Ratio and Proportion

1.

 Ratio

Introduction
The basic application of the concepts involved
in this chapter are comparison of two or more
quantities and changes in their magnitude.
Ratio and proportion is not only important
for quantitative aptitude but also for data
interpretation. Every year one or two problems
from this chapter is /are asked in CAT and other
examinations.

Ratio
Ratio is the relation which one quantity has with
another quantity and it is defined by a multiplier.
Simply, Ratio is the comparison between two
similar quantities in terms of magnitude. The sign
used to denote a ratio is ‘ : ’ . If the values of two
quantities A and B are 2 and 3 respectively, then
we can say that they are in the ratio 2 : 3 (read as
2 is to 3 or 2 ratio 3).
a
A ratio (a : b) can also be written   . So if two
b
items are in the ratio 2 : 3. We can say that their
2
ratio is   . If any integer is written for exmple
3
2 Previous Year’s Question
2, it means that it is in the ratio of   , or (2 : 1)
1
The ratio of two quantities A and B is written as A sum of money is split among
a : b. Here, ‘a’ is ‘antecedent’ and ‘b’ is consequent, Amal, Sunil and Mita so that the
it can also be said that A : B = ax : bx, where x is ratio of the shares of Amal and
any constant known as constant of Sunil is 3:2, while the ratio of the
proportionality, x ≠ 0. shares of Sunil and Mita is 4:5.
“A ratio is said to be a ratio of greater or less If the difference between the
inequality or of equality, antecedent could be largest and the smallest of these
greater than, less than or equals to consequent”. three shares is Rs 400, then
y If the ratio a : b where a > b, then the given ratio is Sunil’s share in rupees is? (TITA)
called the ratio of greater inequality or improper
ratio.
[Example → 5 : 2]
y If the ratio a : b where a < b, then the given ratio
is called the ratio of less inequality or proper
Ratio

2.
 ratio. [Example → 2 : 5]
y If the ratio a : b where a = b, then the given ratio
Keywords
is called the ratio of equality. [Example → 1 : 1]

Ratio As Bridging Element Š The ratio should exist

y Ratio as a bridging element helps us in setting between the quantities of
up the relationship between more than two the same kind.
quantities. Š While comparing two things,
Suppose the conversion rate of Dinar is given the unit should be similar.
with respect to Euro and also with respect to
Pound. If we have to find the conversion ratio of
Euro with Pound, we can do it by making Dinar as
the Bridge between Euro and Pound. Rack your Brain

Let P : Q = a : b, a 2 + b2 b2 + c2
and Q:R =x:y If a ≠ c and = = k ,
a+b b+c
Then P : Q : R = ax : bx : by
find k ?
Graphically, we represent it with the help of
inverted ‘N’

Example 1
The ratio of the salary of Ram and Shyam is 3 : 5
and the ratio of the salary of Shyam and Krishna
is 10 : 7. What is the Ratio of the Salary of Ram,
Shyam, and Krishna?
Explanation
Here, the salary of Shyam is common is both the
ratio, so we will try to make the salary of Shyam
equal in both cases.
Ratio

3.
 The salary of Ram : The salary of Shyam = [3 : 5]×2
The salary of Shyam : The salary of Krishna = [10 : 7]

Now,
The salary of Ram : The salary of Shyam = 6 : 10
The salary of Shyam : The salary of Krishna = 10 : 7

Since, the ratio of salary of Shyam is same in both cases,

Hence, the ratio of salary of Ram, Shyam, and Krishna is (6 : 10 : 7)

Alternate Method

The salary of Ram : The salary of Shyam : The salary of Krishna

⇒ 30 : 50 : 35
Required ratio = (6 : 10 : 7)

Example 2
Given that A : B = a : b, B : C = b : c and C : D = c : d. Find A : B : C : D
Explanation
A:B=a:b
B:C=b:c
C:D=c:d
A : B : C : D = (a × b × c) : (b × c × b) : (c × b × c) : (b × c × d)

Example 3
Given that A : B = a : b, B : C = b : c and C : D = c : d and D : E = d : e.
Find A : B : C : D : E
Rack your Brain
Explanation
A:B=a:b
B:C=b:c If ‘a’ and ‘b’ are two distinct
C:D=c:d positive number and a : b is the
D:E=d:e duplicate ratio of the positive
A : B : C : D : E = (a × b × c × d) : (b × b × c × d) : ratio (a-p) : (b-p), where a, b, p
(b × c × c × d) : (b × c × d × d) : (b × c × d × e) are real numbers then find the
value of ‘p’ in terms of ‘a’ and ‘b’.
Ratio

4.
Comparison of Ratio
Comparison of ratios is useful for quantitative Aptitude and Data
Interpretation. Now we will learn the comparison of ratios and how it is
done in the simplest way.
There are three methods to compare two or more than two ratios.

(i) Percentage Comparison

Understand this with the help of following cases.
Case I :

The percentage change in the Numerator = (200%) ↑

The percentage change in the Denominator = (200%) ↑
So, both the ratios are equal.

Case II :

The percentage change in the Numerator = (100%) ↑

The percentage change in the Denominator = (200%) ↑
So, the first ratio is greater than 2nd Ratio

Case III :

The percentage change in the Numerator = (200%) ↑

The percentage change in the Denominator = (100%) ↑
So, the second ratio is greater than first ratio.
Ratio

5.
 (ii) Cross Multiplication Method
In this method, we have to multiply the numerator of the first fraction to
the denominator of the second fraction and the denominator of the first
fraction to the numerator of the second fraction.

Example 4
9 11
Compare and
13 17
Explanation

9 11 
Since, 153 is greater than 143. Hence  > 
 13 17 
(iii) Denominator comparison
The following steps are involved in this method.
Step 1
Write the given ratios as a fraction.
Step 2
Find the least common multiple (LCM) of the denominators of given ratios.
(If the denominator are not the same).
Step 3
Make the denominators of the given fraction equal to the value of LCM,
using a multiplier(Any number which on multipliyng makes it equal to LCM).
Step 4
After getting the same denominator for the given fraction, compare the
numerator and decide which fraction is greater.
The fraction which has a larger numerator is greater in value.

For Example
Compare 2 : 3 and 4 : 7
Ratio

6.
Explanation
Write the given ratio as fraction. Previous Year’s Question
2
2:3 =
3
In an examination, Rama’s score
4
4:7 = was one-twelveth of the sum of
7
the scores of Mohan and Anjali.
The LCM of the denominators (3, 7) is 21. After a review, the score of each
Now, make the denominator of fraction equal to of them increased by 6. The
21 using multiplier (7, 3). revised scores of Anjali, Mohan,
2 2 × 7 14 and Rama were in the ratio 11:10:3.
= =
3 3 × 7 21 Then Anjali’s score exceeded
4 4 × 3 12 Rama’s score by :
= = (1) 26 (2) 32
7 7 × 3 21
(3) 24 (4) 35
Here, 14 > 21
So, 2 : 3 is greater than 4 : 7.

Standard Results on Ratio

1. The value of a Ratio does not change when the numerator and
denominator both are multiplied or divided by the same quantities.
i.e.
P P×x P×a
= = etc.
Q Q×x Q×a

P P/x P
= =
Q Q/x Q
2. If the ratio a : b where a > b then
(a+x) : (b+x) < a:b
3. If the ratio a : b where a < b then
(a+x) : (b+x) > a:b
4. If the ratio a : b where a = b then
(a+x) : (b+x) = a:b
5. If a/b = original ratio then
a2 a
2
= duplicate ratio of
b b
a3 a
= Triplicate ratio of
b3 b
a a
= Sub duplicate ratio of
b b
Ratio

7.
 3
a a
= Sub triplicate ratio of  
3
b b
a c e g
6. If = = = = ...............K
b d f h
a+ c+ e+ g+ .............
Then =K
b+ d+ f+ h+ .............
3 6 9 3+6+9 18
For example, = = =
.................. =
5 10 15 5 + 10 + 15 30
Example 5
The salaries of Ravi and Kuber are in the ratio of 5 : 7. The sum of their
salaries is Rs. 36000. Find the salary of Kuber.
Explanation
Here, the ratio of their salaries is 5 : 7.
7
Kuber’s salary is th of the total salary,
12
7
Kuber’s salary = × 36000 ⇒ Rs . 21000
12
Example 6
If a : b = 3 : 4 find the value of (a+b) : (a-b)
Explanation
Here, a : b = 3 : 4
Let a = 3k and b = 4k
a+ b 3k+ 4k 7 k
= = = -7
a- b 3k- 4k -k
Example 7
What number shall be added to each term of the ratio 5 : 8, so that it
becomes equal to 19 : 22 ?
Explanation
Let x be the number to be added to each term of the ratio. Then we have,
5 + x 19
= ⇒ 110 + 22 x = 152 + 19 x
8 + x 22
∴ x = 14
So, 14 has to added to each term to make it 19 : 22
Ratio

8.
Example 8
Two numbers are in the ratio of 5 : 8. If 14 is added to each, they will be in
the ratio of 19 : 22. Find the numbers.
Explanation
Let the numbers be ‘x’ and ‘y’.
Gives that- Previous Year’s Question
x 5
= , So x = 5k and y = 8k
y 8 Amala, Bina, and Gouri invest
If 14 is added to each then, money in the ratio 3 : 4 : 5 in
x = 5k+14, and y = 8k + 14 fixed deposits having respective
Given, annual interest rates in the ratio
5k+ 14 19 6 : 5 : 4. What is their total income
=
8k+ 14 22 (in Rs) after a year, if Bina’s income
110k + 308 = 152k + 266 exceeds Amala’s by Rs 250?
–42k = –42 (1) 7000 (2) 6000
∴ k=1 (3) 6350 (4) 7250
Hence, x = 5k = 5 and y = 8k = 8.

Ratio

9.
 Proportion

When two ratios are equal, then the four quantities involved in the two
ratios are said to be proportional. The sign used to denote a proportion
is ‘ :: ’
a c
If = or a : b = c : d , then we can say that a, b, c, d are proportional and
b d
written as a : b : : c : d and it is read as “a is to b and c is to d”. Here, a and
d are called “extreme terms” and (a×d) is called “product of extremes”, b
and c are called “Mean (Middle) terms” and (b×c) is called product of means.
Therefore we can say,
Product of extremes = Product of means.

Standard results on proportion

a b
1. If = then a, b, c are said to be in continued proportion. So, b2 = ac.
b c
Here, b is known as the mean proportion.
Similarly,
a b c
If = = then a, b, c, d are in continued proportion.
b c d
a c
2. If = then,
b d
b d
(a) = [Invertendo]
a c
a b
(b) = [Alternendo]
c d
Both (a) and (b) can be obtained by cross product.
a+ b c+ d
(c) = [Componendo]
b d
a − b c− d
(d) = [Dividendo]
b d
a+ b c+ d
(e) = [Componendo and dividendo]
a- b c- d

Example 9
The first, second, and fourth terms of a proportion are 2, 4, and 14
respectively, find the third term.
Proportion

10.
Explanation
Let the third term be x. Then 2, 4, x, 14 are in proportion i.e., 2 : 4 :: x : 14.
Product of extremes = Product of means
2 × 14 = 4 × x
28 = 4 × x
7=x

Example 10
If 4, x, x, 9 are in proportion, find x.

Explanation

Example 11
What is the least possible number which must be subtracted from 11, 14
and 18 so that the resulting numbers are in continued proportion?

Explanation
Let the number be x, then (11-x)(14-x)(18-x) are in continued proportion, i.e.
(11-x) : (14-x) : : (14-x) : (18-x)
Product of extremes = Product of means
(11–x) × (18–x) = (14–x) × (14–x)
198 – 11x – 18x + x2 = 196 + x2 – 28x
∴ x=2
Proportion

11.
 Variation

If two or more quantitie change with each other according to a mathematical

relation or condition, then they are said to be in Variation.

For Example
If the number of persons increases, then the time taken by them to complete
a certain work also decreases.

There are two types of variation

(i) Direct variation
(ii) Inverse variation

Direct Variation
A quantity ‘A’ is said to vary directly with ‘B’ under the following condition
1. If B increases then ‘A’ will also increase by the same magnitude.
2. If ‘B’ decreases then ‘A’ will also decrease by the same magnitude.
It is expressed as A ∝ B Which means A is Directly Proportional to B
⇒ A = KB
Where K is called proportionality constant.
When two quantities A and B vary directly with each other,
A
then = K [constant]
B
For Example
When the speed of a vehicle is doubled then the total distance covered by
the vehicle will also be twice for a constant time period.

Inverse Variation
A quantity ‘A’ is said to be inversely proportional to B, if A increases by a
certain magnitude then B also decreases by the same magnitude and when
B is decreased by a certain magnitude, then A is also increased by the same
magnitude.
It is expressed as
1 K
A∝ ⇒A=
B B
Where K is called proportionality constant.
When two quantities A and B vary inversely with each other then AB = K
[constant]
Variation

12.
For Example
If the number of women doing a certain work Rack your Brain
increases, then the time taken to do the work
decreases. Conversely when the number of A is proportional to B, B is
women decreases, the time taken to do the work inversely proportional to C, C is
increases. proportional to the square of D
and D is directly proportional to
Joint Variation the cube root of E. A, B, C, D, E
Joint variation is when a quantity varies directly are positive integers. Find when
as the product of two or more quantities. ‘A’ increases then ‘E’ increases or
decreases?
For Example
If there are three quantities A, B, and C such that
A varies with B when C is constant and varies
with C when B is constant. Then A is said to vary
jointly with B and C.
It is expressed as A ∝ B × C ⇒ A = K× B× C
Where K is proportionality constant.

Example 12
If x varies directly to y2 and is equal to 16, when y
= 2. Then find x when y = 16.
Explanation
Here,
x ∝ y2 ⇒ x = ky2 where K is constant
Since x = 16, when y = 2
16 = K × (2)2
K=4
When y=6
x = 4×(6)2
x = 144

Example 13
If variable A varies inversely with B. If B = 2 then
A = 4. Find A when B = 8.
Explanation
Here,
1 K
A∝ ⇒A=
B B
Variation

13.
 Since A = 4, when B = 2
K
4=
2
K=8
When B=8
8
A= ⇒A=
1
8
Example 14
Previous Year’s Question
Variable A varies jointly with the value of B and C.
A = 16 when B = 2 and C = 4, find A when B = 4
Suppose, C1, C2, C3, C4, and C5 are
and C = 6.
five companies. The profits made
by C1, C2, and C3 are in the ratio
Explanation
9 : 10 : 8 while the profits made
Here, A ∝ B× C
by C2, C4 and C5 are in the ratio
So, A = K × B × C 18 : 19 : 20. If C6 has made a profit
Where K is proportionality constant. of Rs 19 crore more than C1, then
Since, A = 16 when B = 2 and C = 4 the total profit (in Rs) made by all
16 = K × 2 × 4 five companies is:
K=2 (1) 438 Crore
When B = 4 and C = 6 (2) 435 Crore
Then, A=2×4×6 (3) 348 Crore
A = 48 (4) 345 Crore
Variation

14.
 Application of Ratio,
Proportion & Variation

1. Partnership :
A partnership is a formal agreement between two or more parties to
manage and operate a business and to share its profit.

There are two types of partners in a business :

(i) Sleeping partner or silent partner :

A partner who just invests his/her money and is not involved in the
day-to-day operation of the Business.

(ii) Working partner :

A partner who invests his/her money and is involved in day-to-day
operations of the Business.

Rules of Partnership
1. When partners invest the same amount for a different time, then the
profits are shared in the ratio of their respective time.
2. When partners invest different amount for the same time, then the
profits are shared in the ratio of their respective investment.
3. When partners invest different amount for different times then profits
are shared in the ratio of the product of investment and time.
(Profit’s share = Investment × Time)

Example 15
A and B started a business by investing an amount of Rs. 5000 and Rs.
6000. Find the share of A, out of the annual profit of Rs. 33000.

Explanation Rack your Brain

Application of Ratio, Proportion & Variation

Ratio of share of A and B
5000 : 6000 ⇒ 5 : 6 If a2 = by + cz, b2 = cz + ax,
c2 = ax + by, then find the value
5 x y z
Share of A = × 33000 ⇒ Rs . 15000
11 of + + ?
=
a+ x b+ y c+ z
Example 16
A and B started a business by investing the same capital. At the end of the
year, they share the profit in the ratio of 4 : 3. If A invested his capital for
the whole year then for how many months B invested his capital.
Explanation
We know that when partners invest same amount for different time, then

15.
 the profits are shared in the ratio of their respective time.
Ratio of share of A and B = 4 : 3
A invested his capital for 12 months
According to the ratio A invested the money for ‘4K’ months
4K = 12
K = 3 months
‘B’ invested his capital for 3K months
3K = 3 × 3 = 9 months.

Ages
Most of the questions based on ages involve the concept of ratio-proportion.

Example 17
The ratio of ages of Ravi and Kavi is 15 : 8. Three years earlier the ratio was
2 : 1. Find the present age of Ravi.

Explanation
Let the present age of Ravi and Kavi be 15x and 8x
3 years ago their ages would have been (15x – 3) and (8x – 3)
15x - 3 2
So, =
8x - 3 1
⇒ 15x–3 = 16x – 6
∴ x=3
Present age of Ravi = 15x = 15×3 = 45 years.

Alternate
Application of Ratio, Proportion & Variation

16.
 Ravi Kavi
Difference
in age = 1 unit ×7
Age 3 years ago 2 Unit 1 Unit

Present Age 15 Unit : 8 unit

Difference
in age = 7 unit

We know that difference between the age of two persons remains constant.

Present age of Ravi = 15 unit = 15 × 3 = 45 years

Example 18
The ratio of Ajeet’s age and his mother’s age is 11 : 18. The difference of their
age is 14 years. The ratio of their ages after 6 years will be :

Explanation

Application of Ratio, Proportion & Variation

Let their ages be 11x and 18x.
18x – 11x = 14
7x = 14
∴ x=2
So their present ages are 22 years and 36 years, respectively.
Their ages after 6 years will be 28 years and 42 years, respectively.
The ratio of the ages of Ajeet and his mother after 6 years will be,

17.
 Incomes-Expenses
When the ratio of incomes and expenses of two or more than two persons
is given and their savings is asked.
(Income = Expenditure + Savings)

Standard Results to Remember

1. If the value of the ratio of incomes is more than the value of the ratio of
expenses, then we cannot determine the ratio of their savings. i.e., we
cannot determine who is saving more.
2. If the value of the ratio of incomes is less than the value of the ratio of
expenses, then we can determine the ratio of their savings i.e., we can
determine who is saving more.

Example 19
The ratio of the incomes of Bulbul and Chulbul is 3 : 5 and the ratio of their
expenses is 1 : 3. Who saves more money ?

Explanation
Let’s understand the question with the help of two cases.

Case-1 :
Incomes Expenses Savings
Bulbul  3  1  2
Chulbul   5   3   2

In this case savings of both of them are equal.

Case-2 :
Application of Ratio, Proportion & Variation

Incomes Expenses Savings

Bulbul 300 150 150
Chulbul 500 450 50
In this case Bulbul saves more than Chulbul.

Case-3 :
Incomes Expenses Savings
Bulbul 300 1 299
Chulbul 500 3 497
In this case Chulbul saves more than Bulbul.
Hence, we can not determine who saves more.

18.
Alternate
3
The value of the ratio of the incomes of Bulbul and Chulbul = ⇒ 0.6
5
1
The value of the ratio of their expenses = ⇒ 0.33
3
The value of the ratio of incomes > Value of ratio of expenses
So, we can’t determine who saves more.

Example 20
The incomes of A and B are in the ratio of 4 : 5. If the expenditures of both
of them is Rs. 35000 each, then the ratio of savings of A and B is 1 : 3. Find
the incomes of each ‘A’ and ‘B’.
Explanation
Let the incomes of A and B is 4x and 5x respectively.
We know that Saving = Income – Expenditure
Savings of A = 4x–35000
Savings of B = 5x–35000
Rack your Brain
The ratio of their savings is 1 : 3.
4 x- 35000 1
= a, b, and c are three positive
5 x- 35000 3
a+b+c
⇒ 12x – 1,05,000 = 5x – 35,000 numbers and S= , if
2
7x = Rs. 70000
(S-a) : (S-b) : (S-c) = 1 : 7 : 4 then
x = Rs. 10,000
ratio of a : b : c = ?
Income of A = 4x
⇒ 4 × 10000 = 40000
Income of B = 5x
⇒ 5×10000 = Rs. 50,000

Application of Ratio, Proportion & Variation

Denomination of coins
The denomination is a proper description of a currency amount for coins
and banknotes. In any currency, there is a main unit and a sub-unit that is
a fraction of the main unit. In our country, “Rupees” is the main unit and
“Paisa” is the sub-unit.
1 Rupees = 100 Paisa.

Example 21
Rs. 100 contained in a box consist of one rupee. 50 paise and 25 paise coins
in the ratio of 2 : 3 : 6. What is the number of 50 paise coins.
Explanation

19.
 Let the number of one rupee, 50 paise, and 25 paise coins are 2x, 3x and
6x respectively.
Value of 1 rupees coins = 1 × 2x = 2x
1 3x
Value of 50 paise coins = × 3 x =
2 2
1 3x
Value of 25 paise coins = × 6 x =
4 2
Total value = 100
5x = 100
x = 20
Number of 50 paise coins = 3x = 3 × 20 = 60 coins

Example 22
A bag contains 2 types of coins; 1 rupee coins and 50 paise coins. If the
total value of the coins of each kind is same, then what is the total amount
in the bag ? It is known the total number of coins is 153.
Explanation
Here, it is given that the total value of the coins of each type is same.
This means if the value of 1 Rupee coins is ‘x’ then the value of 50 Paise
coins will also be ‘x’.
So, number of 1 Rupee Coins = x
Number of 50 Paise coins = 2x
Total number of coins = 153 Previous Year’s Question
x + 2x = 153
3x = 153 A stall sells popcorn and chips
x = 51 in packets of three sizes: large,
Total amount in the bag is = 2x super, and jumbo. The number of
= 2 × 51 = 102 large, super, and jumbo packets
Application of Ratio, Proportion & Variation

in its stock is in the ratio of 7 :

17 : 16 for popcorn and 6 : 15 :
14 for chips. If the total number
of popcorn packets in its stock
is the same as that of chips
packets, then the number of
jumbo popcorn packets and
jumbo chips packets is in the
ratio:
(1) 1 : 1 (2) 8 : 7
(3) 4 : 3 (4) 6 : 5

20.
 Exercise

EASY
4 2 3 1
1. P and Q together have Rs. 1210. If of and R in the proportion of : :
15 3 4 2
2 respectively. How much will Q get?
P’s amount is equal to of Q’s amount.
5 ANSWER : Rs. 1215
How much amount does Q have? SOLUTON :
ANSWER : Rs. 484 2 3 1
Here, P : Q : R = : :
SOLUTION : 3 4 2
4 2 To simplify this ratio, take LCM of
Hence, P= Q
15 5 (3, 4, 2)
3 2 3 1
P= Q P : Q : R = × 12 : × 12 : × 12
2 3 4 2
P 3
= ⇒ (P : Q = 3 : 2) P:Q:R=8:9:6
Q 2
P + Q + R = 3105
 2  8K + 9K + 6K = 3105
Q’s share =  × 1210  = Rs . 484
 5  3105
K= = 135
2. Two numbers are respectively 20% and 23
50% more than a third number. The Q’s share = 9K = 9×135 = Rs. 1215
ratio of the two numbers is :
ANSWER : 4 : 5 5. The compounded ratio of (2 : 3), (5 : 11)
SOLUTON : and (11 : 2)
Let the third number be 100x ANSWER : 5 : 3
Then, first number = 120% of 100x = 120x SOLUTON :
Second number = 150% of 100x = 150x We know that compounded ratio of two
∴ Ratio of first two numbers = 120x: 150x simple ratios
a c ac
= 4: 5 a : b and c : d = × =
b d bd
3. The ratio of Dogs and Cats in a house Similarly, compounded ratio of (2 : 3),
is 3 : 2 and ratio of Cats and Birds in a (5 : 11), and (11 : 2) is
house is 4 : 5. Find the ratio of Dogs, 2 5 11
= × × = 5:3
Cats and Birds. 3 11 2
ANSWER : 6 : 4 : 5 MODERATE
SOLUTION: 6. The value of a diamond varies directly
Ratio of Dogs and Cats = 3: 2 with the square of its weight. A diamond
Ratio of Cats and Birds = 4: 5 broke into three pieces whose weights
In both the ratios Cats must be same. were in the ratio 32 : 24 : 9. The loss
Ratio of Dogs and Cats = (3 : 2)×2 = 6:4 caused due to the breakage was Rs.
Ratio of Cats and Birds = 4 : 5 25.44 lakhs. Find the initial value (In
∴ Dogs : Cats : Birds = 6 : 4 : 5 Rupees) of the diamond.
4. Rs. 3105 is to be divided among P, Q , ANSWER : Rs. 42.25 lakhs
Exercise

21.
 SOLUTON : 8. The ratio of income to the
Let the weights of three pieces be 32x, 24x, expenditure of P is 5 : 3 and that of
and 9x respectively. Q is 7: 6. If the savings of P is twice
Let the value of the diamond be denoted that of Q, then what is the ratio of
by ‘v’ the total income of P and Q to the
v ∝ w2 (where w is the weight) total expenditure of P and Q?
v = kw2 (where k is constant) ANSWER : 4 : 3
Total value of the broken pieces SOLUTON :
= k{(32x)2 + (24x)2 + (9x)2} = 1681 kx2 Let the income of P be 5x, then the
Value of the diamond initially = k(65x)2= expenditure of P will be 3x.
4225 kx2 ∴ Savings of P = 5x – 3x = 2x
Given that 4225 kx2 – 1681 kx2 = 2544 kx2 Let the income of Q = 7y, then the
2544 kx2 = Rs. 25.44 lakhs expenditure of Q will be 6y
⇒ kx2 = 1000 ∴ Savings of Q = 7y–6y = y
The value of the diamond = (4225) × 1000 Given, savings of P is twice that of Q
= Rs. 42.25 lakhs ∴ 2x = 2×y ⇒ x = y
∴ The ratio of total earnings of P and Q
7. Raghav distributed a certain amount
to the total expenditure of P and Q
of money among his friends Ankur,
5x+ 7y
Ashwani and Neeti in the ratio of =
3x+6y
3 : 4 : 5. Neeti distributed the total
12 x
amount among her friends Dheeraj, = =4:3
9x
Sunil and Surendra in the ratio of 6 : 7
: 8. If Dheeraj gets an amount of 3000 9. The mass of the liquid varies directly
Rs., then find the amount that Ashwani with its volume. The mass of a liquid
got from his friend Raghav. is 12gm when its volume is 15 cm3.
ANSWER : Rs. 8400 Find the mass of the liquid if its
SOLUTON : volume is 20 cm3 (in gm).
Let the amounts that Ankur, Ashwani, and ANSWER : 16gm
Neeti receive be 3x, 4x, and 5x respectively. SOLUTON :
Let the amount that Dheeraj, Sunil, Let the mass of liquid be M and the
and Surendra received be 6y, 7y, and 8y volume be V.
respectively. M ∝ V ⇒ M = KV where K is a constant
5x = 21y M1 V
= 1
Given, 6y = 3000 M2 V2
y = 500. = =
M1 12 gm M2 x
21 21 taking
x= y= × 500 = 2100 = =
V1 15 cm3
V2 20 cm3
5 5
The amount that Ashwani received 12 15
= ; x = 16 gm
= 4 × 2100 = Rs. 8400 x 20
Exercise

22.
10. 2x+y : 3x+3y is equal to the duplicate ANSWER : 2 : 1
ratio of 2 : 3. Find x : y SOLUTON :
ANSWER : 1:2 Here, the ratio of mangoes in three
SOLUTON : container is :
2 C1 : C2 : C3 = 3 : 4 : 5
2 x+ y 2
=  So, C1 = 3x, C2 = 4x and C3 = 5x
3 x+ 3 y  3 
After increment in the number of
2 x+ y 4 mangoes in first two containers, the
=
3 x+ 3 y 9 ratio of mangoes becomes:
18x + 9y = 12x + 12y C1 : C2 : C3 =5:4:3
6x = 3y So, C1 = 5y, C2 = 4y and C3 = 3y
x:y=1:2 The number of mangoes remains
DIFFICULT constant in container 3
11. A garden has a rare breed of rose that ∴ 5x = 3y
1 3y
triples in number every minute. If of x=
3 5
the garden is full of rose in 30 min, what Hence,
is the total time taken for the whole
garden to be full :
(1) 31 minutes
(2) 90 minutes
(3) 63 minutes
(4) 62 minutes Increase in the first basket ( C1 ) = 16y
ANSWER : 31 minutes Increase in the second basket ( C2 ) = 8y
SOLUTON : The required ratio = 16y : 8y = (2 : 1)
1
Since in 30 min garden is full. In the next
3
13. Ankur Ji sells Gulabjamun at Rs. 15
1 min, it will triple i.e. the garden will be full.
per kg. A Gulabjamun is made up of
31 minutes is the right answer. flour and sugar in the ratio of 5 : 3.
The ratio of the price of sugar and
12. The number of mangoes in three flour is 7 : 3 (per kg). Thus he earns
containers are in the ratio of 3 : 4 : 5. In 2
what ratio the number of mangoes in the 66 % profit. What is the cost price
3
first two containers must be increased of sugar?
so that the new ratio becomes 5 : 4 : 3 ? ANSWER : Rs. 14/kg
(1) 5 : 4 SOLUTON :
(2) 3 : 1 2 2
Here 66 % =
(3) 2 : 1 3 3
(4) 4 : 1 2
66 % profit means there is a profit of
3
Exercise

23.
 2 units when CP is 3 units. G = 190 and P = 150
So, SP = 5 unit There are 150 Parrots in a zoo.
5 unit = Rs. 15 METHOD 2 : Shortcut method
1 unit = Rs. 3 Head count = 340
CP = 3 unit = 3 × 3 = Rs. 9 Legs count = 1060
CP of Gulabjamun = Rs. 9 1060  53 
Average leg count per head= = 
By using rule of Alligation. 340  17 

9 - 3 x 3 Quantity of sugar
∴ = =
7 x- 9 5 Quantity of flour Number of Goats : Number of Parrots
45 – 15x = 21x - 27 53 53
= −2:4−
–36x = -72 17 17
72 = 19 : 15
x= =2
36 Number of Parrots
Price of sugar = 7x = 7 × 2 = 14 15
⇒ × 340 = 150 Parrots
Hence, Price of sugar = Rs.14/kg 34

14. In a zoo, there are goats and parrots if 15. During Navratri, every devotee
heads are counted, there are 340 heads offers fruits to the orphans. Every
and if legs are counted there are 1060 orphan received Bananas, Oranges,
legs. How many parrots are there? and Mangoes in the ratio of 3 : 2 :
ANSWER : 150 Parrots 7 in terms of dozen. The weight of a
SOLUTON : Mango is 24 gm and the weight of a
METHOD 1 : (Basic equation method) Banana and an Orange are in the ratio
Let the number of goats and parrots be ‘G’ of 4 : 5. If the weight of an Orange is
and ‘P’, respectively. 150 gm, then find the ratio of all the
The number of heads are 340 three fruits in terms of weight, that
So, G + P = 340 ………….(i) an orphan gets :
Now, the number of legs are 1060 SOLUTON :
4G + 2P = 1060 ………….(ii) Ratio of the weight of a Banana and an
(A Goat has 4 legs and a Parrot has 2 legs) Orange = 4 : 5
On solving these two equations we get : Weight of an Orange is 150 gm.
Exercise

24.
∴ 5x = 150
x = 30
Weight of a Banana = 4x = 4 × 30 = 120 gm
Ratio by dozen of
Banana : Orange : Mango = 3 : 2 : 7
Ratio of the weight of
Banana : Orange : Mango= 120 : 150 : 24
Combining the above two statements, the
ratio of total weights
= 3 × 120 : 150 × 2 : 7 × 24
Hence, ratio of weights of
Banana : Orange : Mango
30 : 25 : 14

Exercise

25.
Exercise

26.