Vedanta Excel Maths Book 7

Operations on Whole Numbers

6. Let’s find the square roots of these numbers by division method.

a) 196 b) 256 c) 441 d) 1225 e) 4096 f) 24964

7. Let’s simplify.

a) 25 b) 49 c) 225 d) 256
36 64 256 625

e) 2 × 6 f) 2 3 × 15 g) 7 × 2 14 h) 2 × 3 × 12

i) 3 × 5 × 12 j) 6 × 5 × 10 k) 2 8 × 3 18 × 50

8. Let's simplify and find the values of the following.

a) 22 × 32 × 62 b) 42 × 92 × 32 c) 82 × 92 × 102
42 22 × 62 42 × 52 × 62

9. Let's find the square roots of the following fractions.

a) 49 b) 81 c) 169 d) 111 e) 3 1
64 100 196 25 16

10. Let's find the square roots of the following decimal numbers.

(a) 0.16 (b) 0.81 (c) 1.44 (d) 1.96 (e) 3.24

11. a) In a morning assembly students are arranged in the square form. If there
are 25 students in each row, find the number of students assembled in the
ground.

b) If 45 rose plants are planted along the length and the breadth of a square
floriculture garden, how many plants are there in the garden?

c) The length of a square pond is 30 m. Find its area.
d) When a certain number of children are arranged in a square ground, there

are 14 children along the length and 14 children along the breadth. How
many more children are needed to arrange 15 children along the length and
the breadth?
12. a) 400 students are assembled in the square form. How many students are there
in each row ?
b) The area of a square garden is 625 m2. Find its length.
c) In an afforestation program on the ‘World Environment Day’ every student
from different schools planted as many plants as their number. If they planted
4225 plants altogether, how many students took part in the program?

d) Every student of a school donated as much money as their number to make

a fund for coronavirus victims. If they collected Rs 13,225 altogether, how

many students donated money in the fund?

13. a) Find the smallest number by which each of the following numbers is

multiplied to make them perfect squares.

(i) 32 (ii) 192 (iii) 245 (iv) 448 (v) 720

b) Find the smallest number by which each of the following numbers is

divided to make them perfect squares.

(i) 98 (ii) 125 (iii) 243 (iv) 384 (v) 756

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Operations on Whole Numbers

Creative Section - B

14. a) Find the least number which is a perfect square and exactly divisible by
b) 10, 12 and 15.
Find the least number which is a perfect square and exactly divisible by
15. a)
b) 18, 24 and 36.
c) Find the least number that must be subtracted from 2120 so that the
d) result is a perfect square.
What is the smallest number to be subtracted from 4400 so that the
result is a perfect square?
Find the least number that must be added to 2477 so that the sum will be a
perfect square.
What is the smallest number to be added to 13431 so that the sum will be a
perfect square

It's your time - Project work!

16. a) Let's find the square of the numbers from 1 to 10. Then observe the digits at
ones place of each square number. Now, write a short report about the fact
that you have discovered and present in your class. Can we apply this fact
in square of any other bigger numbers? Discuss in your class.

b) Let's find the square of any five even numbers and odd numbers separately.
What types of square numbers did you find in these two separate cases?
Write a short report and discuss in the class.

c) Let's write any three 2-digit any three 3-digit square numbers. Divide each
square number by 4 and observe the remainders. Write a short report about
your observations and present in the class. Can we use this fact to identify
whether a given number is perfect square or not? Discuss in the class.

d) Let's find the sum of the following consecutive odd numbers.

(i) 1 + 3 (ii) 1 + 3 + 5 (iii) 1 + 3 + 5 + 7 (iv) 1 + 3 + 5 + 7 + 9

(v) 1 + 3 + 5 + 7 + 9 + 11 (vi) 1 + 3 + 5 + 7 + 9 + 11 + 13

Can you discover any fact from these, sums of consecutive odd numbers? Write
a short report and discuss in the class.

17. a) Let's study the tricky process of finding square numbers of the numbers
ending with 5.

152 o 1 × (1 + 1) = 2 and 52 = 25, So, 152 = 225

952 o 9 × (1 + 9) = 90 and 52 = 25, So, 952 = 9025

Now, let's find the square of remaining two-digit numbers ending with 5.

b) A tricky way of finding square root of any bigger number!

Let's take a square number 576.

Let's group the last pair of digits (76) and the rest digit (5).

5 76 o Unit digit of 576 is 6. So, the unit digit of 576 will be either 4 or 6
[42 = 16 or 62 = 36]

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Operations on Whole Numbers

Now, take the rest digit 5 which is in the between the square of 2 and 3.

So, the ten's digit of 576 must be 2.

5 76 4
6

5 is in between Again, 2 × 3 = 6 and 5 < 6, so, the unit digit of 576 must
the square of 2 be the smaller option which is 4.
and 3.

? 576 = 24

Now, let's apply the above tricks and find the square root of following numbers
mentally as soon as possible.

(i) 225 (ii) 256 (iii) 324 (iv) 441 (v) 729 (vi) 1296

3.10 Cube and cube root

Let's study the following illustrations and investigate the idea of cube numbers and
their cube roots.

It is a unit cube. It is a cube of 8 unit cubes. It is a cube of 27 unit cubes.
1 × 1 × 1 = 13 2 × 2 × 2 = 23 3 × 3 × 3 = 33
= 1 is a cube number. = 8 is a cube number. = 27 is a cube number.

Again,
13 = 1 u 1 u 1 = 1 (or 13) is the cube number and 1 is its cube root.
23 = 2 u 2 u 2 = 8 (or 23) is the cube number and 2 is its cube root.
33 = 3 u 3 u 3 = 27 (or 33) is the cube number and 3 is its cube root.

Thus, the product of three identical numbers is the cube of the number (or cubic
number) and one of the identical numbers is the cube root of the cubic number.

The cube root of a number is denoted by the symbol 3 . For example,

3 1 = 1, 3 8 = 2, 3 27= 3, 3 64 = 4, 3 125= 5, and so on.

Worked-out examples

Example 1 : Find the cube of a) 9 b) 12 c) 400

Solution:

Cube of 9 = 93 = 9 × 9 × 9 = 729

Cube of 12 = 123 = 12 × 12 × 12 = 1728

Cube of 400 = 4003 = 400 × 400 × 400 = 64000000

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Operations on Whole Numbers

Example 2 : Find the cube root of 5832.

Solution

Finding the prime factors of 5832,

2 5832 To find the cube root, we
2 2916 should make the group of
2 1458 three identical factors and the
3 729 product of one of the factors
3 243 taken from each group is the
3 81 cube root.
3 27
39

3

Now, 5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

= 23 × 33 × 33

? 3 5832 = 2 × 3 × 3 = 18

Example 3 : Test whether 1080 is a perfect cube number or not.
Solution:
To find the prime factors of 1080,

2 1080 1082 = 2 × 2 × 2× 3 × 3 × 3 × 5
2 540 = 23 × 33 × 5
2 270
3 135 Here, 5 is left to make group of three identical factors.
3 45 So, 1080 is not a perfect cube number.
3 15

5

Example 4 : Find the cube root of 0.125.

Solution:

Here, 0.125 = 125
1000

? 3 0.125 = 3 125 = 3 53 = 5 = 0.5
1000 103 10

Example 5 : Simplify 3 22 × 3 × 3 2 × 32
Solution:

Here, 3 22 × 3 × 3 2 × 32 = 3 22 × 3 × 2 × 32 = 3 23 × 33 = 2 × 3 = 6

Example 6 : If a solid cubical block is 14cm long, find its volume.
Solution:
Here, the length of the cubical block (l) = 14 cm.
Volume of the block = l3

= (14 cm)3 = 14 cm × 14 cm × 14 cm = 2744 cm3

Hence, the required volume of the block is 2744 cm3.

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Operations on Whole Numbers

Example 7 : If the volume of a cubical reservoir water tank is 8000 m3 , find its
height.

Solution:
Here, volume of the cubical tank = 8000 m3 ,

or, l3 = 8 × 1000 m3
or l3 = 23 × 103 m3

? 3 l3 = 3 23 × 103

l = 2 × 10 m = 20 m
In a cube, length = breadth = height
Hence, the required height of the reservoir tank is 20 m.

Example 8 : What is the smallest number by which 875 must be multiplied so
that the product is a perfect cube?

Solution:

Finding the prime factors of 875,

5 875 Here, the factor 7 is only one time.
5 175 So, to make the group of three
5 35 sevens, it should be multiplied by
7 × 7, which is 49.
7

? 875 = 5 × 5 × 5 × 7

= 53 × 7

Hence, the required smallest number is 72 = 49.

Example 9 : What is the smallest number by which 3087 must be divided so that

the quotient becomes a perfect cube? 3 3087
3 1029
Solution:

Finding the prime factors of 3087, 7 343
7 49
7
3087 = 3 × 3 × 7 × 7 × 7 = 3 × 3 × 73

Thus, when we divide 3087 by 3 × 3 = 9, the quotient is 73, which is a perfect cube.

Hence, the required number is 9.

EXERCISE 3.4

General Section - Classwork
1. a) 33 = 27 303 = 27000, 3003 = 27000000, 30003 = 27000000000 so on.

30 has one zero. So its cube 27000 has three zeros. 300 has two zeros. So its cube
27000000 has six zero and so on.
Let's investigate the tricky idea from the above facts. Tell and write the cubes
of these numbers as quickly as possible.

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Operations on Whole Numbers

(i) 23 = .................... 203 = ....................... 2003 = .................................
(ii) 43 = .................... 403 = ...................... 4003 = .................................
(iii) 53 = .................... 503 = ........................ 5003 = .................................
(iv) 63 = .................... 603 = ........................ 6003 = .................................

b) Again, 3 8 = 2, 3 8000 = 20, 3 8000000 = 200 and so on.

8000 has three zeros. So its cube root 20 has one zero. 8000000 has six zeros.
So, its cube root 200 has two zeros.

Let's investigate the tricky idea from the above facts. Tell and write the cube
roots of these numbers as quickly as possible.

(i) 3 27 = ............. 3 27000 = ............. 3 27000000 = .............

(ii) 3 64 = ............. 3 64000 = ............. 3 64000000 = .............

(iii) 3 216 = ............. 3 216000 = ............. 3 216000000 = .............

(iv) 3 512 = ............. 3 512000 = ............. 3 512000000 = ............

Creative Section - A

2. Let’s find the cubes of: a) 6 b) 8 c) 9 d) 12 e) 25
e) 114
3. Let’s find the cube of: a) 2 b) 3 c) 2 d) 7
3 4 5 9
4. Let’s find the cube roots of the following numbers.

a) 64 b) 512 c) 729 d) 1000 e) 5832

5. Let’s find the cube roots of (a) 8 (b) 64 (c) 125 (d) 343 e) 512
27 125 216 1728 3375

6. Let’s, find the cube roots of the following decimal numbers.

a) 0.008 b) 0.027 c) 0.125 d) 0.343 e) 1.331

7. Let’s simplify. b) 3 23 × 73 c) 3 53 × 73
a) 3 33 × 53 e) 3 64 × 125 f) 3 2 × 3 × 3 22 × 32
d) 3 8 × 125 h) 3 4 × 3 × 3 2 × 9 i) 3 25 × 2 × 3 4 × 5
g) 3 32 × 52 × 2 3 3 × 5

8. a) A cubical die is 6cm long. Find its volume.
b) A cubical water tank is 90 cm tall.
(i) Find the volume of the tank.
(ii) If 1000 cm3 = 1 litre, how many litres of water does it hold?

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Operations on Whole Numbers

9. a) If the volume of a cube is 512 cm3, find its length.

b) The capacity of a cubical water tank is 343000cm3.

(i) What is the volume of the tank ? (ii) How high is the tank?

c) The capacity of a cubical tank of drinking water is 8000 litres. Find the
length of the tank. (1m3=1000l)

10. Test, whether these numbers are perfect cubes or not.

a) 64 b) 216 c) 500 d) 2744 e) 1458

Creative Section - B

11. a) Find the least number by which the following numbers must be multiplied,
so that the products are prefect cubes?

(i) 72 (ii) 128 (iii) 288 (iv) 675

b) Find the least number by which the following numbers must be divided,
so that the quotients are perfect cubes.

(i) 54 (ii) 192 (iii) 500 (iv) 1080

12. a) Find the least cube number which is exactly divisible by 4 and 6.

b) Find the least number which is a perfect cube and exactly divisible by 6
and 9.

It's your time - Project work!

13. a) Let's find the cube of any five even numbers and the cube of any five odd
numbers. What types of cube numbers, did you find in these two cases?
Write a short report and discuss in the class.

b) Let's find the cubes of 1, 4, 5, 6, 9 and 10. Observe the digit at ones place of
each cube number. What did you notice? Write a short report and discuss
in the class.

14. Let's study the following intersecting facts about the operations on cube and
square numbers.

13 + 23 = 1 + 8 = 9 and (1 + 2)2 = 32 = 9

13 + 23 + 33 = 1 + 8 + 27 = 36 and (1 + 2 + 3)2 = 62 = 36

Now, let's complete the following operations.

a) 13 + 23 + 33 + 43 = .......... and (1 + 2 + 3 + 4)2 = ..........

b) 13 + 23 + 33 + 43 + 53 = .......... and (1 + 2 + 3 + 4 + 5)2 = ..........

c) 13 + 23 + 33 + 43 + 53 + 63 = .......... and (1 + 2 + 3 + 4 + 5 + 6)2 = ..........

d) What did you discover from the above given facts? Write a short report and
present in your class.

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Unit Real Numbers

4

4.1 Integers – Looking back
Classroom - Exercise

1. Let’s tell and write ‘true’ or ‘false’ for the following statements.

a) 0 is the less than any negative integer. .......................

b) –1 is the greatest negative integer. .......................

c) –5 is greater than 1. .......................

d) The set of integers is the universal set of whole numbers. .......................

e) The set of integers is a subset of the set of natural numbers. .......................

f) The difference of any two whole numbers is always a .......................
whole number.

g) The quotient of 2 ÷ 4 is an integer. .......................

h) The quotient of 4 ÷ 2 is an integer. .......................

2. Let's tell and write the correct answers as quickly as possible.

a) 9 – 4 = ........, 4 – 9 = ........, 9+4 = .........

b) – 7 + 2 = ........, 7 – 2 = ........, –7–2 = .........

c) 3 × (– 6) = ........, – 3 × 6 = ........, (–3 ) × (–6) = .........

d) 18 ÷ 3 = ......, – 18 ÷ 3 = ......, (–18) ÷ (–3) = .........

Now, let's take a set of whole numbers, W = {0, 1, 2, 3, …} and a set of natural
numbers, N = {1, 2, 3, …}.

Here, the set of whole numbers is the universal set of the set of natural numbers. In
other words, the set of natural numbers is the subset of the set of whole numbers.

Now, let’s take any two whole numbers 3 and 6.

Here, 3 + 6 = 9 (9 is a member of the set of whole numbers.)
6 – 3 = 3 (3 is a member of the set of whole numbers.)
6 × 3 = 18 (18 is a member of the set of whole numbers.)

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Real Numbers

But, now to subtract 6 from 3, let’s study the following number line.

–6
+3

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

–3

Thus, from the number line, 3 – 6 = – 3
Here, – 3 is not the member of the set of whole numbers

– 3 is the member of the set of Integers

Thus, the set of all numbers both positive and negative including zero (0) is called
the set of integers.

The set of integers is denoted by the letter ‘Z’.

?Z = {…, – 3, – 2, – 1, 0, 1, 2, 3, …}is the set of integers.

Z+ = {+ 1, + 2, + 3, + 4, + 5, …} is the set of positive integers.

Z– = {– 1, – 2, – 3, – 4, – 5, …} is the set of negative integers.

In the above number line, the positive integers right to the zero mark are in increasing
order. However, the integers left to the zero mark are in decreasing order. In other
words, every integer on a number line is always greater than the integers on its left
side. For example:

5 > 4 > 3 > 2 > 1 > 0 – 1 > – 2 > – 3 > – 4 > – 5, and so on.

Further more, let’s consider two integers + 4 and – 4.

+ 4 is four units right from the zero mark and – 4 is four units left from the zero
mark. Here, + 4 and – 4 are called the opposite integers.

4.2 Absolute value of integers P Q

Let's study the adjoining number line. Let P and Q – 2 km – 1 km 0 1 km 2 km
are two places. The place P is – 2 km left from the 4 km

zero mark and place Q is + 2 km right from the zero mark.

Now, what is the distance between P and Q?

Is it (+ 2 km) + (– 2 km) = 0 km? It is impossible.
It must be 2 km + 2 km = 4 km.
Here, we do not consider the directions of P and Q from the zero mark. In such cases
the numerical value of + 2, and – 2 is the same which is 2.

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Real Numbers

Therefore, 2 km is known as the absolute value of – 2 km and + 2 km.
We can write it as |+ 2| = |– 2| = 2

Thus, the absolute value of an integer is only the numerical value of the integer.
For any integer x: (+x) = ( x) = x

EXERCISE 4.1

General Section - Classwork

1. Let's tell and write ‘True’ or ‘False’ as quickly as possible.

a) The sum of 2 + 5 is a whole number .................

b) The product of 3 × 4 is a whole number .................

c) The difference of 3 – 7 is a whole number .................

d) The difference of 2 – 5 is an integer. .................

e) 0 is the smallest integer. .................

f) –1 is the greatest negative integer. .................

2. Let's tell and write the correct answers as quickly as possible.

a) Integers between – 3 and + 3 are .................

b) Integers less than 0 and greater than – 4 are .................

c) Integers that lie 1 units right and 1 units left to – 1 are .................

d) The integer which is neither negative nor positive is .................

e) The absolute value of

(i) |7| = ............. (ii) |– 7| = .................

3. Let's insert ‘>’ or ‘>’ sign between each pairs of integers and compare them.

a) – 4 ........ 0 b) – 3 ........ – 6 c) 3 ........ – 8 d) – 5 ........ –2

Creative Section - A

4. List the members of the sets of integers and find the members of the given set
operations.

a) Zfi1n=d { integers between –2 and +2 } and Z2= { integers between 0 and –3},
(i) Z1 ‰ Z2 (ii) Z1 ˆ Z2

b) Z(i3) = {x : –3 < x < 3, x  z} and Z4 = { x : –3 ≤ x ≤ 3, x  Z }, find
Z3 – Z4 (ii) Z4 – Z3

5. Let’s simplify

a) |+ 2|+ |+ 2 | b) |– 3|+| +3| c) |– 5| +| – 5 |

d) |– 7|– |+ 4 | e) |+6 |– |– 9| f) |– 3|–|–5|

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Real Numbers

Creative Section - B
6. a) Roji's house is 4 km east and Dikshya’s house is 1km west from their school

in the same straight line.
(i) Mark their houses as A and B on a number line by using integers.
(ii) Find the distance between their houses.
b) Mt. Everest is 8849 m above the sea-level. The Marianas Trench, a deep
depression under the Pacific Ocean, is at 11,032 m below the sea-level. Find
the vertical distance between them.
c) On a winter day, the maximum temperature of the Jomsom Valley was
recorded 15.5° C and the minimum temperature was recorded – 2.5q C. What
is the difference between the maximum and the minimum temperature on
that day?
7. a) Shopkeeper gains Rs 540 by selling some fruits but loses Rs 612 by selling
some vegetables. Find his net profit or loss.
b) A stationer loses Rs 1,350 by selling a few number of pens, but he gains
Rs 1,525 by selling a few number of staplers. Find his profit or loss.

4.3 Operations on integers

It is easier to understand the fundamental operations (addition, subtraction,
multiplication and division) of integers on the number lines. Let's study the following
examples and learn about the operations on integers by using number lines.

(a) Addition of integers

(i) Add: (+3) + (+2) = ? (+3) + (+2)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

+5

When we move 3 units to the right from 0 and again 2 units to the right from 3, we
reach at 5 units from 0. Thus, (+3) + (+2) = +(3+2) = +5

(ii) Add: (+7) + (– 3) (+7)

( 3)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

+4

When we move 7 units to the right from 0 and move 3 units back to the left from 7.
We reach at 4 units right from 0. Thus (+7) + (–3) = + (7 – 3) = +4

(iii) Add: (–2) + (–4) ( 4 + ( 2

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
6

When we move 2 units to the left from 0 and again 4 units to the left from –2, we
reach at the point 6 units to the left from 0. Thus, (–2) + (–4) = – (2 + 4) = – 6

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Real Numbers

(iv) Add : (+2 ) + (–9) ( 9)

( 2)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
7

When we move 2 units to the right from 0 and then move 9 units to the left from 2,
we reach at 7 units to the left from O.

Thus, (+2) + (–9) = (–9) + (+2) = – (9 – 2) = – 7.

(b) Subtraction of integers (+6)
(i) Subtract: (+6) – (+4) ( 4)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

+2

When we move 6 units to the right from 0 and then move 4 units left from 6, we
reach at the point 2 unit right from 0.

Thus, (+6) – (+4) = + (6 – 4) = + 2

(ii) Subtract: (–2) – (+6) = (–2) + (–6)

( 6) ( 2)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

8

When we move 2 units to the left from 0 and again 6 units left from –2, we reach
at –8.
Thus, (–2) – (+6) = (–2) + (–6) = –2 – 6 = – (2 + 6) = – 8

(iii) Subtract: (+4) – (–3) (+4) (+3)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
+7

When we move 4 units to the right from 0 and change the direction two times, then
the direction remains right from 0. Again move 3 units right from 4. We reach at 7.

Thus, (+4) – (–3) = (+4) + (+3) = + (4 + 3) = + 7

(iv) Subtract: (– 8) – (–5)

( 8) ( 5)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
3

When we move 8 units to the left from 0 and change the direction two times, then
the direction is right. Again, move 5 units to the right from (–8), we reach at – 3.
Thus, (–8) – (– 5) = (–8) + ( + 5) = – (8 – 5) = – 3

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Real Numbers

4.4 Sign rules of addition and subtraction of integers

From the above illustrations operations of addition and subtraction of integers on
number lines, we can summarise the following sign rules.

(i) The positive integers are always added. (+) + (+) = + sum
The sum holds the positive (+) sign.

For example:

(+ 3) + (+ 6) = + 9 or 9

(ii) The negative integers are always added. (–) + (–) = – sum
The sum holds the negative (–) sign.
For example:

(– 4) + (– 3) = – 7

(iii) The positive and negative integers are always subtracted.
The difference holds the sign of the bigger integer.

For example:

(+ 8) + (– 3) = + 5 or 5 (+) + (–) = + difference if (+) > (–)
(+ 8) + (– 9) = – 1 (+) + (–) = – difference if (+) < (–)

4.5 Properties of addition of integers

Let’s study the following properties of addition of integers.

(i) Closure property
It states that the sum of two integers is also an integer. For example:
(+ 2) + (+ 4) = + 6, which is also an integer.
(+ 8) + (– 3) = + 5, which is also an integer.
(– 2) + (– 3) = – 5, which is also an integer.

Thus, if a and b are any two integers and Z is the set of integers then a + b  Z.

(ii) Commutative property
It states that the sum of two integers remains unchanged if their places are
interchanged. For example,
(+ 4) + (+ 3) = (+ 3) + (+ 4) = + 7
(– 5) + (+ 2) = (+ 2) + (– 5) = – 3
Thus, if a and b are two integers, then a + b = b + a.

(iii) Associative property
It states that the sum of three integers remains unchanged, if the order in which
they are grouped is altered. For example,
[(+ 2) + (+ 3)] + (+ 1) = (+ 2) + [(3) + (+ 1)] = + 6
[(– 4) + (+ 9)] + (– 2) = (– 4) + [(+ 9) + (– 2)] = + 3
Thus, if a, b, and c are three integers, then (a + b) + c = a + (b + c).

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Real Numbers

(iv) Additive property of zero (0)
If zero is added to any integers, the sum is equal to the integer itself. For example:
(+ 5) + 0 = + 5, 0 + (– 9) = – 9
Thus, if a is any integer, then a + 0 = a.
Here, zero (0) is known as the identity element of addition.

(v) Additive inverse
For every integer, there exists an integer such that their sum is zero (0). Here, each
integer is said to be the additive inverse of the other. For example,
(+ 2) + (– 2) = 0.
Here, (+ 2) is the additive inverse of (– 2) or (– 2) is the additive inverse of (+ 2).
Thus, if a is any integer, then (+ a) + (– a) = 0, where (+ a) is the additive inverse
of (– a) and (– a) is the additive inverse of (+ a).

EXERCISE 4.2
General Section – Classwork

1. Let's tell and write the answers as quickly as possible.

a) If p and q are any two integers, then according to commutative property,

.................................................................................................................................

b) If x, y and z are any three integers, then according to associative property,

.................................................................................................................................

c) What is the identity element of addition? ....................

d) What is the additive inverse of (+5)? ....................

e) What is the additive inverse of (–9)? ....................

f) What is the additive inverse of the absolute value of –4? ....................

2. Let’s write operations shown by the number lines, then find the sums.
a) b)

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

c) d)

-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

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e) Real Numbers
f)
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Creative Section - A

3. Let's show these operations by using number lines. Then simplify.

a) (+3) + (+4) b) (+7) + (–4) c) (–5) + (+1) d) (–8) + (–3)

e) (+9) – (+2) f) (–4) – (+5) g) (+8) – (–3) h) (–2) – (–3)

4. Let's simplify. b) (– 3) + (+ 7) + (+ 1)
a) (+ 2) + (+ 6) + (+ 3) d) (– 5) – (– 7) + (– 4)
c) (+ 6) – ( + 9) + (+ 3) f) (– 7) – (– 5) – (– 8)
e) (– 1) + (– 2) + (– 3) h) (– 13) – (– 6) + (+2) – (+ 7)
g) (+ 10) + (– 7) – (+ 2) + (– 4) j) (+ 3) + (– 11) – (– 15) – (+ 9)
i) (– 10) – (– 12) + (– 8) – (+ 3)

5. Let's use commutative property to add the following integers.

a) (+ 8) + (+ 2) b) (+ 6) + (– 2) c) (– 9) + (+6) d) (– 5) + (– 4)

6. Let's use associative property to simplify the following sums.

a) (+ 2) + (+ 3) + (+ 6) b) (+ 4) + (+ 3) + (– 5)

c) (+ 9) + (– 7) + (+ 3) d) (– 6) + (– 8) + (+ 10)

7. a) What should be added to (– 10) to get (+10)?
b) What should be added to ( + 15) to get (–20) ?
c) What should be subtracted from (+ 25) to get (– 10) ?
d) What should be subtracted from (– 8) to get (– 16) ?

8. a) Subtract (– 40) from the sum of (–20) and (–30).
b) Subtract the sum of ( + 10) and (–25) from (– 30)

Creative section - B

9. a) On a winter day, the temperature of Namche Bazar in the morning was
–5°C. If it is increased by 10°C in the day time and again decreased by 6°C
in the evening, calculate the temperature in the evening.

b) Bimlesh is a fruit- seller. He made a profit of Rs 400 on Sunday, a loss of
Rs 180 on Monday and again a loss of Rs 350 on Tuesday. Find his net
profit or loss in three days.

c) In a mathematics quiz, positive marks are awarded for correct answers
and negative marks are given for incorrect answers. If Yellow-house
scores 20, – 5, – 10 and 15 in four successive rounds, find the total marks
obtained by Yellow-house.

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4.6 Multiplication and division of integers
(c) Multiplication of integers

Multiplication of integers can also be performed by using number lines. Let’s study
the illustrations given below:

(i) Multiply: (+ 4) × (+ 2) 4 times (+2) = +8

(+ 4) × (+ 2) = 4 × (+2) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
=+8 +8

(ii) Multiply: (+ 2) × (– 3) 2 times (–3) = –12

(+ 2) × (– 3) = 2 × (–3) -14 -13-12-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
=–6 –6

(iii) Multiply: (– 3) × (+ 3) 2 times (–3) = –6

(– 3) × (+ 3) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
= (+ 3) × (– 3) = – 9 –9

(iv) Multiply: (– 2) × (– 3) 2 × (–3) (–2) × (–3)

(– 2) × (– 3) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
If (+ 2) × (– 3) = (– 6), +6
Then, (– 2) × (– 3) = (+ 6)

To make it more convenient to understand, let's study the following patterns of
operations.

2×–3 =–6

multipliers are 1×–3 =–3 products are
decreasing 0 × –3 = 0 increasing
–1×–3 =3

–2 × – 3 = 6

(d) Division of integers
Division is the inverse operation of multiplication. So, the same sign rules of
multiplication are used in division of integers. Let's study the examples given below:

(i) Divide: (+ 6) ÷ (+ 2)

(+ 6) ÷ (+ 2) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
= (+ 3) m +2 o m + 2 o m + 2 o

(ii) Divide: (+ 12) ÷ (– 4)

(+ 12) ÷ (– 4) -14 -13-12-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

= (– 12) ÷ (+ 4) = (– 3) –4 –4 –4

(iii) Divide (– 12) ÷ (– 4)
(– 12) ÷ (– 4)

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If (– 12) ÷ (+ 4) = (– 3), (–12) ÷ (–4)
Then (– 12) ÷ (– 4) = (+ 3)

(–12) ÷ (+4)

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

+4 +4 +4

4.7 Sign rules of multiplication and division of integers

From the above illustrations of multiplication and division of integers on number
lines, we can summarise the following sign rules.

(i) The product or quotient of two positive integers is always positive. For
example:

(+ 3) × (+ 4) = (+ 12) (+) × (+) = (+) product
(+ 15) ÷ (+ 5) = (+ 3) (+) ÷ (+) = (+) quotient

(ii) The product or quotient of two negative integers is always positive. For
example:
(–) × (–) = (+) product
(– 4) × (– 5) = (+ 20) (–) ÷ (–) = (+) quotient

(– 21) ÷ (– 3) = (+ 7)

(iii) The product or quotient of two positive and negative integers is always
negative. For example:

(+ 5) × (– 2) = (– 5) × (+ 2) = (– 10) (+) × (–) = (–) × (+) = (–) product
(+ 18) ÷ (– 6) = (– 18) ÷ (+ 6) = (– 3) (+) ÷ (–) = (–) ÷ (+) = (–) quotient

4.8 Properties of multiplication of integers
Let's study the following properties of multiplication of integers

(i) Closure Property
It states that the product of any two integers is also an integers.
For example: (+ 2) × (+ 3) = + 6 Which is also an integer
(+ 4) × (– 2) = – 8 Which is also an integer
(– 5 ) × (+6) = – 30 Which is also an integer
(– 6) × (– 7) = + 42 Which is also an integer

Thus, if a and b are any two integers and z is the set of integers then a × b = z.

(ii) Commutative Property
It states that the product of two integers remain unchanged if their places
are interchanged. For example :
( +3) × (– 2) = ( – 2) × ( 3) = – 6
(–8) × (–3) = (–3) × ( – 8)= + 24
Thus, if a and b are any two integers, then a × b = b × a.

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(iii) Associative property
It states that the product of three integers remains unchanged, if the
order in which they are grouped is altered. For example:
[( +2) × (+3)] × (+4) = (+2) × [(+3) × (+4)] = + 24
[( –5) × (–2)] × ( +3) = (–5) × [(–2) × (+3)] = + 30
Thus, if a, b and c are any three integers, then (a × b) × c = a × ( b ×c)

(iv) Distributive property
If a, b and c are any three integers then a × (b + c) = ab + ac
For example :
[(+2) × (+3) + (+4)] = (+2) × (+7) = +14 and
[(+2) × [(+3) + (+4)] = (+2) × (+3) + (+2) × (+4) = (+6) + (+8) = +14

(v) Multiplicative property of 1
If any integer is multiplied by 1, then the product is the integer itself.
For example :
(+2) × 1 = 1 × (+2) = +2, (–5) × 1 = 1 × (– 5) = –5
Thus, if a is an integer then a × 1 = 1 × a = a
Here, 1 is known as identity element of multiplication.

EXERCISE 4.3
General Section - Classwork
1. a) If x and y are two integers, then according to the closure property of

multiplication of integers: ................................................................................

b) If p and q are two integers, then according to the commutative property of
multiplication of integers: ................................................................................

c) If x, y and z are three integers, then according to the associative property of
multiplication of integers: ................................................................................

d) If p, q, and r are three integers, then according to the distributive property of
integers: .............................................................................................................

e) What is the identity element of multiplication of integers? ...........................

2. Let's tell and write the correct answer in the blank spaces.
a) The sign of product of two positive integers is always .........................
b) The sign of product of a positive and a negative integers is always. ..................
c) The sign of product of two negative integers is always .........................
d) The sign of product of three negative integers is always .........................

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e) The sign of quotient of dividing a negative integers by a positive integers is
always .........................

f) The sign of quotient of dividing a negative integers by a negative integers is
always .........................

3. Let's tell and write the products or quotients as quickly as possible.

a) (+3) × (–2) = .................... b) (–2) × (+5) = ....................

c) (–4) × (– 5) = .................... d) (+ 12 ) ÷ (+4) = ....................

e) (–24) ÷ (–8) = .................... f) (+56) ÷(– 7) = ....................

4. Let's study the following illustrations.

(–2)2 = (– 2) × (–2) = 4

(–2)3 = (–2) × (–2) × (–2) = (+4) × (–2) = –8

(–2)4 = (–2) × (–2) × (–2) × (–2) = (+4) × (+4)= 16
Investigate the idea from the above facts, then tell and write the values as
quickly as possible.
a) (–1)2 = ................., (–1)3 = ..............., (–1)4 = ................, (–1)5 = ...................

b) (–2)2 = ................., (–2)3 = ................., (–2)4 = ................., (–2)5 = ...............

c) (–3)2 = .................. (–3)3 = ................., (–3)4 = ................., (–3)5 = ................

Creative section - A
5. Let’s show these operations in number lines and simplify.

(a) (+2) × (+3) (b) (–2) × ( 4) (c) ( 5) × (–2) (d) ( 4) × (–3)
(e) (+6) y (+2) (f) (–8) y ( 4) (g) ( 15) y (–3) (h) ( 12) ÷ (–4)

6. Let’s find the products or quotients.

a) (+5) × (+6) b) (+7) × (–4) c) (–9) × (+3) d) (–10) × (– 8)
h) (–63)÷(–9)
e) (+48) ÷ (+6) f) (– 15) ÷ (+ 5) g) (+28) ÷ (–7)

7. Let’s simplify. b) (–2) × (–4) × (+5)
a) (+5) × (–2) × (+3) d) (+9) × (–4) × (–2)
c) (– 4) × ( + 6) × (–2) f) (–10) × (–5) × (–1)
e) (–2) × (–3) × (– 4) h) (+8) × (– 1) × (– 2) × (– 3)
g) (– 1) × (+2) × (–3) × (+4) (j) (– 2) × (– 3) × (– 4) × (–5)
i) (– 4) × (–5) × (+2) × (–1)

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Creative section - B

8. a) The product of two integers is (+40). If one of them is (–5), find the other
integer.

b) The product of two integers is (–72). If one of them is ( +12), what is the
other integer?

9. a) By what integer should (–4) be multiplied to get the product (+20)?
b) By what integer should (+7) be multiplied to get the product (–28)?

10. a) In a test, +2 marks are given for every correct answer and – 1 is given for
every incorrect answer. Priya attempted all the questions but 20 answers
were correct and 5 answers were incorrect. What marks does she get in
total ?

b) In an objective test containing 50 questions, a student is to be awarded
+ 3 marks for every correct answer, – 2 for every incorrect answers and
0 for not supplying any answers. If Samir had 30 correct answers and
5 questions were left to supply any answer. Find the marks secured by him
in this test.

c) A shopkeeper gains Rs 5 on each pen and loses Rs 2 on each pencils. He
sells 48 pens and 60 pencils on a day. Find his net profit or loss.

It's your time - Project work!

11. a) Let's write any two integers and show the closure property and commutative
property of multiplication of integers separately.

b) Let's write any three integers and show the associative property and
distributive property of multiplication of integers separately.

c) Let's take any two integers between 0 and –4. Then show that the even
number powers of negative integers are always positive and odd number
powers of negative integers are always negative.

4.9 Order of operations

We simplify a mathematical expression containing mixed operations (addition,
subtraction, multiplication and division) in order. The rule of this order is
well- known as 'BODMAS'.

Here,

B stands for removal of Brackets in the order ( ), { } and [ ].
O stands for Of (multiplication)
D stands for Division (÷)
M stands for Multiplication ( × )
A stands for Addition (+)
S stands for Subtraction (–)

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At first, let's study the following illustrations given below and learn about the use of
brackets in simplification of mixed operation.

Worked-out examples

Example 1: What is the result when 40 is divided by 5 times the difference
between 10 and 6?

Solution

Here, the mathematical expression is 40 ÷ 5 ( 10 – 6)
Now, 40 ÷ 5 (10 – 6) = 40 ÷ 5 (4)

= 40 ÷ 20 = 2
Example 2: Simplify : 48 ÷ 6 [ – 52 ÷ 2 { 4 – 3 ( 2 – 15 ÷ 3)}]
Solution

Here,
48 ÷ 6 [ – 52 ÷ 2 { 4 – 3 ( 2 – 15 ÷ 3)}]

= 48 ÷ 6 [ – 52 ÷ 2 { 4 – 3 ( 2 – 5)}] Inside ( ), 15 ÷ 3 = 5

=48 ÷ 6 [ – 52 ÷ 2 { 4 – 3 (– 3)}] Inside ( ), 2 – 5 = – 3

= 48 ÷ 6 [ – 52 ÷ 2 { 4 + 9}] –3 ( – 3) = ( – 3) of (– 3) = + 9

= 48 ÷ 6 [ – 52 ÷ 2 { 13 }] Inside { } 4 + 9 = 13

= 48 ÷ 6 [ – 52 ÷ 26] 2 { 13} = 2 of 13 = 26

= 48 ÷ 6 (–2) Inside [ ] – 52 ÷ 26 = – 2

= 48 ÷ ( – 12) 6 ( – 2) = 6 of ( – 2) = – 12

= –4

Example 3: Shaswat has a 1000 rupee note. He purchases 6 notebooks at
Rs 80 each, 2 pens at Rs 20 each, 1 diary milk at Rs 25 and he donates
Rs 100 to a charity. How much money is left with him now? Solve it
by making mathematical expression.

Solution

Here, Total Expenditure = ( 6 × 80 ) + ( 2 × 20) + (1 × 25) + 100

? The mathematical expression = 1000 – {( 6 × 80 ) + ( 2 × 20) + (1 × 25) + 100}

= 1000 – (480 + 40 + 25 + 100)

= 1000 – 645 = 355

Hence, Rs 355 is left with him.

EXERCISE 4.4
General Section - Classwork

1. Let's simplify mentally. Then, tell and write the answer as quickly as possible.

a) 3 + 5 – 6 = .......... b) 2 – 7 + 9 = .......... c) 15 – 6 – 4 = ..........

d) 8 + 3 × 4 = .......... e) 7 – 2 × 3 = .......... f) 3 × 6 + 7 = ..........

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g) 2 × 18 ÷ 3 = .......... h) 3 × 42 ÷ 7 = .......... i) 28 ÷ 4 + 3 = ..........

j) 4 + (7 – 2) = .......... k) 3 × (2 + 6) = .......... l) {4+(8 –5)}×3 = ..........

Creative Section - A
2. Let’s simplify.

a) 30 ÷ 3 (9 – 28 ÷ 7) b) 42 – 2 (6 + 4 × 9 ÷ 3)

c) 18 – { 5 – (7 + 8) } d) 25 – { 23 – 2 ( 3 – 7 )}

e) – 40 + 5 { 3 – 2(4 – 8) } f) 6 + 6 { 4 – 2(6 ÷ 3 – 5 )}

g) 13 + [9 – { 6 + (12 – 3)}] h) 2 [{2 – 2(2 ÷ 2 – 2)} ÷ 2 ] ÷ 2

i) 21 ÷ [ 5 + 16 ÷ 2{6 – 18 ÷ (7 + 2)}]

j) 48 ÷ [3 + 15 ÷{4 + 10 ÷ (3 – 13)}]

k) 45 ÷ 3 [ 90÷ 2 {6 + 3(19 – 16)}]

l) 61 – 60 ÷ 2 [35 – 3{– 12 ÷ 6 – 4 (15 ÷ 3 – 8)}]

3. Let's make the mathematical expressions, then simplify.

a) 7 times the sum of 6 and 5

b) 5 times the difference of 6 and 2

c) 4 times the product of 2 and 3

d) 8 is subtracted from 2 times the sum of 7 and 3.

e) 60 is divided by the sum of 6 and 9.

f) 4 times the sum of 8 and 1 is divided by 6.

g) 50 is divided by 2 times one more than the difference of 7 and 3.

h) 4 times 5 less than the sum of 13 and 7 is divided by the difference of 11
and 6.

i) 8 times the difference of 9 and 6 is divided by the product of 2 and 4.

j) The sum of 75 and 45 is divided by 5 and the difference of quotient and 10
is multiplied by 2.

Creative section - B

4. Let's make the mathematical expressions. Simplify them and find the correct
answers.

a) Mr. Dahal earns Rs 8,000 in a week. He spends Rs 450 on food and Rs 125
on transportation everyday. How much money does he save in a week?

b) Sunayana had 28 strawberries. She ate 7 strawberries and she divided the
rest among her 3 friends. How many strawberries does each of them get?

c) There are 27 apples in a basket and mother puts 9 more apples. You eat
6 apples and you divide the rest of them among your 10 friends. How many
apples does each of them get?

d) Bishwant has a 1000-rupee note. He buys 2 packets of crayons at Rs 56 each,
10 pens at Rs 25 each and he donates Rs 251 to a charity. How much money
does he have now?

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5. Diyoshana, Hemant and Rupesh got different answers for this problem:
{–10 – 3 ( – 10 ÷ 2) } – 4.

Diyoshana's answer was 1, Hemant's answer was – 29, and Rupesh's answer was 21.

a) Which student had the correct answer?

b) Show and explain how other two students got their answers. What
errors did they make?

It’s your time - Project work!

6. You have number cards with integers – 3, – 2, –1, 0, + 1, + 2, + 3.

Now, work out the following sums and find the results.
a) two integers having the least sum.
b) two integers having the greatest product.
c) two pairs of integers that have a quotient of – 3.

7. a) Let's use three sevens with proper signs (+, –, ×, ÷) and brackets, and
make four separate expressions to get the result 98, 2, –7, and –6.

b) Let's make any two mixed operations of your own using all four signs
(+, –, ×, ÷) inside middle bracket { } and small bracket ( ).Then, simplify
and get the correct answers.

4.10 Rational numbers – review

Let's recall the following different sets of numbers.

set of natural numbers (N) = { 1, 2, 3, … }

set of whole numbers (W) = {0, 1, 2, 3, …}

set of integers (Z) = {…, – 3, – 2, –1, 0, 1, 2, 3, …}

Now, let's take any two integers (–2) and (+4) and study the following
operations on these integers.

(i) Addition: (– 2) + ( +4) = + 2 which is an integer

(ii) Subtraction: (–2) – (+4) = –2 – 4 = – 6 and

(+4) – (–2) = + 4 + 2 = + 6 which is an integer

(iii) Multiplication: (–2) × (+4) = – 8 which is an integer

(iv) Division: (+4) ÷ (–2) = –2 which is an integer

but, (+2) ÷ (–4) = – 2 = – 1 which is not an integer.
4 2

Thus, when an integer is divided by another integer the quotient is not always

an integer. The fact indicates another set of numbers that can also include such

quotients which are not integers. The set of such numbers is called the set of

Rational numbers.

Any numbers which can be expressed in the form rabat,iownhalerneuamabnerds b are integers
and b ≠ 0, are called rational numbers. The set of is denoted by

the letter ‘Q’.

? Q = {…, – 3, – 5 , – 2, – 3 , – 1, – 1 , 0, 1 , 1, 3 , 2, 5 , 3, …}
2 2 2 2 2 2

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The set of rational numbers is the wider set that includes the sets of natural numbers
(N), whole numbers (W) and integers (Z). So, the sets of natural numbers, whole
numbers and integers are the proper subsets of the set of rational numbers.

? N  Q, W  Q and Z  Q

4.11 Properties of Rational numbers
(i) Closure property

When two rational numbers are added, subtracted or multiplied, the result is always
a rational number.

For example:

1 + 1 = 5 o 5 is also a rational number
2 3 6 6

1 – 1 = – 1 o 112 is also a rational number
3 4 12

1 × 2 = 1 o 31 is also a rational number
2 3 3

Thus, for any two rational number a and c , a + c and a u c are also rational
numbers. b d b d b d

(ii) Commutative property

Two rational numbers can be added or multiplied in any order.

For example,

3 + 1 = 1 + 3 = 5
4 2 2 4 4

5 × 6 = 6 × 5 = 10
9 7 7 9 21
a c
Thus, for any two rational numbers b and d , we have

a + c = c + a and a × c = c × a .
b d d b b d d b

(iii) Associative property

The sum or product of three rational numbers remain unchanged if the order in
which they are grouped is altered.

For example,

1 + 1 + 2 = 1 + 1 + 2 = 17
4 2 3 4 2 3 12

3 × 2 × 1 = 3 × 2 × 1 = 1
4 9 5 4 9 5 30
a c e
Thus, for any three rational numbers b , d and f , we have

a + c + e = a + c + e and a × c × e = a × c × e
b d f b d f b d f b d f

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(iv) Identity property

0 is a rational number such that the sum of any rational number and 0 is the rational
number itself.

For example:

1 + 0 = 0 1 =12 , 0 + 3 = 3 0 = 3
2 2 8 8 8
a a a a
Thus, for every rational number b , b + 0 = 0 + b = b

Here, 0 is called the additive identity for rationals.

On the other hand, 1 is a rational number such that the product of any rational
number and 1 is the rational number itself.

For example:

2 × 1 = 1 u 2 = 2 , 1 × 4 = 4 u 1 = 4
3 3 3 9 9 9
a a a a
Thus, for every rational number b , b × 1 = 1 × b = b

Here , 1 is called the multiplicative identity for rationals.

(v) Inverse property

For every rational number a , there exists a rational number – a such that
a a b b
b + – b = 0.

For example:

1 + – 1 = – 1 1 =0, – 5 + 5 = 5 – 5 =0
4 4 4 4 7 7 7 7
a a
Here, – b is called the additive inverse of b .

On the other hand, for every non-zero rational number a , there exists a rational
b a b b
number a such that b × a = 1.

For example:

2 × 3 = 1, 1 × 4 = 1
3 2 4 1

Here, b is called the multiplicative inverse of a . The multiplicative inverse of a
a b
rational number is also called its reciprocal.

4.12 Terminating and non-terminating rational numbers

When a rational number is decimalised, the decimal may be terminating or
non-terminating decimal. If the decimal is non–terminating, a digit or a block of
digits after the decimal point repeat after certain intervals. Such decimals are called
non-terminating recurring decimals. For example:

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1 = 0.5 (Terminating decimal)
2 (Terminating decimal)
1 (Terminating decimal)
4 = 0.25 (Non-terminating recurring decimal)
(Non-terminating recurring decimal)
3 = 0.375 (Non terminating recurring decimal)
8
13= 0.333…
5
6 = 0.8333…

9 = 1.285714 285714285…
7

The non-terminating recurring decimals can be indicated by putting dots just above
the beginning and the end of repeated digit or block of digits. For example,

1 = 0.333… = 0.•3 (It indicates 3 is the recurring digit.)
3

1 = 0.1666… = 0.1•6 (It indicates 6 is the recurring digit.)
6

5 = 0.41666… = 0.41•6 (It indicates 6 is the recurring digit.)
12

8 = 1.142857142… =1.1•4285•7 (Indicates the block of digits 142857
7
recurring periodically.)

4.13 Irrational numbers

Let's study the following illustrations and try to investigate the idea of irrational
numbers.

4 =2 (2 is a rational number)

9 =3 (3 is a rational number)

5 = 0.3125 (156 is a rational number. The decimal is terminating)
16
3 (131 is a rational number. The decimal is non-terminating
11 = 0. 27272727..

On the other hand, recurring)

2 = 1.414213562… (Decimal is non-terminating non-recurring. So, 2 is not a
rational number. It is an irrational number.)

3 = 1.7320508070… (Decimal is non-terminating non-recurring. So, 3 is not a
rational number. It is an irrational number.)

Thus, the numbers which are not rational are called irrational numbers. 2 , 3 ,
5 , 6 , 7 , etc. are a few examples of irrational numbers.

When irrational numbers are decimalised, the decimals are non-terminating non-
a
recurring. So, irrational numbers cannot be expressed in b form.

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Real Numbers

In this way, all the sets of number system, for example, natural numbers, whole
numbers, integers, rational numbers, and irrational numbers are defined under the
set of Real number system.

Real numbers

Rational numbers Irrational numbers
0, 1, 2, –1, –5, 21, 23, … 2, 3, 5, …

Fractional numbers Integers
12, 23, 14, 57, … –2, –1, 0, 1, 2, 3, …

Terminating Non-terminating Positive integers Negative integers
decimals recurring decimals 0, 1, 2, 3, 4, … –4, –3, –2, –1, …
21, 14, 53, 87, … 31, 56, 67, …

Natural numbers Whole number
1, 2, 3, 4, … 0, 1, 2, 3, 4, …

Worked-out examples

Example 1: Find any two rational numbers between 3 and 4.

Solution:

Rational number at the middle of 3 and 4 = 1 (3 + 4) = 7
2 2

Rational number at the middle of 3 and 7 = 1 (3 + 7 )
2 2 2

= 1 × 13 = 13
2 2 4

Hence, 7 and 13 are any two rational numbers between 3 and 4.
2 4

EXERCISE 4.5
General Section - Classwork
1. Let's identify and write whether these numbers are ‘Rational’ or ‘Irrational.’

(a) 2 ................. (b) 7 ................. (c) 4 .................
(f) 3.•4 .................
(d) 25 ................. (e) – 18 .................

(g) 0.2•8571•4 ................. (h) 5 ................. (i) 2 .................
8 3

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Real Numbers

2. Let's observe the venn-diagram representing the relationship of different
types of number and write ‘True’ or ‘False’ as quickly as possible.

a) Every natural number is a rational number. ................. Q R
b) Every rational number is a natural number. ................. Z Q
c) Every whole number is a rational number. .................
N
W

d) Every rational number is a whole number. .................

e) Every integer is a rational number . .................

f) Every rational number is an integer. .................

g) Every rational number is a real number. .................

h) Every real number is a rational number. .................

i) Set of irrational numbers is a subset of real numbers. .................

j) Set of real numbers is a subset of rational numbers. .................

3. a) If m and p are any two rational numbers, then according to
n q

commutative property of addition ...................................................................

commutative property of multiplication ..........................................................

b) If a , c and e are any three rational numbers, then according to
b d f

associative property of addition ........................................................................

associative property of multiplication ..............................................................

c) The additive inverse of p is ...........................
q

d) The multiplicative inverse of p is ...........................
q

e) The additive identity for a rational number is ...........................

f) The multiplicative identity for a rational number is ...........................

4. Let's tell and write whether the decimals of these fractions are terminating or
non – terminating recurring.

a) 1 .................................... b) 1 ......................................
5 3

c) 3 ....................................... d) 7 .........................................
4 8

e) 5 ...................................... f) 4 ...........................................
9 7

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Real Numbers

Creative Section - A

5. Let's answers the following questions

a) What do you mean by rational numbers? Give an example of rational
number.

b) Write down a rational number whose numerator is the smallest number of
two digits and denominator is the greatest number of three digits.

c) What is the multiplicative inverse of 2 ?
3
d) Write down the additive inverse of − 87.

e) Write the rational numbers whose multiplicative inverses are themselves.

f) Write a rational number which has no multiplicative inverse.

g) What are the additive and multiplicative identities for any rational number?

6. Let's find any two rational numbers between each of the following pairs of
rational numbers.

a) 2 and 3 b) 4 and 5 c) 7 and 8 d) 0.2 and 0.6 e) 0.25 and 0.75

7. Let's express these rational numbers in decimal. State whether they are
terminating or non-terminating recurring. Also represent the non-terminating
recurring decimals by using dots above the digits at appropriate places.

a) 1 (b) 3 (c) 11 (d) 2 (e) 7 (f) 5 (g) 17
2 5 8 3 9 7 13

It's your time - Project work!

8. a) Let's write any five rational numbers with the denominators 2, 4, 5, 8, and
10 such that the numerator in each case is not the multiple of denominator.
Then, identify whether they are terminating or non-terminating recurring
decimals.

b) Let's write any five rational numbers with the denominators 3, 6, 7, 9, and
11 such that the numerator in each case is not the multiple of denominator.
Then, identify whether they are terminating or non-terminating recurring
decimals.

c) The decimals of rational numbers with denominators prime numbers
greater than 2 and numerator is not the multiple of denominator are always
non-terminating recurring. Verify it by four examples.

d) Now, write a short report about the denominators of any rational numbers
that give the terminating decimals and non-terminating recurring decimals.
Then, present in your class.

9. Let's draw a diagram showing all types of real numbers with examples on a chart
paper.

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Unit Fraction and Decimal

5

5.1 Fraction – Looking back
Classroom - Exercise

1. a) Let's identify and list the like and unlike fractions separately.

2 , 2 , 3 , 1 , Like fractions: (i) .............................. (ii) ..............................
3 5 4 3 Unlike fractions: ................................................................

3 , 47, 1 , 5
5 5 6

b) Let's identify and list the proper and improper fractions and mixed numbers
separately.

3 , 8 , 183 , 6 , 5 , Proper fractions: ..........................................
8 3 5 6 Improper fractions: .....................................

221 , 3 , 7 , 331
7 3

Mixed numbers: ..........................................

2. Let's write an equivalent fraction of each of the following fractions.

a) 1 = ........... b) 2 = ........... c) 3 = ........... d) 5 = ...........
2 3 4 6

3. Let's tell and write the sum or difference as quickly as possible.

a) 3 + 2 = ........... b) 4 + 2 = ........... c) 1 + 1 = ...........
7 7 9 9 3

d) 7 – 3 = ........... e) 7 – 2 = ........... f) 1 – 1 = ...........
10 10 8 8 4

4. Let's tell and write the products or quotients as quickly as possible.

a) 2 × 1 = ............... b) 1 × 2 = ............... c) 1 × 6 = ...............
3 2 3 3 7 = ...............

d) 1 ÷ 1 = ............... e) 2 ÷ 1 = ............... f) 1 ÷ 3
4 3 6

5.2 Addition and subtraction of fraction - revision

Case I: While adding or subtracting the like fractions, 2 3
we simply add or subtract the numerators 7 7
leaving the denominator as it is. For example:

Add: 2 + 3 = 2+3 = 5 5
7 7 7 7 7

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Fraction and decimal

Case II: In the case of addition and subtraction of unlike fractions, at first we should
convert them into like fractions with the least common denominator (LCD).
Then we add or subtract their numerators. For example:

Subtract: 3 – 1 3
4 6 4
9
3 – 1 = 3 × 3 – 1×2 12
4 6 4 × 3 6×2

= 9 – 2
12 12

= 9–2 = 7 9 – 2 = 7
12 12 12 12 12

Direct process:

3 – 1 = 3×3–2×1 = 9–2 = 7
4 6 12 12 12

Worked-out examples

Example 1: Simplify 534 – 285 – 1172
Solution
Alternative Process

543 – 258 –1172 543 – 258 –1172

= 23 21 19 = (5 − 2 − 1) + (43 − 5 − 172)
4 8 12 8

= 23 × 6 – 21 × 3 – 19 × 2 = 2 ( 3 u 6 5u3 7 u 2)
24 24

= 138 – 63 – 38 = 2 (18 – 15 – 14)
24 24

= 138 – 101 = 2 11 )
24 24

= 37 = 1 11 )
24 24

= 1 13 = 1 24 – 11 =1 13 = 1 13
24 24 24 24

Example 2: A water tank has two pipes of different sizes to fill it. One pipe
3 1
can fill 10 parts of the tank and another pipe can fill 4 part of the

tank in one hour. If both pipes are opened at once, what parts of

the tank would be filled in one hour? What parts of the tank is left

to be filled if both the pipes are closed after one hour?

Solution of the tank is filled by two pipes in one hour = 3 + 1 = 6+5
Here, parts 10 4 20

= 11 parts
20

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Fraction and decimal

Hence, 11 parts of the tank is filled in one hour.
20
11 20 – 11 9
Again, the remaining parts of the tank = 1 – 20 = 20 = 20
parts of the tank is left
Hence, 9 to be filled.
20

Example 3: On the occasion aotfeA14napmarotl'asnbdirhthisdmayo,thheerataete25 parts of the birthday
cake, his father 3
10 parts of the cake.

(i) What part of the cake was left?

(ii) Who ate the greatest parts of the cake?

Solution 2 1 3
5 4 10
(i) Here, the total parts of the cake eaten by the three persons = + +

= 8+5+6 = 19
20 20

Now, the remaining parts of the cake = 1 – 19 = 20 – 19 = 1
20 20 20

(ii) Again, the L.C.D. of 5, 4 and 10 is 20.

Then, 2 = 2×4 = 8 , 1 = 1× 5 = 5 and 3 = 3×2 = 6
5 5×4 20 4 4× 5 20 10 10 × 2 20

Between 8 , 5 and 6 , the greatest fraction is 280.
20 20 20

Hence, Anamol ate the greatest part of the cake.

EXERCISE 5.1

General Section -Classwork

1. Let's insert the appropriate sign ‘ >’ or ‘ < ‘ in the blanks.

(a) 3 ........... 2 (b) 3 ........... 5
5 5 7 7
We can mentally convert each unlike
(c) 10 ........... 13 (d) 9 ........... 7 fractions into one decimals place and we
17 17 20 20 can compare decimals.

(e) 1 ........... 2 (f) 3 ........... 5 25
2 3 4 6 5 = 0.4, = 0.8...
6 5 2
(g) 54........... 5 3 1
8 (h) 10 ........... 5 0.8... > 0.4 .... So, 6 > 5

2. Let's investigate the idea of quick calculation from given illustrations. Then,
tell and write the sums or the differences as quickly as possible.

1 + 1 + = 3 ( 2 + 1 ) Interesting! When a fraction is added to 1
2 2 2 or subtracted from 1, I should simply add
the numerator and denominator of the
1 + 3 + = 10 ( 7 + 3 ) fraction, or numerator is subtracted from
7 5 7 the denominator of the fraction.

1 – 3 – = 2 ( 5–3 )
5 5 5

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Fraction and decimal

a) 1 + 1 = .......... b) 1 + 3 = .......... c) 1 + 2 = .......... d) 1 + 4 = ..........
3 4 5 9
1 3 3 11
e) 1 – 4 = .......... f) 1 – 8 = ........... g) 1 – 11 = ........... h) 1 – 15 = ..........

Creative section - A

3. Let's add or subtract.

a) 2 + 3 b) 121 + 2 3 c) 5 – 4 d) 354 – 132
3 4 4 6 7

4. Let's simplify.

a) 1 + 1 + 1 b) 3 + 2 – 1 c) 5 – 7 + 11 d) 3 – 1 – 5
3 4 6 4 5 10 6 8 12 4 6 8

e) 113 + 214 – 3 5 f) 212 – 313 +441 g) 323 – 265 – 189 h) 456 – 272 – 1 3
12 14

i) 2+ 3 – 5 j) 125 – 2 3 + 3 k) 519 – 2 5 – 1 l) 9 – 183 – 116
4 8 10 18

5. a) Bhurashi did 8 out of 10 math problems and Bishwant did 11 out of 15
similar math problems. Express the number of problems solved by each of
them in fractions and identify who did better performance.

b) A pipe fill 7 parts of a tank in 1 hour and another pipe can fill 9 parts of
10 15
the same tank in 1 hour. Which pipe can fill the tank faster?

6. a) Mrs. Mahato bought 414 kg of potatoes and 212 kg of tomatoes from a vendor.

Find the total weight of vegetables bought by her.
b) Kohalpur is 120110 km west from Lamahi and Attariya is 15534 km west from

Kohalpur. Find the distance between Lamahi and Attariya.
7. a) A picture is 735 cm wide. By how much should it be trimmed to fit in a frame
7130
b) The cm wide? Pinky is 3043 kg and the weight of Dolma is 3014 kg. Who has
weight of

more weight and by how much?
c) A movie file is of 221 GB. After downloading 143 GB of the file, the internet
stopped functioning. What size of the file was left to be downloaded?

8. a) A water tank has two pipes of different sizes to fill the tank. One pipe can fill
2 3
5 parts of the tank and another pipe can fill 8 parts of the tank in 1 hour.

(i) What parts of the tank would be filled in 1 hour if both the pipes are

opened at once?

(ii) What parts of the tank would be left to be filled if both the pipes are

closed after 1 hour?

(iii) Which pipe would fill the tank faster?

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Fraction and decimal

b) On the occasion of Pratik's birthday, Pratik ate 3 parts of a cake, his mother
ate 1 1 10 parts to his brother
4 , father ate 5 parts, and he gave the remaining

Bishwant.

(i) What parts of the cake did Pratik give to Bishwant?

(ii) Who ate the greatest parts of the cake?

c) Mr. Gurung gave 1 part of his property to his daughter, 3 parts to his son and
2 8
he donated the remaining parts to a charity.

(i) What parts of his property did Mr. Gurung donate to charity?

(ii) Between the daughter and the son, who got greater parts of the property?

5.3 Multiplication of fractions

Case I: Multiplying a fraction by a whole number.

Let’s multiply 3 × 14. It means 3 times 1 . 3 × 1 means 1 1 1
4 4 4 4 4

3 × 1 = 3×1 = 3
4 1×4 4

1 1 1 3
4 4 4 4
Case II: Multiplying a whole number by a fraction

Let's multiply 1 × 6. It means one-third of 6.
3

1 × 6 = 1×6 = 6 = 2 2
3 3×1 3

Case III: Multiplying a fraction by a fraction

Let's multiply 1 u 3 . It means half of 3 .
2 4 4

Here, 1 u 3 = 1×3 = 3 1 of 3 = 3
2 4 2×4 8 2 4 8

5.4 Division of fractions

Case I: Dividing a whole number by a fraction.

Let’s divide, 3 ÷ 1 . It means how many 111
2 222
halves are there in 3? 111
222
Now, let’s think how can we get 3 ÷ 1 = 6? 2 halves + 2 halves + 2 halves = 6 halves
2

It must be 3 ÷ 1 = 3 × 2 = 6 = 6.
2 1 1

Thus, to divide a whole number by a fraction, we should multiply the whole number
by the reciprocal of the fraction.

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Fraction and decimal

Case II: Dividing a fraction by a whole number.

Let’s divide, 3 ÷ 2. It means divide 3 parts of a bread equally between 2 persons.
4 4
3 3
4 ÷ 2 = = Each person will share 8 parts of the bread.

Now, let's think how we can get 3 from 3 ÷ 2? It must be 3 × 1 = 38.
8 4 4 2

Case III: Dividing a fraction by another fraction.

Let’s divide: 1 ÷ 1 . It means how many one-quarter there are in 12.
2 4

1 ÷ 1 = 11 = 2 number of one-quarters.
2 4 44

1
2

1 ÷ 1 = 1 × 4 = 2 Similarly, 3 ÷ 9 =34 × 16 = 4 = 131
2 4 2 1 4 16 9 3

Worked-out examples

Example 1: Simplify a) 121 241 7 9 1
8 10 1
y u 1 b) 27 c) 1+ 1+ 1
1+
35 1
2
Solution:

a) 112 y 214 u 187 = 3 y 9 u 15 The reciprocal of 9 is 4
2 4 8 4 9

= 3211× 4923×11 155 = 5 = 141
84 4

b) 9 = 9 ÷ 27 = 9 u 35 = 7 =116 The reciprocal of 27 is 35
10 10 35 10 27 6 35 27
27
35

1 = 1 = 1 = 1 1 = 1
1 1+ 1 +
C) 1+ 1 1 1+ 1+2 1 1 1 + 1 2 1+ 1
+ 1+ + 1+ 3 3 3
1 1 3
1
2 1 22 1 5
1 8 8
= 1 = 3 = 1 = =
1+ 5 5 5
1+ 5 5

3 4
9
Example 2: Find the value of of Rs 450.

Solution:

4 of Rs 450 = 4 1 × Rs 45500 = Rs 200.
9 9

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Fraction and decimal

Example 3: Simplify 7 ÷ 1 of 1 + 3
8 6 2 4

Solution:

7 ÷ 1 of 1 + 3 = 7 ÷ 1 of 2+3 = 7 ÷ 1 of 5 234
8 6 2 4 8 6 4 8 6 4 5
1 × 5
= 7 ÷ 6 × 4 = 7 y 5 =187 × = 21 = 415
8 8 24 5

Example 4 : Mrs. Magar earns Rs 36,000 in a month. She spends 1 part of money
3
on her children's education, 1 part on food, and 1 part on rent. If
4 6
she deposits the rest of the money in a bank, how much does she

deposit in a month?

Solution:

Here,

Mrs. Magar’s monthly earning = Rs 36,000

Expenditure on education = 1 of Rs 36,000
Expenditure on food 3
Expenditure on rent 1
= 3 × Rs 36,000 = Rs 12,000

= 1 of Rs 36,000
4
1
= 4 × Rs 36,000 = Rs 9,000

= 1 of Rs 36,000
6
1
= 6 × Rs 36,000 = Rs 6,000

Total expenditure in a month = Rs. 12,000 + Rs 9,000 + Rs 6,000 = Rs 27,000

? Total saving in a month = Rs 36,000 – Rs 27,000 = Rs 9,000
Hence, she saves Rs 9,000 in a month.

Example 5: If 4 part of a sum is Rs 300, what is its 2 parts?
5 3
Solution:

Let the required sum be Rs x. Alternative process:
4
Now, 5 of x = Rs 300 4 parts of the sum = Rs 300
5
or, 4 ×x = Rs 300
or, 5 4x = 5 × Rs 300 Whole (1) sum = Rs 300 = Rs 300 × 5
4 4

or, x = 5 × R4s130075= Rs 375 5
Hence, the required sum is Rs 375. = Rs 375

2 2 parts of the sum = 2 of Rs 375
3 3 3

Again, part of Rs 375 = 2 × Rs 375 = Rs 250
3
2 317255
= 3 u Rs = 2 × Rs 125 = Rs 250

Thus, the required sum is Rs 375 and its 2 part is Rs 250.
3

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Fraction and decimal

Example 6: A computer can download 2 parts of a game file in 1 minute. After
5
downloading the file for 45 seconds, the internet was interrupted

and after a while it again started downloading. If the size of the

file was 2 GB, what was the size of the file downloaded before and

after the interruption of the internet?

Solution: 2
5
Here, in 1 minute, i.e. in 60 seconds, parts of the file can be downloaded.
2
62550×pa61r0tspoafrtths e=fi1le150capnarbtes
In 1 second, downloaded. be downloaded.
In 1 second, of the file can

In 45 second, 1 × 45 parts = 3 parts of the file can be downloaded.
150 10
3 7
Now, parts of the file downloaded after internet interruption = 1 – 10 = 10 parts.

Again, 3 parts of 2 GB = 3 × 2000 MB = 600 MB
10 10
7 7
10 parts of 2 GB = 10 × 2000 MB = 1400 MB

Hence, 600 MB of the file was downloaded before and 1400 MB was download after

the internet interruption.

Example 7: Simplify 3 y 4 5 ÷ 531 3 5 1
8 6 4 6 8
Solution:

Here, 3 y 4 5 ÷ 513 3 5 1
8 6 4 6 8

= 3 y 4 5 ÷ 16 18 20 3
8 6 3 24

= 3 y 4 5 ÷ 16 of 1
8 6 3 24

= 3 y 4 5 ÷ 126 u 1
8 6 3 u 243

= 3 y 4 5 y 29
8 6
652u 293
= 3 y 4 = 3 y 4 145 = 3 y 16 15= 3 y 1 = 3 u 411= 3 = 121
8 8 8 4 8 4 8 2
2

EXERCISE 5.2
General Section - Classwork

1. Let's tell and write the reciprocal of these numbers as quickly as possible.
1 5
a) 5 ............ b) 4 ............ c) 7 ............ d) 10 ............
9

2. Let's tell and write the correct answers as quickly as possible.
1 1 3 1
a) 3 × 5 = ............. b) 3 × 2 = ............. c) 4 × 2 = .............

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Fraction and decimal

d) 1 × 2 = ............. e) 1 × 6 = ............. f) 1 × 10 = .............
2 3 3 7 5 13

g) 2 ÷ 1 = ............. h) 3 ÷ 1 = ............. i) 1 ÷ 2 = .............
2 3 2

j) 1 ÷ 3 = ............. k) 1 ÷ 1 = ............. l) 1 ÷ 1 = .............
3 3 2 6 9

m) 1 of Rs 50 = ............. n) 1 of 30 m = ............. o) 1 of 120 kg = ..............
2 3 4

Creative section - A

3. Let's simplify.

a) 16 × 3 b) 5 × 9 c) 5 × 8 d) 2 × 5 × 9
4 6 6 15 3 6 10

e) 4 × 374 × 1156 f) 6 × 159 × 338 g) 3 ÷ 1 h) 4 ÷ 8
5 7 4 5

i) 8 ÷ 4 j) 347 ÷ 1114 k) 432 ÷ 321 l) 3 × 16 ÷ 9
15 5 4 25 15

m) 3 ÷ 9 × 343 n) 173 ÷ 241 × 285
5 10

4. Let's find the values.

a) 1 of Rs 260 b) 2 of Rs 720 c) 3 of 2m (in cm)
4 5 8

d) 3 of 1 km (in m) e) 3 of 1kg (in g) f) 4 of 540 students
10 4 9

5. 30% of 40 means 30 of 40 = 3 of 40 = 12. Apply the similar process and
100 10
evaluate the following.

a) 10% of Rs 50 b) 20% of Rs 350 c) 25% of 200 students

d) 16% of 1 m (in cm) e) 45% of 1 km (in m) f) 8% of 1l (in ml )

6. Let's simplify. 2 1 4 18 8
6
9 2 15 25 18
a) 3 b) 10 c) 3 d) 10 e) 3 f) 10 g) 12 h) 51
6 4 4
5 5 4 21 35 81

i) 1 j) 1 1 k) 1 1
+1 1 1 1
1 1+ 1 1 1 + +1
2 2 1+
1
3

7. Let's investigate the idea from the given illustration and simplify the following
problems.

15 ÷ 1 of 6 = 15 ÷ 1 × 6 3 = 15 ÷ 3 = 15 5 71 = 5 = 141
28 2 7 28 2 × 7 28 7 28 31 4
×
1
4

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Fraction and decimal

a) 3 ÷ 1 of 3 b) 6 of 5 ÷ 10 c) 1 + 1 of 2
4 8 5 7 9 21 2 4 3
2 1 8 4 1 4 1 1 5 3
d) 3 – 2 of 9 e) 5 + 2 of 5 – 3 f) 2 of 3 + 6 8
10

8. Let’s simplify.

a) 11 5 3 – 1 b) 5 2 + 3 – 1
12 8 4 2 6 7 14 4

c) 3 4 32 16 y 4 1 d) 1 7 54 5 1 y 3 y 9
10 15 25 5 10 6 10 6 2 8 20

e) 159 y 231 y 3 – 1 2 – 1 1 f) 9 y 118 9 1 – 4 423 – 11
4 2 3 6 8 16 17 4 9 12

9. a) Mr. Limbu earns Rs 28,000 in a month. He spends 4 parts of his earning to
7
run his family and he saves the rest in a bank every month.

(i) How much money does he spend in a month ?

(ii) How much money does he save in the bank in a month?

b) The monthly income of Mr. Gupta is Rs 32,500. He earns 3 parts of his
income from 3 parts from fishery and the 10
5
poultry, rest from vegetables

farming.

(i) Find his total earning from poultry and fishery.

(ii) How much does he earn from vegetables ?

(iii) If he spends 2 parts of his earning to run the family every month, how
5

much does he save in a month?

10. a) If 2 parts of a sum is Rs 420, find the sum and 3 parts of the sum.
3 5
3
b) There are 18 girls in a class. If this number is 7 parts of the total number of

students in the class, find the total number of students in the class.
3
c) Bipina has some money. When she spends 5 parts of her money on her

birthday, Rs 1500 is left with her. How much money does she have in the
beginning ?
7
d) Mr. Magar spends 10 parts of his income every month which amounts to
Rs 28000.

(i) Find his income every month. (ii) How much does he save in a year?

e) During the rapid diagnostic test for coronavirus among some people in an
3
urban area, 900 of them were found negative result and it was 4 of the

number of tested people. Rest of the people were found to be suspected
of COVID-19 and their throat swab was tested. From the tested swab,

3 of them were found infected. (i) Find the tested number of people
20

(ii) Find the number of suspected people

(iii) Find the number of infected people of COVID-19.

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Fraction and decimal

Creative section - B

11. a) A garden is in the shape of a square of length 3525 m. Find its (i)perimeter
b) and (ii) area.

A rectangular ground is 3041 m long and 2012 m broad. Find the perimeter
and area of the ground.

12. a) A car travels 3114 km in 1 hour. How many kilometres does he travel in 351
b) hours ?
10 jars each with a capacity of 512 litres are required to fill a drum completely.

Find the capacity of the drum in litres.

c) 18 litres of milk is required to distribute equally among all the students of

d) a hostel. If each student gets 3 litre of milk, find the number of students in
8
e)
f) the hostel. 2041 3
13. a) 4
b) A jar contains litres of water. A small jug has a capacity of litre. How
c)
d) many times the jug has to be filled with water from the jar to get the jar
emptied?

Mrs. Thapa is a theearchscehr.oSohl ewchaicnhwisal4k212k14mkmforinfroomnehheoruhro. mHeo?w long does
she take to go to

It takes 231 m c9l13otmh to make a shirt. How many shirts can be made from a
piece of cloth long?
Dipak bought a book and read 243 hours everyday. If he read the entire book
8
in 11 weeks, how many hours in all did he require?

The weight of an object on moon is 1 of its weight on the Earth. If weight of
6
is 353 kg, what is its weight on moon?
an object on the Earth If a girl needs 2 pieces of ribbon each of 5 m long, to
A ribbon is 15m long. 6

how many girls can the ribbon be distributed equally?

A patient is given 521 ml of medicine three times a day. How long does a
bottle of 132 ml of medicine last?

14. a) A water tank has two pipes of different sizes to fill water. One pipe can fill
1 1
5 part of the tank in 1 hour and another pipe can fill 4 part in 1 hour.

(i) If both the pipes are opened at a time, what parts of the tank would
be filled in 2 hours?

(ii) If the capacity of the tank is 10,000 litres, how much water would be
filled in 2 hours?

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Fraction and decimal

b) A computer can download 3 parts of a movie file in 1 minute. After
10

downloading the file for 50 seconds, the internet was interrupted and after a
while, it again started downloading. If the size of the movie file was 2.4 GB,
what were the sizes of the file downloaded before and after the interruption
of the internet?

It’s your time - Project work!

15. a) Let’s write any two pairs of like fractions. Then, show the addition and
b) subtraction of each pair by shading the diagrams.

Let’s find the sum of (i) 1 + 1 (ii) 1 + 1 (iii) 1 + 1 (iv) 1 + 1
2 3 2 4 3 4 3 6
Show the sums by shading diagrams.

16. a) Let’s multiply a proper fraction by a whole number and show the product
b) by shading a diagram.

c) Let’s multiply a proper fraction by another proper fraction and show the
d) product by shading a diagram.
e)
Let’s divide a whole number by 1 , 1 and 14. Then show the result by drawing
diagrams. 2 3

Let’s draw 12 circles and colour 3 of the circles.
4
2 3
Let’s draw 20 circles and colour 5 of them with green and 10 of them with

blue.

5.5 Decimal - revision
Let's study the illustrations given below and learn about the decimals.

3 = 0.3 are coloured blocks 4 = 0.04 are coloured blocks
10 100

5 = 0.5 are coloured blocks 12 = 0.12 are coloured blocks
10 100

7 = 0.7 are coloured blocks
10
3 5
In this way, 10 , 10 , 4 , 12 , 75 , etc. are the fractions having denominators
100 100 1000

10 or power of 10 are called decimal fractions and 0.3, 0.5, 0.04, 0.12, and 0.075 are

the decimal numbers.

Here, 3 = 0.3 (It is read as zero point three.)
10
4
100 = 0.04 (It is read as zero point zero four.)

75 = 0.075 (It is read as zero point zero seven five.)
1000

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Fraction and decimal

5.6 Terminating and non-terminating recurring decimal

Terminating decimals

When a fraction is expressed in decimal and the decimal part is terminated (ended)
with a certain number of digits, it is called a terminating decimal. For example,

1 = 0.5 (The decimal part is terminated with one digit. It is a terminating
2 decimal.)

1 = 0.25 (The decimal part is terminated with two digits. It is a terminating
4 decimal.)

Non-terminating decimals

When a fraction is expressed in decimal and the decimal part is never terminated
(ended), it is called a non-terminating decimal. In such decimals, a digit or a block of
digits in the decimal part repeats periodically. So, they are called non-terminating
recurring decimals. for example:

1 = 0.333… (The decimal part is never terminated. It is a non-terminating
3 decimal.)

1 = 0.166… (The decimal part is never terminated. It is a non-terminating
6 decimal.)

3 = 0.428571… (The decimal part is never terminated. It is a non-terminating
7
decimal.)

The non-terminating recurring decimals are indicated by putting a dot just above the

repeated digit. In the case of repeated block of digits, dots are put just the beginning

and end of repeated digits. For example,

1 = 0.333… = 0.•3 (It indicates 3 as the recurring digit.)
3
11 = 1.5714285714… = 1.•57142•8 (It indicates the block of digits 571428 as recurring)
7
8 =0.7•2•
11 = 0.727272... (It indicates the block of digits 72 as recurring)

Worked-out examples

Example 1: Express the following fractions in decimals. State whether the
Solution:
decimals are terminating or non-terminating recurring. Indicate

the recurring digits by using dots.

a) 3 b) 7 c) 5 d) 11 e) 10
8 32 6 7 33

a) 3 = 0.375 It is a terminating decimal.
8 It is a terminating decimal.
7
b) 32 = 0.21875

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Fraction and decimal

c) 5 = 0.8333… = 0.8•3 It is a non-terminating recurring decimal.
6 It is a non-terminating recurring decimal.
11 It is a non-terminating recurring decimal.
d) 7 = 1.57142857… = 1. •57142•8

e) 10 = 0.303030… = 0.•3•0
33

Example 2: Multiply the following non-terminating recurring decimals by 10,

100 and 1000.
a) 0.3• b) 0.1•6 c) 0.•5•4

Solution: = 0.33• × 10 = 3.•3 Decimal point is shifted one digit to the right
a) 0.•3 × 10
= 0.333• × 100 = 33.3• Decimal point is shifted two digits to the right
0.•3 × 100 = 0.333•3 × 1000 = 333.3• Decimal point is shifted three digits to the right
0.•3 × 1000

b) 0.1•6 × 10 = 0.16• = 1.6•
0.1•6 × 100 = 0.16•6 × 100 = 16.•6
0.1•6 × 1000 = 0.1666• × 1000 = 166.6•

c) 0.5•4• × 10 = 0.54•54• × 10 = 5.4•5•4
0.5•4• × 100
0.5•4• × 1000 = 0.54•54• × 100 = 54.•5•4

= 0.5454•54• × 1000 = 545.4•5•4

Example 3: Express the following decimals in fractions.

a) 0.24 b) 0.5• c) 0.•27•

Solution:

a) 0.24 = 0.24 × 100 = 24 = 6 c) Let 0.2•7• = x ............. (i)
100 100 25 Multiplying (i) by 100. we get
0.2•7• × 100 = 100 x
b) Let 0.•5= x ............. (i)
or, 0.272•7• u 100 = 100 x
Multiplying both sides by 10, or, 27.•2•7 = 100 x .................. (ii)
0.•5 × 10 = 10x
Now, subtracting (i) from (ii), we get
or,0.5•5 × 10 = 10 x 27.•2•7 = 100x
or, 5.•5 = 10x .................. (ii) – 0.2•7• = – x

Now, subtracting (i) from (ii), we get 27 = 99x
5.•5 = 10x 27
99
– 0.•5 = – x

5 = 9x or, x =

0.•5 5 ?x =95 ? x = 3
9 11
Thus, =
Thus, 0.2•7• = 3
11

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Fraction and decimal

EXERCISE 5.3

General Section - Classwork

1. Let’s observe the denominator of each fraction. Identify and list the non-
terminating recurring decimals without actual division..
1 5 1 2 3 1 5 4 3
a) 2 , 2 , 3 , 7 , 4 , 5 , 6 , 9 , 8 , 3 , 5 , 4 , 5 , 7 , 3
25 16 15 14 20 40

Fractions that give terminating Fractions that give non-terminating
decimals recurring decimals

2. Let's tell and write the correct product as quickly as possible.
a) 0. •4 × 10 = ............ 0.•4 × 100 = ............ 0.•4 × 1000 = ............

b) 0.1•8 × 10 = ............ 0.1•8 × 100 = ............ 0.1•8 × 1000 = ............

c) 0.7••2 × 10 = ............ 0.7••2 × 100 = ............ 0.•7•2 × 1000 = ............

d) 0.•21•6 × 10 = ............ 0.•216• × 100 = ............ 0.•21•6 × 1000 = ............
Creative section

3. Answer the following questions.
a) Write the meanings of 0.5, 0.05 and 0.005. Between these decimals, which
is the greatest one?
b) Do the decimals 0.4 and 0.40 have the same meaning? In what way are they
same and in what way are they different?
c) What are the place names and place values of 7 and 9 in 0.749?

4. Express the following fractions in decimals. State whether the decimals are

terminating or non-terminating recurring. Indicate the recurring digits by

using dots.

a) 3 b) 2 c) 6 d) 5 e) 9 f) 5 g) 4 h) 8
2 3 5 6 5 9 7 11

5. a) Let's write the following decimal numbers in ascending order.

(i) 0.02, 0.2, 0.002, 0.0002 (ii) 0.25, 0.05, 0.2, 0.025

b) Let's write the following decimal numbers in descending order.

(i) 0.003, 0.3, 0.03, 0.0003 (ii) 0.54, 0.4, 0.5, 0.054

6. Multiply the following non-terminating recurring decimals by 10, 100 and

1000. b) 0.•5 c) 0.1•6 d) 0.208•3 e) 0.•8•1 f) 0.4•86•
a) 0.•3

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Fraction and decimal

7. Express the following decimals in fractions.

a) 0.3 b) 0.7 c) 0.16 d) 0.05 e) 0.125 f) 0.008
k) •21•6 l) •324•
g) 0.•4 h) 0.•7 i) 0.•5•4 j) 0.•1•2

5.7 Four fundamental operations on decimals

(i) Addition and subtraction of decimals

While adding or subtracting decimals, we should arrange them in such a way
that the digits at the same places should lie in the same column.

Worked-out examples

Example 1: Add a) 0.4 + 0.005 b) 3.6 + 2.75
Solution:

a) 0.4 0.400 b) 3.6 3.60
+ 0.005 + 0.005 + 2.75 + 2.75

0.405 6.35

Example 2: Subtract a) 0.01 – 0.008 b) 18.79 – 7.843
Solution:

a) 0.01 0.010 b) 18.79 18.790
– 0.008 + 0.008 – 7.643 – 7.843
10.947
0.002

EXERCISE 5.4

General Section - Classwork

1. Let's add, then tell and write the answers as quickly as possible.

a) 0.4 + 0.5 = ............ b) 0.6 + 0.2 = ............ c) 0.9 + 0.6 = ............

d) 0.2 + 0.45 = ............ e) 0.36 + 0.4 = ............ f) 0.05 + 0.4 = ............

g) 0.07 + 0.02 = ............ h) 0.04 + 0.005 = ............ i) 0.005 + 0.2 = ............

2. Let's subtract, then tell and write the answers as quickly as possible.

a) 0.4 – 0.2 = ............ b) 0.9 – 0.4 = ............ c) 1.5 – 0.8 = ............

d) 0.6 – 0.04 = ............ e) 0.8 – 0.07 = ............ f) 0.6 – 0.09 = ............

g) 0.8 – 0.008 = ............ h) 0.02 – 0.005 = ............ i) 0.04 – 0.008 = ...........

Creative Section - A

3. Let’s find the sum or difference of the following decimal numbers.

a) 0.142 + 0.816 b) 0.975 + 0.589 c) 0.2 + 0.846

d) 5.36 + 9.4 e) 134.7 + 2.635 f) 0.246 – 0.124

g) 10.025 – 4.85 h) 5.55 – 0.555 (i) 35.04 – 9.365

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Fraction and decimal

4. Let’s simplify. b) 0.58 – 0.74 + 0.2 c) 400 – 225.8 – 10.86
a) 0.36 + 0.85 –0.9 e) 15.04 – (2.5 + 3.8) f) 99.9 – (9.99 – 0.999)
d) (0.65 – 0.8) + 1.5

5. a) Subtract 6.9 from the sum of 4.8 and 8.4.
b) Subtract 0.567 from the sum of 2.3 and 3.45.
c) Add 13.95 and 1.002, then subtract the result from the sum of 8.04 and 25.6.
d) Add 6.85 and 0.685, then subtract the result from the sum of 8.25 and 0.825.

Creative Section - B

6. Let's read the price list of different items Menu

displayed in a restaurant menu. Then solve the Veg. Mo:Mo 150.50
given problems.
Chicken Mo:Mo 210.75

a) Calculate the total cost of a plate of Chow Mein 140.80

veg. Mo:Mo and a cold drink. French fries 75.25

b) Find the total cost of a plate of chicken Cold drink 40.50

Mo:Mo and a plate of French fries.

c) You and your friend had two plates of chow mein and two bottles of cold
drink. If you gave a Rs 500 note to the waiter to clear the bill, how much
change did the waiter return you?

d) If you have Rs 300 pocket money, what two items do you choose to have in
the restaurant? What is the total cost of these two items? How much money
was left with you after paying the bill?

7. a) A side of an equilateral triangle is 4.6 cm, find its perimeter.

b) One of the two equal sides of an isosceles triangle is 5.3 cm and the
remaining side is 6.8 cm. Find the perimeter of the triangle.

c) The sides of a scalene triangle are 4.5 cm, 4.7 cm, and 5.9 cm respectively.
Find its perimeter.

8. The exchange rates of the currencies of a few countries on a certain day are
given below. Answer the following questions.

US $ 1 = Rs 115.65, 1 Euro = Rs 132.80 1 Australian dollar = Rs 78.96

a) By how much is the Euro more expensive than US dollar?

b) By how much is the Australian dollar cheaper than US dollar?

c) If you exchange 1 US dollar, 1 Euro and 1 Australian dollar in Nepali
currency, how much Nepali currency do you get in total?

(ii) Multiplication of decimals

Study the following examples and learn to multiply decimal numbers by any
other numbers.

Worked-out examples

Example 1: Multiply: a) 2.59 by 6 b) 4.216 by 5.4

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Fraction and decimal

a) 2.59 Multiplicand has two decimal places.
×6 Product has two decimal places.

15.54

b) 4.216 Multiplicand and multiplier have four decimal places in total.
× 5.4 ? The product has four decimal places.
16864

210800
22.7664

In the case of multiplying a decimal number by 10 or the power of 10, we should
simply shift the decimal point to the right of as many number of digits as there are
number of zeros in 10, 100, 1000, etc. For example,

Example 2: Multiply 7.2564 by 10, 100 and 1000.

7.2564 × 10 = 72.564 10 has one zero. So, the decimal point is shifted one digit to
the right.

7.2564 × 100 = 725.64 100 has two zeros. So, the decimal point is shifted two digits to
the right.

7.2564 × 1000 = 7256.4 1000 has three zeros. So, the decimal point is shifted three digits
to the right.

(iii) Division of decimals

While dividing a decimal number by another decimal number, we should first
eliminate the decimal point from the divisor multiplying it by some power of 10.
In the same time, the dividend should also be multiplied by the same power of 10.
Then, we should proceed the division.

Example 3: Divide: 3.888 by 0.08 In 0.08, there are two decimal places. So, to
Multiplying 0.08 by 100 0.08 × 100 = 8 eliminate decimal, it is multiplied by 100.

Multiplying 3.888 by 100 3.888 × 100 = 388.8

Now, dividing 388.8 by 8. 8 ) 38 (4 8 ) 388 (48 8 ) 388.8 ( 48.6
–32 –32 –32
8 ) 388.8 ( 48.6 68
–32 6 68 – 64
68 48
– 64 – 64 – 48
48 0
– 48 4
0

∴ 3.888 ÷ 0.08 = 48.6

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 95 Vedanta Excel in Mathematics - Book 7

Fraction and decimal

In the case of division of a decimal number by 10 or the power of 10, we should
simply shift the decimal point to the left of as many numbers of digits as there are
numbers of zeros in 10 or power of 10.

Example 4: Divide: 72.49 by 10, 100, and 1000.

72.49 ÷ 10 = 7.249 10 has one zero. So, the decimal point
is shifted one digit to the left.

72.49 ÷ 100 = 0.7249 100 has two zeros. So, the decimal
point is shifted two digits to the left.

72.49 ÷ 1000 = 072.49÷ 1000 = 0.07249 1000 has three zeros. So, the decimal
= 0.07249 point is shifted three digits to the left.

EXERCISE 5.5
General Section - Classwork
1. Let's tell and write the products as quickly as possible.

a) 1.728 × 10 = ............ 1.728 × 100 = ............ 1.728 × 1000 = ............
b) 3.005 × 10 = ............ 3.005 × 100 = ............ 3.005 × 1000 = ............
c) 0.923 × 10 = ............ 0.923 × 100 = ............ 0.923 × 1000 = ............
d) 0.0006 × 10 = ............ 0.0006 × 100 = ............ 0.0006 × 1000 = ............

2. Let's tell and write the quotients as quickly as possible.
a) 231.5 ÷ 10 = ............ 231.5 ÷ 100 = ............ 231.5 ÷ 1000= ............
b) 538 ÷ 10 = ............ 538 ÷ 100 = ............ 538 ÷ 1000 = ............
c) 86 ÷ 10 = ............ 86 ÷ 100 = ............ 86 ÷ 1000 = ............
d) 0.7 ÷ 10 = ............ 0.7 ÷ 100 = ............ 0.7 ÷ 1000 = ............

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Fraction and decimal
3. Let's tell and write the products or quotients as quickly as possible.

a) 0.3 × 2 = .............. 0.3 × 0.2 = .............. 0.3 × 0.02 = ..............

b) 8 × 0.3 = .............. 0.8 × 0.3 = .............. 0.08 × 0.03 = ..............

c) 0.9 ÷ 3 = .............. 0.9 ÷ 0.3 = .............. 0.9 ÷ 0.03 = ..............

d) 2.5 ÷ 5 = .............. , 2.5 ÷ 0.5 = .............. , 0.25 ÷ 5 = ..............

Creative Section - A

-a

4. Let's find the products or quotients.

a) 1.25 × 12 b) 2.36 × 1.4 c) 6.78 × 0.27 d) 0.93 × 0.05

e) 24.92 ÷ 7 f) 525.6 ÷ 12 g) 23.04 ÷ 2.4 h) 0.884 ÷ 0.26

5. Let's simplify.

a) 2.7 × 5 + 3.75 b) 3.8 × 1.2 – 2.63 c) 14.08 – 2.5 × 0.75

d) 3.08 × 1.2 + 0.7 × 0.4 e) 4.8 ÷ 0.24 × 0.35 f) (2.29 + 1.07) ÷ 1.4

g) 0.96 ÷ (1.05 – 1.17 ) h) (0.85 + 0.59) ÷ (2.59 – 0.99)

6. 1 cm = 10 mm

a) How many millimetres are there in i) 4.5cm ii) 9.3 cm iii) 0.4cm?

b) How many centimetres are there in i) 2mm ii) 7mm iii) 18 mm?

7. 1 m = 100 cm

a) How many centimetres are there in i) 1.2 m ii) 3.6 m iii) 0.5 m?

b) How many metres are there in i) 7 cm ii) 72 cm iii) 180 cm?

8. 1 km = 1000 m

a) How many metres are there in (i) 1.6 km (ii) 4.32 km (iii) 0.99 km?

b) How many kilometres are there in (i) 4 m (ii) 45 m (iii) 840 m?

9. 1 kg = 1000 g

a) How many grams are there in i) 1.8 kg ii) 7.35 kg iii) 0.05 kg iv) 0.48 kg?

b) How many kilograms are there in (i) 50 g (ii) 200 g (iii) 1260 g (iv) 2575 g?

10. 1 l = 1000 ml

a) How many millilitres are there in i) 2.3 l ii) 4.55 l iii) 0.075 l iv) 0.125 l?
b) How many litres are there in i) 5 ml ii) 20 ml iii) 350 ml iv) 1590 ml?

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Fraction and decimal

11. a) Find the cost of 10.5 kg of rice if the rate of cost of rice is Rs 110.75 per kg.

b) If the cost of 7.5 kg of sugar is Rs 566.25, find the rate of cost of sugar.

Creative Section - B

12. Area of rectangle = l × b, area of square = l2 = l × l, perimeter of rectangle
= 2(l + b) and perimeter of square = 4 × l. Find the area and perimeter of the
following rectangles and squares.

b=3.2cm
l=2.7cm

b=4.3cm
l=3.2cm
a) b) c) d)

l = 5.8 cm l=2.7cm l=2.5cm l=3.2cm

13. a) A rectangular park is 60.48 m long and 40.75 m wide.

(i) Find its perimeter (ii) Find its area.

b) A square garden is 50.5 m long. Find its perimeter and area.

c) A rectangular field is 36.75 m long and its area is 937.125 m2, find

(i) its breadth. (ii) Its perimeter

d) The perimeter of a square ground is 147.2 m. Find its length and area.

14. a) The circumference of the wheel of a bus is 2.8 m. How many revolutions
does it make to cover 3.5 km ?

b) The diameter of a circular coin is 1.14 cm. How many coins of the same
size are required to place in a straight row to cover 2.85 m length (without
leaving any gap between each coin) ?

c) The radius of a circular plate is 6.25 cm. How many plates of same size
are required to place in a straight row to cover 10.625 m length (without
leaving any gap between each plate.)

It's your time - Project work!

15. Let's search today's exchange rates of the following foreign currencies from
any daily news papers or by visiting reliable website.

a) US dollars ($) b) Sterling pound (£) c) Eure (¤)

d) Indian currency e) Chinese Yuan f) Australian dollar (AUD)

(i) Now, calculate, how much Nepali currency is required to exchange 100
units of each currency?

(ii) How many rupees do you need to exchange $ 1,500?

(iii) How many Indian rupees can you exchange with Nepali Rs 8,000?

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