) 11.14% b) 28.14% c) 14.28% d) 18%
ALLIGATION BASED QUESTION
A sum of ₹12000 is invested partly at 8% per annum and the remaining at 11% per
annum simple interest. If the total interest at the end of 2.6 years is ₹ 3120, how much
money was invested at 11% per annum?
a) 2000 b) 3250 c) 8000 d) 3000
A sum of Rs.1550 was lent partly at 5%and partly at 8% p.a. simple interest. The total interest
received after 3 years was Rs.300. The ratio of the money lent at 5% to that lent at 8% is …..
(a) 5:8 (b) 8:5 (c) 16:15 (d) 31:61
Ravi gave Rs. 1200 on loan. Some amount he gave at 4% per annum simple interest and
remaining at 5% per annum simple interest. After two years, he got Rs. 110 as interest. Then,
the amounts given at 4% and 5% per annum simple interest respectively are.
a) 500, 700 b) 400, 800 c) 800, 400 d) 1100, 100
Arun lends Rs. 20,000 to two of his friends for a year. He gives Rs. 12,000 to the first at 8% p.a.
simple interest. Arun wants to make a profit of 10% on the whole. The simple interest rate at
which he should lend the remaining sum of money to the second friend is
a) 8% b) 16% c) 12% d) 13%
A person invested one-fourth of the sum of Rs 20000 at a certain rate of simple interest and the
rest at 3% p.a. higher rate. If the total interest received for 2 years is Rs. 4100, what is the rate
at which the second sum was invested?
a) 8% b) 11% c) 6% d) 9%
Peter invested an amount of Rs. 12,000 at the rate of 10% simple interest and another
amount at the rate of 20% simple interest. The total interest at the end of one year on the
total amount invested became 14%. Find amount invested.
a) Rs.20,000 b) Rs.22,000 c) Rs.24,000 d) Rs.25,000
A person invested a sum of Rs.90000 in 3 Schemes A, B & C at the rate of 16%, 19% &
31%per annum respectively. The amount invested in scheme C is 50% more than the
amount invested in scheme A. Find the total amount invested in scheme B, if he gets a total
amount of Rs.150300 in three years.
a) 30000 b) 40000 c) 50000 d) 35000
INSTALLMENT BASED QUESTION
Shivam buys a web cam for his personal computer costs Rs.40000. He pays 125/4% at
once and the rest amount 15 months later, on which he is charged simple interest at the rate
of 16% per annum. If Shivam pays whole money at once, then by what approximate per cent
he would have to pay less amount from the amount he is paying with the interest?
A. 10 % B. 16 % C.12% D. 14%

Principal Down Remain Annual ‘R’ 15 month ‘R’ S.I.

payment
 A one plus 5T cell phone is available for Rs. 6000 cash payment or for Rs. 3000 cash down
payment together with Rs. 3600 to be paid after two months. Find the annual rate of interest
charged under this scheme.
a) 20% b) 50% c) 120% d) 100%

Principal Down Remain Monthly ‘R’ S.I. Amount paid

payment
 Atul can purchase a wine bottle for Rs 2400 cash or for Rs 1000 cash down payments and
two monthly instalment of Rs 800 each. Find Rate of interest per annum.
a) 20% b) 60% c) 100% d) 120%

P D Remain 1st 2nd Monthly ‘R’ Amount S.I.

A toy bicycle can be purchased on cash payment of Rs. 1500. But the same cycle can also
be purchased on the cash down payment of Rs. 350 and rest can be paid in three equal
monthly installments of Rs. 400 for next three months. Find rate of SI per annum.
a) 90/7% b) 80/3% c) 75/2% d) 62/5%

P D R 1st 2nd 3rd Monthly S.I. A

‘R’
Ankit borrowed Rs. 8000 from bank and returned Rs.3000 after 5 years. After 11 years from
starting he returned Rs. 7100 and settle his account. What was the rate of interest ?
a) 6% b) 5% c) 4% d) 3%

P AFTER 5 yr AFTER 11 yr ‘R’ S.I. A

A person borrowed a sum at 12% per annum and return Rs 5400 after 1 year. Now the rate
of interest becomes 10% per annum on rest of the amount. If the interest of the 2nd year is
4/5 of the 1st year. Find the amount borrowed?
a) 135000 b) 140000 c) 125700 d) 120000

P 1st yr ‘R’ Remain 2nd yr ‘R’ S.I.

A sum lent out at simple interest amounts to ₹6076 in 1 year and ₹7504 in 4 years. The sum
and the rate of interest p.a. are respectively:
a) ₹5,600 and 8.5% b) ₹5,600 and 9% c) ₹5,400 and 9% d) ₹5,400 and 10%

Sum 1st yr 4th yr 3 yr 1 yr ‘R’

Compound Interest
Principal = Rs. ‘P’
Time = 'T' years
Rate of interest = ‘R%’
Compound Interest = ‘C.I.’
Amount = ‘A’
 DIFFERENCE BETWEEN S.I. & C.I.

 In Sl, the interest is calculated

SIMPLE INTEREST COMPOUND INTEREST
only on PRINCIPAL.
P R T S.I. A P R T C.I. A  In Cl, the interest is calculated on
Amount (Principal + Interest).

100 10% 1 10 110 100 10 1 10 110  SI & CI for the first year remains
same.

100 10% 2 10 120 110 10 2 11 121  SI for every year remains same.
 Cl for every year increases.

100 10% 3 10 130 121 10 3 12.1 133.1

 CI on the present is year is
calculated on the Amount of the
Previous Year.
 𝑹 𝑻
A = P (1 + )
𝟏𝟎𝟎

A = P + C.I.
C.I. = A – P

𝑹 𝑻
C.I. = P (1 + ) -P
𝟏𝟎𝟎

𝑹 𝑻
C.I. = P [(1 + ) - 1]
𝟏𝟎𝟎
1) Formula

2) Successive %

3) Tree

4) Base method
Akshay invested an amount of Rs. 10000 at compound interest rate of 10% per annum for a
period of three years. Calculate the amount and total C.I. which Akshay will get.

Method 1 :
P R T CI A

𝑹 𝑻
A = P (1 + ) C.I. = A – P
𝟏𝟎𝟎
Akshay invested an amount of Rs. 10000 at compound interest rate of 10% per annum for a
period of three years. Calculate the amount and total C.I. which Akshay will get.

Method 2 :
P R T CI A

𝒂𝒃
successive % = a + b + 𝟏𝟎𝟎
Akshay invested an amount of Rs. 10000 at compound interest rate of 10% per annum for a
period of three years. Calculate the amount and total C.I. which Akshay will get.

Method 3 :
P R T CI A

1st year 2nd year 3rd year

Akshay invested an amount of Rs. 10000 at compound interest rate of 10% per annum for a
period of three years. Calculate the amount and total C.I. which Akshay will get.

Method 4 :
P R T CI A

Year P A
1st
2nd
3rd
Total