MATHEMATICS

Chapter 1: Knowing Our Numbers

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KNOWING OUR NUMBERS

Natural Numbers

Counting numbers 1, 2, 3, 4, ...... etc. are called Natural numbers. The smallest natural number
is 1 and there is no largest natural number.

Digits and Place value

Numbers are formed using the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These symbols are called
digits or figures.

To find the place value of a digit in a number, multiply the digit with the value of the place it
occupies.

1. Indian place value chart for 9-digit number.

Crores Lakhs Thousands Ones Period

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TC C TL L TTH Th H T O Place

2. International place value chart for a 9 digit number.

Millions Thousands Ones Period

HM TM M HTh TTh Th H T O Place

3. Inserting commas: Inserting commas helps in reading and writing large numbers. A
comma is inserted after each period in both the systems.
Comparing numbers

i. When numbers of digits are different, then the number having more number of digits is
greater.
ii. When numbers of digits are the same, compare the leftmost digits in the two
numbers. The number with the greater leftmost digit is greater. In case the
leftmost digits are the same, then compare the next left digits in both the
numbers. The number with greater digit is greater. Continue this process, until
you come across unequal digits at the corresponding places. Then compare
those two digits to find the greater number.

4. Place value of a digit in a number is the digit multiplied by its place.

5. Face value of a digit remains the same, whatever place it may be occupying in the
place value chart. In other words, face value is the actual value of the digit.

6. Numbers can be arranged in ascending or descending order. Ascending order

means arrangement from smallest to the greatest. Descending order means

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arrangement from largest to smallest.

7. The number which is one more than the given number is called its successor.

8. The number which is one less than the given number is called its predecessor.

9. There are certain situations where we have to give an idea about a number without
actual calculation. This is called estimation.

10. Rounding a number to nearest ten

i. See the ones digit of the given number.
ii. If ones digit is less than 5, replace ones digit by 0, and keep the other digits as they are.
iii. If ones digit is 5 or more, increase tens digit by 1, and replace ones digit by 0.

Some important terms

• The arrangements of numbers from the smallest to the greatest is called ascending order.

• The arrangement of numbers from the greatest to the smallest is called descending order.

Place Value of digit

Let’s discuss the place value of digits in the number and how a number can be written in that
form

Indian system of numeration

Values of the places in the Indian system of numeration are Ones, Tens, Hundreds, Thousands,
Ten thousand, Lakhs, Ten Lakhs, Crores and so on.

The following place value chart can be used to identify the digit in any place in the Indian
system.

Periods Crores Lakhs Thousands Ones

Places Tens Ones Tens Ones Tens Ones Hundreds Tens Ones

We can use below table format for easily reading and writing the Number

Number Tens Ones Tens Ones Hundreds Tens Ones Number

lakh lakh thousand thousand Name

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5,46,851 5 4 6 8 5 1 Five lakh

forty-six
thousand
eight
hundred
fifty one

32,75,829 3 2 7 5 8 2 9 Thirty
two lakh
Seventy-
five
thousand
eight
hundred
twenty
nine

Use of Commas

Commas added to numbers help us read and write large numbers easily. As per Indian
Numeration, Commas are used to mark thousands, lakhs and crores. The first comma comes
after hundreds place (three digits from the right) and marks thousands. The second comma
comes two digits later (five digits from the right). It comes after ten thousand place and marks
lakh. The third comma comes after another two-digits (seven digits from the right). It comes
after ten lakh place and marks crore

International system of numeration

Values of the places in the International system of numeration are Ones, Tens, Hundreds,

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Thousands, Ten thousands, Hundred thousands, Millions, Ten millions and so on.

1 million = 1000 thousands,

1 billion = 1000 millions

Following place value chart can be used to identify the digit in any place in the International
system.

Periods Billions Millions Thousands Ones

Places Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones

Estimation of the Numbers

A reasonable guess of the actual value is called an estimate.

A quick, rough estimate of the result of number operations can be done by rounding off the
numbers is involved.

Rules of Estimation

• Estimating numbers to the nearest tens is done by rounding off numbers 1, 2, 3 and 4 to 0
and number 6, 7, 8, 9 to 10.

• Estimating numbers to the nearest hundreds is done by rounding off numbers 1 to 49 to 0

and numbers 51 to 99 to 100.

• Estimating numbers to the nearest thousands is done by rounding off numbers 1 to 499 to 0

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and the numbers 501 to 999 to 1000.

Estimation involves approximating a quantity to an accuracy required. We can apply the

above rules depending on the accuracy required.

We can estimate Sum, difference and Multiplication by applying the rules of estimation also.
We can apply the above rules depending on the accuracy required and how quickly answer
can be find out

Roman Numerals
Roman Numerals system is another system used apart of Hindu-Arabic system.

The Roman numerals are

Rules of the system

• In Roman numerals a symbol is not repeated more than three times, but the symbols V, L

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and D are never repeated.

• Roman numerals are read from left to right and the letters of Roman numerals are arranged
from the largest to the smallest.

• If a symbol of smaller value is written to the right of a symbol of greater value, then its value
gets added to the value of greater symbol.

VI = 5 + 1 = 6

• If a symbol of smaller value is written to the left of a symbol of greater value, its then value
is subtracted from the value of the greater symbol.

IV = 5 – 1 = 4

• The symbol I can be subtracted from V and X only.

The symbol X can be subtracted from L, M and C only.

Importance of Brackets
Brackets help in simplifying an expansion with more than one mathematical operation.

In an expression that includes brackets, the numbers inside the brackets must be simplified into
a single.

11. Rounding a number to nearest hundred

i. See the tens digit of the given number.
ii. If tens digit is less than 5, replace each of tens and ones digit by 0, and keep the
other digits as they are.
iii. If tens digit is 5 or more, increase hundreds digit by 1, and replace each digit on its right
by 0.
12. Rounding a number to nearest thousand
i. See the hundreds digit of the given number.
ii. If hundreds digit is less than 5, replace each of hundreds, tens and ones digit by
0, and keep the other digits as they are.
iii. If hundreds digit is 5 or more, increase thousands digit by 1, and replace each
digit on its right by 0.

13. Estimating sum or difference

i. Select the smaller number.
ii. Round off the given numbers to the highest place value of that of the smaller number.
iii. Add or subtract the rounded numbers as per the question.

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14. Estimating product

i. Round off each factor to its nearest greatest value.
ii. Multiply the rounded off factors.
15. Estimating quotient
i. Round off the dividend to the nearest multiple of the divisor so that division becomes
easy.
ii. Divide to get the quotient.

16. Roman numeral system is one of the systems in which certain symbols are
used to represent numbers. There are seven symbols to represent numbers
of Hindu-Arabic system in Roman numeration.

Roman Numeral I V X L C D M

Hindu-Arabic Numeral 1 5 10 50 100 500 1000

17. While writing numbers in Roman system, certain rules are to be followed. They are:
i. Repetition of a symbol in Roman numeral means addition.
ii. A symbol is not repeated more than three times. But the symbols V, L and D are never
repeated.
iii. If a numeral of lesser value is written to the left of a greater value, the resulting
value is obtained by finding their difference.
iv. If a numeral of lesser value is written to the right of a greater value, the
resulting value is obtained by finding their sum.
v. The symbols V, L and D are never written to the left of the symbol of
greater value. In other words, V, L and D are never subtracted.
The symbol I can be subtracted from V and X only. The symbol X can be subtracted

from L, M and C only.

18. For simplifying a given expression containing brackets, we strictly perform

simplification in the following order (VBODMAS):
i. Vinculum
ii. Brackets
iii. of
iv. Division
v. Multiplication
vi. Addition
vii. Subtraction

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Important Questions
Multiple Choice Questions:
1. The smallest 8 digit number is called:
a) Ten lakh
b) One lakh
c) Ten crore
d) One crore
2. Which of the following is not a symbol to write Roman numerals?
a) D
b) V
c) M
d) N
3. If we add 1 more to the greatest 5-digit number we get:
a) Smallest 5-digit number
b) Smallest 4-digit number
c) Smallest 3-digit number
d) Smallest 6-digit number
4. The smallest 4-digit number using any one digit twice from the numbers 2, 8, 9 is:
a) 2889
b) 2289
c) 2298
d) 2898
5. The smallest 4-digit number having three different digits is:
a) 1000
b) 1203
c) 1320
d) 1002
6. Which of the following options is equal to MMMCCLXXV − MCCCXXIV?
a) MLMCI
b) MMCLI

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c) MCMLI
d) MCMIL
7. Which pair has same digits at hundreds place?
a) 4232, 4341
b) 5432, 6922
c) 6524, 7823
d) 2334, 2340
8. Which of the following options is true?
a) XCVIII > MCVIII
b) DCVIII < MCVIII
c) DCVIII = MCVIII
d) XCVIII < LXXXVII
9. The greatest 4-digit number using any one digit twice from the numbers 7, 1, 2 is:
a) 7221
b) 7721
c) 7122
d) 7112
10. The difference between face value and place value of 5 in 210517 is:
a) 4995
b) 5005
c) 495
d) 395
11. Make the greatest four digit number by using any one digit twice by 3, 8, 7?
a) 8378
b) 8873
c) 3387
d) 7783
12. Write the greatest 4-digit number using different digits with 6 in the tens place.
a) 9876
b) 9867
c) 9687

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d) 6987
13. Which of the following is the Roman numeral for the number 2765?
a) MMDCCXLV
b) MMDCXLXV
c) MMDCCLXV
d) MMCDCLXV
14. The smallest 4-digit number using the digits 2, 1, 3, 7 without repetition is:
a) 1237
b) 2137
c) 2371
d) 3217
15. Identify the greatest and the smallest in 2853, 7691, 9999, 12002, 124:
a) 12202, 124
b) None of these
c) 12202, 7691
d) 9999, 124
Match The Following:
Column I Column II

1. Estimation A. International System

2. Billion, Million B. Rough Calculation

3. Crore C. 90,030

4. Nine Thousand Thirty D. Indian System

Fill in the blanks:

1. 1 million = _____ hundred thousand.
2. 1 crore = _____ ten lakh.
3. 1 crore = _____ million.
4. 1 million = _____ lakh.
True /False:
1. 1 thousand = 10 hundred.

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2. There are many Roman Numbers.

3. You are using estimation to get rough idea.
4. Standard unit of mass is centimeter.
Very Short Questions:
1. How many crores makes a billion?
2. Find the greatest number from below number 763298 and 764298.
3. Place commas correctly and write the numerals:
(a) Seventy three lakh seventy five thousand three hundred seven.
(b) Nine crore five lakh forty one.
(c) Seven crore fifty two lakh twenty one thousand three hundred two.
(d) Fifty eight millions four hundred twenty three thousand two hundred two.
4. On Sunday 6000 people visited a zoo. Among them 2615 were children then how
many adult visited zoo?
5. Kirti Bookstore sold books worth Rs.2,85,891 in the first week of June and books
worth Rs.4,00,768 in the second week of the month. How much was the sale for the
two weeks together? In which week was the sale greater and by how much?
6. Find the greatest and the smallest number. 42375, 42367, 42329, 42338.
7. Write and solve the expression: Thirteen multiplied by sum of four and eleven.
Now reverse the result and add it to earlier result, what you obtain multiply it by
13.
8. Write the smallest three digit number whose value does not change on reversing
its digits.
9. Write the greatest three digit number which does not change on reversing its
digits.
10. What must be added to 203 to get a number whose digits are reversed of the
given number?
11. Write 438 in its expanded form.
12. Write the greatest five-digit number using the digits 4, 2 and 0.
13. The capacity of a water tank is 300 liters. Express its capacity in milliliters.
14. What is the successor of greatest 6-digit number?
15. What is the place value of 7 in 1743?
Short Questions:

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1. Of 7,12,540 and 71,25,400 which number is greater and by how much?

2. Write the smallest and the greatest 5-digit numbers using the digits 0,2,4,6,8
(Repetition of digits is not allowed).
3. Write the following numbers in ascending order. How many of them are even
numbers?
63,854, 63,584, 65,348, 68,543, 64,835
4. Round the given numbers to the nearest tens.
(a) 48
(b) 59
(c) 64
(d) 215
5. Estimate the following products:
(а) 86 × 316
(b) 898 × 786
6. Divide 2,63,175 by 275.
7. A student multiplied 3759 by 231 instead of multiplying by 213. How much was his
product greater than the correct product?
8. Estimate: 25,148 + 7394 + 9343 + 752
9. Write all the even numbers between 90 and 100 in Roman Numerals.
Long Questions:
1. Write the missing digits in the following sums:

2. Write Hindu-Arabic numerals for:

(a) LXXXVI
(b) LXXV
(c) XCIX
(d) XCI

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3. The distance between the school and Reena’s house is 1 km 480 m. Everyday she
walks both ways. What distance does she cover in 6 days of a week?
4. Simplify: 36 ÷ [5 + {4 x 5 ÷ 2}]
5. To stitch a pant 1 m 15 cm cloth is needed. Out of 36 m cloth, how many pants can
be stitched and how much cloth will remain?
6. There are two factories located at place P and the other at place Q. From these
factories, a certain commodity is to be delivered to each of the depots situated at A,
B and C. Weekly production of commodity by P and Q are 120 kg and 150 kg
respectively. Weekly requirement of commodity by A, B and C are 80 kg, 90 kg and
100 kg respectively. P delivers 60 kg to A, 40 kg to B and 20 kg to C. How much
amount of the commodity should Q deliver to A, B and C to meet their
requirement? If the rate of the commodity is? 20 per kg, find the total amount to be
paid to P and Q.
Assertion Reason:
1.) Assertion (A) –92 is greater number than 52.
Reason (R) – When a number is bigger or larger than the second or rest quantities or
numbers it is known as greater number.
a) Both A and R are true and R is the correct explanation of A
b) Both A and R are true but R is not the correct explanation of A
c) A is true but R is false
d) A is false but R is true
2.) Assertion (A) – 59785 is the greatest number among 382, 4972, 18, 59785, 750.
Reason (R) – When a number is bigger or larger than the second or rest quantities or
numbers it is known as greatest number.
a) Both A and R are true and R is the correct explanation of A
b) Both A and R are true but R is not the correct explanation of A
c) A is true but R is false
d) A is false but R is true

ANSWER KEY-
Multiple Choice questions:
1. D. One crore
2. D. N
3. D. Smallest 6-digit number

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4. B. 2289
5. D. 1002
6. C. MCMLI
7. D. 2334, 2340
8. B. DCVIII < MCVIII
9. B. 7721
10. C. 495
11. B. 8873
12. B. 9867
13. C. MMDCCLXV
14. C. 1237
15. A. 12202, 124
Match The Following:
Column I Column II

1. Estimation B. Rough Calculation

2. Billion, Million A. International System

3. Crore D. Indian System

4. Nine Thousand Thirty C. 90,030

Fill in the blanks:

1. 1 million = ten hundred thousand.
2. 1 crore = hundred lakh.
3. 1 crore = ten million.
4. 1 million = ten lakh.
True /False:
1. True
2. False
3. True
4. False
Very Short Answer:
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1. 1 billion= 100 crore

2. 764298 is the greatest number
3. a) 73,75,307
b) 9,05,00,041
c) 7,52,21,302
d) 58,423,202
4. Total number of people visited the zoo = 6000 people
Total number of children = 2615 children
Then number of adults = 6000 – 2615 = 3385
5. Sale of books in the first week = Rs.2,85,891.
Sale of books in the second week = Rs.4,00,768
∴ Sale for the two weeks together = Rs.2,85,891 + Rs.4,00,768 = Rs.6,86,659.
The sale was greater in the second week by Rs. 4,00,768 – Rs.2,85,891 i.e., by Rs.
1,14,877.
6. The greatest number is 42375
The smallest number is 42329
7. Expression 13 × (4 + 11) = 13 × 15 = 195
On reversing the answer = 591
Adding reversing number to the result i.e., 591 + 195 = 786
Multiplying it by 13 i.e., 786 × 13 = 10218.
8. The required number is 101.
9. The required number is 999.
10. The number obtained by reversing the digits of 203 = 302.
∴ Difference = 302 – 203 = 99
Hence, the required number is 99.
11. 438 = 4 × 100 + 3 × 10 + 8.
12. The greatest five-digit number using the digits 4, 2 and 0 is 44420.
13. We know that
1 liter = 1000 mL
∴ 300 liters = 300 × 1000 mL = 3,00,000 mL
Hence, the capacity of water tank = 3 lakh milliliters.

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14. Greatest 6-digit number = 999999

Successor of it = 999999 + 1 = 1000000
i. e., smallest 7-digit number.
Hence, the required successor = 10,00,000.
15. Let us write 1743 in its expanded form
1743 = 1000 + 700 + 40 +3
Place value of 7 = 700
Hence, the place value of 7 = 700.
Short Answer:
1. Since 71,25,400 is a seven-digit number and 7,12,540 is a six-digit number.
So, 71,25,400 is greater than 7,12,540.

Hence 71,25,400 is greater than 7,12,540 by 64,12,860.

2. Given digits are 0, 2, 4, 6, 8
5 – digit greatest number = 86420;
5 – digit smallest number = 20468.
3. The given numbers are 63,854, 63,584, 65,348, 68,543 and 64,835.
Ascending order is 63,584; 63,854; 64,835; 65,348; 68,543
Even numbers are 63,584, 63,854 and 65,348
4. Given number Rounded off to tens
(a) 48 → 50
(b) 59 → 60
(c) 64 → 60
(d) 215 → 220
5. (a) 86 × 316
∵ 86 → 90 [Rounding off to tens] and 316 → 320 [Rounding off to tens]
So, the estimated product is 90 × 320 = 28800
6. We have

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Hence, quotient = 957 and remainder = 0.

7. First Method:
(3759 x 231) - (3759 x 213) = 868329 – 800667 = 67662
Second Method: 3759 x (231 – 213) = 3759 x 18 = 67662
Hence, the product difference is 67662.
8. Estimated values are
25,148 → 25100
7394 → 7400
9343 → 9300
752 → 800
So, the estimated sum is 25100 + 7400 + 9300 + 800 = 42600
Hence, the estimated sum is 42600.
9. Even numbers between 90 and 100, we have 92, 94, 96, 98.
∴ 92 = XCII,
94 = XCIV,
96 = XCVI,
98 = XCVIII
Long Answer:
1.

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2. (a) LXXXVI = 50 + 30 + 6 = 86
(b) LXXV = 50 + 20 + 5 = 75
(c) XCIX = (100 – 10) + 9 = 99
(d) XCI = (100 – 10) + 1 = 91
3. Distance covered when she walks one way = 1 km 480 m = 1480 m
Therefore, the distance covered when she walk both ways in a day = 1480 × 2 m = 2960
m
Total distance covered by Reena in 6 days = 2960 × 6 m = 17760 m or 17 km 760 m.
4. Given:
36 ÷ [5 + {4 × 5 ÷ 2}]
Using B, O, D, M, A, S
5
= 36 ÷ [5 + {4 × } ] = 36 ÷ [5 + {2 × 5}]
2
= 36 ÷ [5 + 10] = 36 ÷ 15
1 12 2
= 36 × = 𝑜𝑟 2
15 5 5

Cloth required to stitch 1 pant = 1 m 15 cm

= 100 cm + 15 cm [∵ 1 m = 100 cm]
= 115 cm

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Total cloth = 36 m = 36 × 100 cm = 3600 cm

3600
Therefore number of pants stitched =
115
Hence, 31 pants can be stitched and cloth left over is 35 cm.
5.

Amount of commodity delivered by P to A = 60 kg

Amount of commodity delivered by Q to A = 80 – 60 = 20 kg
Amount of commodity delivered by P to B = 40 kg
Amount of commodity delivered by Q to B – 90 – 40 = 50 kg
Amount of commodity delivered by P to C = 20
Amount of commodity delivered by Q to C = 100 – 20 = 80 kg.
Now Amount of money to be paid to P by A, B and C = ₹ (60 × 20 + 40 × 20 + 20 × 20)
= ₹ (1200 + 800 + 400) = ₹ 2400
and amount of money to be paid to Q by A, B and C
= ₹ (20 × 20 + 50 × 20 + 80 × 20)
= ₹ (400 + 1000 + 1600) = ₹ 3000
Hence, the total amount
= ₹ 2400 + ₹ 3000 = ₹ 5400.
Assertion Reason Answers:
1.) a) Both A and R are true and R is the correct explanation of A

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2.) a) Both A and R are true and R is the correct explanation of A

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