Math5

The greatest and the smallest number of different digits

The Hindu Arabic system consists of 10 digits which are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Study the given table properly and get the idea about the greatest and the smallest
number of different digits.

Digit Smallest number Greatest number

1 1 9
2 10 99
3 100 999
4 1000 9999
5 10000 99999
6 100000 999999
7 1000000 9999999
8 10000000 99999999
9 100000000 999999999
10 1000000000 9999999999

I understand, greatest number of I got an idea, 9 is the greatest number of one
preceeding digit + 1 = smallest number of digit. The greatest number of other digits
can be obtained by repeating 9 in all places.
one more digit.

Greatest number of one digit = 9
9 + 1 = 10 = smallest number of two digits.
Greatest number of two digits = 99
99 + 1 = 100 = smallest number of three digits.

Forming the smallest and the greatest number with given digits:
Let's select three digits 0, 1, 8. The possible 3 digit numbers from the given
digits are 108, 801, 810, 180,
Among these numbers,
The greatest number = 810
The smallest number = 108
Again, select the digits 2, 6, 4. The possible 3 digit numbers are 264, 246, 462,
426, 624, 642.
Among these numbers,
Greatest number = 642
Smallest number = 246

Oasis School Mathematics Book-5 51

Remember !
To form the smallest number, arrange the given digits in ascending order.
To form the greatest number, arrange the given digits in descending order.
‘0’ should not be placed first.

Example : I have to arrange the digits in
descending order to form the greatest

number.

Find the greatest and the smallest numbers 0 should not be kept at first.
formed from digits 2, 5, 0, 7, 8.

Solution:
Given digits: 2, 5, 0, 7, 8
Greatest number formed from the given
digits = 87520

Again, smallest number formed from the
given digits = 20578

Class Assignment

1. Complete the given table:

Greatest number of 9 Smallest number of 1
1-digit 1-digit

Greatest number of Smallest number of
2-digits 2-digits

Greatest number of 5 Smallest number of
-digits 5-digits

2. Write the greatest and the smallest number formed by the given digits.

Digits Smallest number Greatest number
4, 0, 1, 3, 8 10348 84310
2, 5, 7, 0 3
2, 5, 7, 0,1
7, 2, 3, 4, 5, 1
3, 0, 1, 6, 4, 2

52 Oasis School Mathematics Book-5

Devanagari number

Our script to write a number is the Devanagari script.

Hindu Arabic numbers: 0 123456789
Devanagari numbers : ) !@#$%^&*(
Number name in Nepali: zG" o Ps b'O{ tLg rf/ kfFr 5 ;ft cf7 gf}

Notation of number in Hindu Arabic and Devanagari is different but the system
of counting is same in both cases.

Note: Hindu Arabic : Ones Tens Hundreds Thousands

Devanagari : Ps bz ;o xhf/

Example :

Write the given number in Devanagari with its number name in Nepali.
a. 56742
Solution:

Given number: 56742
Number in Devanagari : %^&$@
Number name in Nepali : 5kGg xhf/ ;ft ;o aofnL;

Exercise 3.2

1. Show the following numbers in place value chart and write them in expanded
form:

a. 2684301 b. 321569325 c. 6318946230

d. 53064193272 e. 60507123701

2. Write the following in the standard form:
a. 400 + 30 + 5
b. 3 × 1000 + 5 × 100 + 6 × 10 + 7 × 1
c. 9 × 1000000 + 6 × 100000 + 5 × 10000 + 4 × 1000 + 3 × 100 + 2 × 10 + 7 × 1
d. 7×10000000+5×1000000+2×100000+4×10000+3×1000+2×100+1×10+5×1
e. 6 × 1000000000 + 4 × 100000000 + 3× 10000000 + 2 × 1000000 + 1 ×
100000 + 7 × 10000 + 4 × 1000 + 7 × 100 + 8 × 10 + 9 × 1
f. 5 × 1000000000 + 6 × 10000000 + 5× 100000 + 4 × 10000 + 3× 1000 + 2 ×
100 + 5 × 10 + 8 × 1

Oasis School Mathematics Book-5 53

3. a. Write the greatest and the smallest number of 5 digits.

b. Write the greatest and the smallest number of 8 digits.

c. Write the greatest and the smallest number of 7 digits.

d. Write the smallest number of 6 digits and the greatest number of 5
digits. Also find their difference.

e. What is the difference between the smallest number of 10 digits and
the greatest number of 9 digits?

4. Write the greatest and the smallest number formed by the following
digits:

a. 4, 0, 7, 2, 8 b. 1, 7, 9, 6, 4, 2 c. 8, 1, 4, 6, 3, 2

d. 2, 1, 4, 6, 3, 7 e. 9, 3, 6, 4, 0 f. 7, 0, 8, 6, 5, 4, 9

5. Write the following numbers in Devanagari with their name in Nepali:

a. 46842 b. 37452 c. 82641

d. 972351 e. 493067

Answers : Consult your teacher.

Worksheet

Read the following instruction and colour the answers with the given colour code.

• Place value of 3 in the number 24631278 (Red)
• Number of thousands in one million (Blue)
• Greatest number of 5 digits (Sky blue)
• Smallest number of 3 digits (Dark green)
• Smallest number formed by the digits. 2, 3, 5, 1, 0 (Light green)
• 1 more than the greatest number of 5 digits (Purple)
• 1 less than smallest number of 3 digits (Pink)
• Greatest number formed by the digits 2, 5, 7, 6, 4, 1 (Brown)

99999 100 10235

1000 30000 100000

99 765421

54 Oasis School Mathematics Book-5

Objective Questions

Colour the correct alternatives:

1. One hundred thousand is equal to

one lakh ten lakh one crore

2. One million is equal to ten lakh one crore
one lakh

3. The smallest number formed by the digits 2, 3, 0, 1, 4 is

01234 43210 10234

4. 2 × 1000000 + 5 × 100000 + 0 × 1000 + 3 × 10000 + 2 × 100 + 0 × 10 + 7 is equal to

25327 2503207 250327

5. The greatest number of five digits is

99999 10000 11111

6. The smallest number of 7 digits is

9999999 1111111 1000000

7. Place value of the digit 3 in the number 237642 is

30000 10000 300000

8. 1 more than greatest number of 3 digits is equal to

999 1000 9999

Number of correct answers

Oasis School Mathematics Book-5 55

Do you know?
Before the invention of zero (0), there was no place value system. Ancient
Egyptians used

for 14

Where, for 423
= 1, = 10, = 100

Unit Test

Full marks: 15

1. Show a number 215701406 in the place value chart according to Nepali

place value system and write its number name. 2

2. Show a number 3746207136 in the place value chart according to

international place value system and write its number name. 2

3. Write the number 2376401492 in expanded form. 2

4. Write in the standard form: 2

4 × 1000000000 + 3 × 10000000 + 5 × 1000000 + 4 × 100000 + 7 × 10000 +
6 × 1000 + 2 × 100 x 1 × 10 + 5 × 1

5. Write the greatest and the least number formed by 2,0,7,1,4,6. 2

6. Write the greatest and the smallest number of 6 digits. 2

7. Answer the following questions. 3
a. How many lakhs are there in one million?
b. How many arabs are there in one billion?
c. How many millions are there in one crore?

56 Oasis School Mathematics Book-5

UNIT Four Fundamental
Operations
4

12 Estimated Teaching Hours: 10
93

6

Contents • Addition and subtraction of large numbers

• Simple verbal problems on addition and
subtraction

• Multiplication and division of large
numbers

• Simple verbal problems on multiplication
and division

• Sign rule of +, – , × and ÷

• Order of operations

Expected Learning Outcomes

Upon completion of the unit, students will be
able to develop the following competencies:

• To add and subtract the larger numbers

• To solve simple verbal problems on addition and subtraction

• To multiply the larger numbers (4-digit number by 3-digit
number)

• To divide large numbers

• To simplify the expressions using DMAS rule

Materials Required : Addition table, multiplication table, A4 size paper,
flash card, etc.

Oasis School Mathematics Book-5 57

Addition and subtraction

Review:

- When the numbers are added, the result is the sum: 7 + 5 = 12 (sum)

- When the numbers are subtracted, the result is the difference: 12 - 8 = 4 (difference)

In the relation 7 + 5 = 12, 12 is the sum.

12 – 8 = 4, 4 is the difference.
Add:

7 + 9 + 3 = 10 + 9 = 19 7 + 3 = 10, 10 + 9 = 19

14 + 8 + 6 = 20 + 8 = 28 14 + 6 = 20, 20 + 8 = 28

Class Assignment

1. Complete the following as above:
a. 12 + 7 + 8 = ........ + ........ = ........ b. 25 + 6 + 5 = ........ + ........ = ........
c. 13 + 7 + 6 = ........ + ........ = ........ d. 27 + 3 + 8 = ........ + ........ = ........
e. 14 + 18 + 2 = ........ + ........ = ........

2. a. What should be added to 18 to make 25?
b. What should be subtracted from 12 to make 7?
c. What should be subtracted from the sum of 12 and 18 to get 20?
d. What should be added to the difference of 18 and 12 to make 15?
e. What should be subtracted from the difference of 18 and 6 to make 8?
3. Complete the given table:

+ 32 – + –
25 57 38 42 18

4. A set of numbers is given
12, 18, 16, 20, 25, 30

• Take any three numbers whose sum is 50. .......... + ........... + ............ = 50

• Take any two numbers whose sum is 50. + = 50

• Take any three numbers whose sum is 75. + + = 75

• Take any four numbers whose sum is 80. ......... + ......... + ........ + ....... = 80

58 Oasis School Mathematics Book-5

Addition and subtraction of large numbers

Addition:

Addition of very large numbers is same as the addition of smaller numbers.
Start to add from ones and regroup if necessary.

Add: 1 1 1 1 1 1 Carry over I understand, while adding I have to
3 4 7 6 4 8 7 add the digits of same place.

+ 2 9 4 5 7 6 3

6 4 2 2 2 5 0
Look at one more example,

3746489 + 589642 + 57829

1 1 2 1 1 2 Carry over

3 7 4 6 4 8 9

+ 5 8 9 6 4 2

+ 5 7 8 2 9

4 3 9 3 9 6 0

Subtraction:
Subtraction of very large numbers is same as the subtraction of the smaller
numbers. Start subtracting from ones and regroup if necessary.

4 12 11 15 13 8 14
5 3 2 6 3 9 4
- 2 6 5 7 6 8 7
2 6 6 8 7 0 7
Look at one more example.
8000000 - 2563827

7 9 9 9 9 9 10 Alternative method Subtract 1 from 8000000 =
8 0 0 0 0 0 0 7 9 9 9 9 9 9 7999999
- 2 5 6 3 8 2 7 Subtract 1 from 2563827 =
5 4 3 6 1 7 3 - 2 5 6 3 8 2 6 2563826
Subtract them
5 4 3 6 1 7 3

Addition and subtraction together:

23576408 + 57146374 - 21857431 First
Solution: 1 1 1 1 Add two numbers having positive sign.

2 3 5 7 6 4 0 8

+ 5 7 1 4 6 3 7 4

8 0 7 2 2 7 8 2

Oasis School Mathematics Book-5 59

Again, 7 9 16 11 12
8 0 7 2 2 7 8 2
- 2 1 8 5 7 4 3 1 Subtract the numbers having
5 8 8 6 5 3 5 1 negative sign from the sum.

Class Assignment

1. Add: b. 3 2 0 8 9 6 4 9 c. 5 7 2 4 6 3 7
a. 3 4 7 6 8 7 9 + 5 8 6 9 4 7 3 6 + 5 6 8 3 9 6

+ 8 2 7 4 9 6 7

d. 45 879 2 3 e. 5 7 9 4 1 3 2
+ 569467 + 8 5 9 2 1 6
+ 85926 + 3 9 8 4 2

2. Add the following:

a. 256376 + 54321 + 2964 b. 6357871 + 567835 + 42631

c. 2387641 + 586435 + 287637 + 4873 d. 4587326 + 38645731 + 56473 + 678

3. Subtractthefollowing:

a. 5 6 8 7 3 4 2 b. 4 7 6 3 5 2 1 c. 3 2 0 4 3 7 0
- 6 4 2 1 7 5
- 2 1 6 5 9 7 3 - 2 8 7 4 7 6 5

d. 8 3 0 7 2 1 5 e. 3 7 5 2 8 4 2 1
- 2 3 4 8 2 3 - 2 3 6 4 9 6 5 7

4. Subtract the following: b. 5365042 - 876358
a. 3257639 - 1283768 d. 72548692 - 379689
c. 63577804 - 29765496

5. Simplify: b. 58324604 - 2304325 + 28410842
a. 2368407 + 5346218 - 1423857 d. 45730123 + 23571241 - 15762132
c. 6576465 - 3785461 + 2341732

Answers: Consult your teacher.

60 Oasis School Mathematics Book-5

Verbal problems on addition and subtraction

Example :

The male and female population of a town are 1,56,97,642 and 1,37, 14, 572

respectively. Find the total population of the town.

Solution:

Male population = 1 5 6 9 7 6 4 2

Female population = + 1 3 7 1 4 5 7 2 Read the questions
Total population = 2 9 4 1 2 2 1 4 properly and decide what
\ Total population of the town is 2,94,12,214. you have to do, addition
or subtraction.

Example :

The cost of a piece of land is Rs. 45,25,742 and the cost of another piece is
Rs.35,46,827. By how much is the cost of first piece more than the second?
Solution:

The cost of first piece of land = Rs 4 5 2 5 7 4 2

The cost of second piece of land = - Rs 3 5 4 6 8 2 7

Their difference = Rs 9 7 8 9 1 5

\ The cost of first piece of land is Rs 9, 78, 915 more than that of the second.

Exercise 4.2

1. a. There are 56,692 men, 78,342 women and 22,683 children in a town.
Find the total population of the town.

b. A man earns Rs 32,635 in the month of Baishakh, Rs 45,645 in the
month of Jestha and Rs 56205 in the month of Asad. What is his total
income in three months?

2. a. Find the number that is 3,852 greater than 18,296?
b. Find the number which is 5625 more than 12,682?

c. By how much 63,72,045 is more than 28,76,314?
d. Find the number which is 27,32,451 more than 53,84,271.

3. a. What is the sum of the greatest number of 7 digits and the smallest
number of 8 digits?

b. What is the sum of the greatest number of 7 digits and the greatest
number formed by the digits 3, 8, 9, 2, 6, 0?

Oasis School Mathematics Book-5 61

4. a. Find the number that is 12,632 less than 24,864?
b. Find the number which is 14,696 less than 18,782?

5. a. What is the difference of the greatest number of 6 digits and the
smallest number of 6 digits?

b. What is the difference between the greatest number of 7 digits and
the smallest number formed by the digits 2, 3, 6, 4, 2, 1, 5?

6. a. The sum of two numbers is 57,28,631. If one of the numbers is
27,81,624, find the other number.

b. Which number should be subtracted from 74,26,817 to make 34,08,962?

c. The difference of two numbers is 37,26,841. If the greater number is
5728313, find the smaller number.
Answers: Consult your teacher.

Multiplication and Division

Review:

When two numbers are multiplied, the result is called the product and the numbers

are its factors. 5 × 7 = 35 Product

factors
Multiplication of a number by 10, 20, 30, etc.

45 × 30 = 1350 45 × 3 = 135, write one zero after 135

55 × 40 = 2200 55 × 4 = 220, write one zero after 220

Multiply: 32 × 20 = 46 × 40 =
24 × 30 =

52 × 50 = 56 × 30 = 83 × 60 =

Multiplication of a number by 100, 200, 300, etc. 35 × 2 = 70
Study the given multiplications, \ 35 × 200 = 7000

48 × 300 = 14400 Multiply 48 and 3

35 × 200 = 7000 48 × 3 = 144

Write 2 zeros at the end

= 144000

62 Oasis School Mathematics Book-5

Class Assignment 12 × 100 = ......... 15 × 1000 = .........
32 × 200 = ......... 17 × 2000 = .........
Multiply: 56 × 300 = ......... 28 × 3000 = .........
43 × 400 = ......... 36 × 4000 = .........
42 × 10 = ......... 38 × 500 = ......... 42 × 6000 = .........
46 × 20 = ......... 45 × 600 = ......... 45 × 5000 = .........
82 × 30 = .........
96 × 40 = .........
57 × 50 = ..........
84 × 60 = .........

Multiplication of large numbers

We have already learnt the multiplication of 4 digit numbers by 3 digit numbers.
Let's revise it.

Let's see an example,

Multiply 4263 × 524 Multiply by ones Multiply by tens Multiply by hundreds
Solution:
4263 4263 4263
4 2 6 3 × 4 × 2 0 × 5 0 0
× 5 2 4 17052 85260 2131500

1 7 0 5 2

8 5 2 6 0

+ 2 1 3 1 5 0 0

2 2 3 3 8 1 2

\ 4263 × 524 = 2233812

In short method,
4 2 6 3

× 5 2 4

17052

85260

+ 2131500

2233812

Oasis School Mathematics Book-5 63

Exercise 4.3

1. Multiply: b. 363 × 30 c. 462 × 200 d. 568 × 400
a. 62 × 20 f. 882 × 7000 g. 498 × 5000 h. 3254 × 4000
e. 637 × 6000
c. 3402 × 45 d. 2315 × 54
2. Multiply: g. 8093 × 265 h. 5264 × 374
a. 2315 × 54 b. 9621 × 36
e. 1968 × 305 f. 5021 × 412

Answers : Consult your teacher.

Division

When a number is divided by the other,

Divisor 5) 36 (7 Quotient
- 35 Remainder

1

Division of a number by 10, 20, 30, 100, 200, 300, etc.

50 ÷ 10 = 5

60 ÷ 20 = 6 ÷ 2 = 3 When the dividend ending with zeros is
divided by another number ending with
80 ÷ 40 = 8 ÷ 4 = 2 zeros cancel the same number of zeros
Again, from the end of both dividend and divisor.

400 ÷ 20 = 40 ÷ 2 = 20 Cancel same number of zeros from
600 ÷ 40 = 60 ÷ 4 = 15 the dividend and divisor.
900 ÷ 30 = 90 ÷ 3 = 30

Class Assignment

200 ÷ 20 = 500 ÷ 50 = 800 ÷ 40 =
400 ÷ 10 = 400 ÷ 20 = 600 ÷ 20 =
300 ÷ 30 = 60 ÷ 20 = 80 ÷ 10 =
600 ÷ 30 = 80 ÷ 40 = 400 ÷ 80 =
600 ÷ 20 = 900 ÷ 30 = 600 ÷ 40 =

64 Oasis School Mathematics Book-5

Division of larger numbers Remember !

We have already learnt about the • 0 ÷ any number = 0
division of four digit numbers by • any number ÷ 1 = the number itself
two digit numbers. Let's divide a 6
digit number by a 3 digit number.

Divide: 475286 ÷ 125 Steps:
Solution:
125) 4 7 5 2 8 6 (3802 Quotient • 125 is a three digit number. So take three
- 375 digits 475 from the dividend.
1002
- 1000 • 125 × 3 = 375, write 3 as the quotient
2 8 6
- 2 5 0 • 475 - 375 = 100, bring down 2, now the new
3 6 Remainder dividend is 1002.

\ Quotient = 3802 125 × 8 = 1000, write 8 as the quotient
Remainder = 36
• 1002 - 1000 = 2, bring down 8, now the new
Exercise 4.4 dividend is 28.

• 28 is not divisible by 125, write 0 as the
quotient.

• Bring down 6. Now, the new dividend is
286.

• 125 × 2 = 250, write 2 as the quotient.

• 286 - 250 = 36 (Remainder)

1. Divide:

a. 140 ÷ 20 b. 270 ÷ 90 c. 840 ÷ 40 d. 950 ÷ 50

e. 2800 ÷ 70 f. 1600 ÷ 400 g. 35000 ÷ 7000 h. 54000 ÷ 9000

i. 7200000 ÷ 80000

2. Divide:

a. 2225 ÷ 15 b. 3325 ÷ 25 c. 4618 ÷ 35 d. 2275 ÷ 25

e. 3360 ÷ 35 f. 5832 ÷ 84 g. 12686 ÷ 51 h. 49872 ÷ 68

i. 18468 ÷ 22 j. 4650 ÷ 265 k. 37654 ÷ 365 l. 67632 ÷ 575

m. 53824 ÷ 675 n. 76532 ÷ 615 o. 35824 ÷ 415

Answers : Consult your teacher.

Word problems on multiplication and division
While solving the word problems, follow the following steps:

Steps:
• Understand the questions
• Decide what to do. (+, –, ×, ÷)
• Solve the problem.

Oasis School Mathematics Book-5 65

Example :

The cost of a refrigerator is Rs 15,875. What is the cost of 12 refrigerators?

Solution:

The cost of a refrigerator = Rs 15875 Cost of 12 refrigerators is more
The cost of 12 refrigerators = Rs 15,875 × 12 than the cost of one. So I have to
Now, 1 5 8 7 5
multiply.

× 1 2

31750

+ 15875

190500

\ The cost of 12 refrigerators = Rs 1,90,500.

Example :

A train travels 3696 km in 24 hours. How many kilometers does it travel in one hour?

Solution:

Distance covered by train in 24 hours = 3696 km
Distance covered by train in 1 hour = 3696 km
24
Now,
24 ) 3 6 9 6 (154 In 1 hour a train travels less
- 2 4 distance than in 24 hours. So, I

1 2 9 have to divide.

- 1 2 0

9 6

- 9 6

0 \ Distance covered by a train in 1 hour = 154 km

Exercise 4.5

1 Solve the following problems:
a. The cost of a mobile set is Rs 1,875. What will be the cost of 18 mobile sets?
b. There are 346 chocolates in a packet. How many chocolates are there in
525 packets?
c. A dealer bought 3,465 meters of clothes at a rate of Rs 425 per metre. How
much did he pay altogether?
d. A man earns Rs 2,235 a day. How much does he earn in 365 days?

2. a. How many times can a number 36 be subtracted from 7344?
b. The product of two numbers is 14,875. If one number is 425, find the other number.
c. Yearly income of a man is Rs 1,65,710. How much does he earn in 1 day?
d. A school needs 24,500 pencils in a year. How many boxes of 25 must the
school buy?

Answers : Consult your teacher.

66 Oasis School Mathematics Book-5

Activity

Multiplication using doubling and halving skills.
Multiply:

35 × 27 First Second
Steps: column column
• Write two numbers in two columns.
• Double the first column and half the 35 27

second column. Ignore the remainder 70 13
while halving.
• Keep finding halves until you reach 1. 140 6
• Cross both the numbers if there are even
numbers in both column. 280 3
• Add the remaining numbers from the
column where you double the numbers. 560 1

Using this method, multiply: 35 + 70 + 280 + 560 = 945

Checking:

35
× 2 7
245
+70
945

(a) 42 × 24 (b) 21 × 35 (c) 52 × 23

Game of addition and division

• Take any three single digit numbers

3 5 and 6

• Make all the possible 2 digit numbers

35 36 53 56 63 and 65

• Add these numbers.

35 + 36 + 53 + 56 + 63 + 65 = 308 It's interesting! Every time final

• Add the first 3 single digit numbers. result is 22.

3 + 5 + 6 = 14

• Divide : 308 by 14

308 ÷ 14 = 22

Let's try this, with three single digit numbers.

a. 3, 5, 9 b. 2, 4, 8 c. 3, 6, 8

Oasis School Mathematics Book-5 67

Sign rules of Plus (+), Minus (-), Multiplication (×) and Division (÷)

+ 5 is a positive number.

5 is a positive number. The number with no sign just
before it is a positive number.
- 5 is a negative number.
+5+3=+8
The mathematical operations (addition,
subtraction, multiplication and division) follow
the following rules to use + or - sign.

Addition rule:

- The addition of two numbers with positive sign gives
the sum with positive sign.

(+ number) + (+ number) = (+ sum) (-3)+ (-6) = (-9)

- The addition of two numbers with negative sign gives
the sum with negative sign.

(- number) + (- number) = (- sum)

Two numbers with same signs are always added.

- The addition with two numbers with a positive and (+6) + (-5) = (+1)
negative gives the sum with the sign of larger number. (+3) + (-7) = (-4)

(+ number) + (- number) = (+ sum) if (+ number) > (-
number)

(+ number) + (- number) = (- sum) if (+ number) < (-
number)

Two numbers with different signs are always subtracted.
Sign of the result is the sign of the greater number.

Multiplication rule: (+2) × (+3) = ( +6)

• Multiplication of two numbers with positive sign

each gives the product with positive sign.

(+number) × (+ number) = (+ product)

• Multiplication of two numbers with negative sign (-4) × (-5) = (+20)
each gives the product with positive sign.

(- number) × (- number) = (+ product)

• Multiplication of two numbers with different signs

gives the product with negative sign. (+3) × (-5) = (-15)

(+ number) × (- number) = (- product) (-4) × (+6) = (-24)

(- number) × (+ number) = (- product)

68 Oasis School Mathematics Book-5

Division rule: (+6) ÷ (+2) = ( +3)
• The division of two numbers with positive sign
(+6) ÷ (-2) = (-3)
each gives the quotient with positive sign. (-12) ÷ (+3) = (-4)

(+ number) ÷ (+ number) = (+ quotient)

• The division of two numbers with negative sign
each gives the quotient with positive sign.

(- number) ÷ (- number) = (+ quotient)

• The division of two numbers with different signs
gives the quotient with negative sign.

(-number) ÷ (+ number) = (- quotient)

(+number) ÷ (- number) = (- quotient)

Summary (+) ÷ (+) = (+)
(+) × (+) = (+)

(+) × (-) = (-) (+) ÷ (-) = (-)

(-) × (+) = (-) (-) ÷ (+) = (-)

(-) × (-) = (+) (-) ÷ (-) = (+)

Exercise 4.8 b. (+5) + (+7) + (+3) c. (+6) + (+9) + (+3)
e. (-8) + (+3) f. (-6) + (+9)
1. Simplify: h. (-9) + (+5) i. (+7) + (-9) + (+3)
a. (+6) + (+8) k. (-8) + (-3) l. (-7) + (-6)
d. (-7) + (-6) n. (-3) + (-7) + (+6) o. (-6) + (-7) + (+8)
g. (+7) + (-3) q. (-7) + (-8) + (+3)
j. (+9) + (-8) + (+3)
m. (-5) + (-9) b. (+8) × (+4) c. (+7) × (-3)
p. (-9) + (+6) + (-3) e. (+7) × (-2) f. (-9) × (+3)
h. (-3) × (-4) i. (-6) × (-2)
2. Simplify: k. (-8) × (-5) l. (-2) × (-6)
a. (+6) × (+3) n. (+3) × (-6) × (-8) o. (-6) × (+3) × (-2)
d. (+5) × (-6) q. (-2) × (+3) × (-4)
g. (-3) × (+4)
j. (-7) × (-4)
m. (-2) × (+6) × (+3)
p. (+8) × (-3) × (-2)

Answers : Consult your teacher.

Oasis School Mathematics Book-5 69

Order of operations

An expression may contain two or more mathematical operations. Such expressions
can be solved by following these operations in proper order. This process is called
simplification.

Addition, subtraction, multiplication and division are four fundamental operations.
Such mixed operations in a problem are performed in the following order.

First Division (D)

Second Multiplication (M)

Third Addition (A)

Fourth Subtraction (S)

This rule is called DMAS rule.

Example 1:

Simplify: I have to operate ÷, ×, +
and - in order.
49 ÷ 7 × 4 - 6 + 15
Solution: [operating ÷ sign]
49 ÷ 7 × 4 - 6 + 15 [operating × sign]
= 7 × 4 - 6 + 15 [operating + sign]
= 28 - 6 + 15 [operating - sign]
= 43 - 6
= 37

Example 2:

Divide 75 by 5, then subtract 10 from the quotient.
Solution:
Mathematical expression for the given statement is
75 ÷ 5 - 10
= 15 - 10 = 5

Exercise 4.7

1. Simplify: b. - 7 + 6 + 8 c. 18 - 7 - 6 + 8 - 2
a. 6 + 3 - 2 e. - 20 + 6 - 3 + 15 - 7
d. 15 - 5 + 6 - 8 - 2 c. 5 × 7 - 3 + 2
b. -6 + 3 × 7 - 5 f. 7 × 2 - 3 × 5
2. Simplify: e. 25 + 13 × 3 + 4
a. 6 × 3 - 2
d. 20 × 2 - 3 - 5
g. 15 × 5 - 13 × 3 + 9

70 Oasis School Mathematics Book-5

3. Simplify: b. 35 ÷ 5 - 2 × 3
a. 20 - 6 ÷ 3 d. 25 + 20 ÷ 5 - 8
c. 15 ÷ 3 + 12 ÷ 4 f. 3 × 6 + 15 ÷ 5 - 2 × 5 + 65
e. 18 ÷ 3 - 15 ÷ 5 + 2 × 3 - 2 h. 32 + 99 ÷ 9 × 3 - 14 × 4 - 11
g. 18 × 3 - 5 × 6 - 35 ÷ 7 + 10 j. 18 × 3 - 20 × 2 - 8 × 200 ÷ 25 + 10
i. 20 ÷ 4 + 3 × 7 - 8 × 6

4. Translate the following statements into mathematical expression and simplify:
a. 12 is subtracted from the sum of 13 and 18.
b. 7 is added to the product 3 and 5.
c. The product of 2 and 7 is added to 9 and 15 is subtracted.
d. The product of 4 and 5 is added to the product of 2 and 3.
e. 24 is divided by 3 and the quotient is multiplied by 4.
f. 95 is divided by 19 and 3 is subtracted from the quotient.
g. The quotient of 120 divided by 6 is added to the product of 5 and 7.

5. Start with the number given and perform the operations in order shown by
arrows:

Answers :

1. a) 7 b) 7 c) 11 d) 6 e) -9
2. a) 16 b) 10 c) 34 d) 32 e) 68 e) 68 f) -1
3. a) 18 b) 1 c) 8 d) 21 e) 7 f) 76 g) 29 h) -2 i) -22 j) -40
4. a) 19 b) 22 c) 8 d) 26 32 f) 2 g) 55

Use of brackets in simplification Follow DMAS rule
inside the brackets.

The simplification inside the First in order
brackets also follow the same rule,
while simplifying such expression. Second in order
Third in order
• Remove brackets ( ), { }, [ ]
in order by simplifying all the
operations within it.

Oasis School Mathematics Book-5 71

• Perform the operations involving division (D)
• Perform the operations involving multiplication (M)
• Perform the operations involving addition (A)
• Perform the operations involving subtraction (S)

Example : operation inside ( )

Simplify: operation inside { }
10 - [20 ÷ {20 - 5(4 - 1)}] operation inside [ ]
Solution:
10 - [20 ÷ {20 - 5(4 - 1)}]
= 10 - [20 ÷ {20 - 5 × 3}]
= 10 - [20 ÷{20 - 15}]
= 10 - [20 ÷ 5]
= 10 - 4
= 6

Exercise 4.8 b. (3 × 8) ÷ 12 c. 5 × (6 ÷ 2)
e. (24 - 9) ÷ 5 f. (15 × 6) ÷ 10
1. Simplify: h. 20 ÷ (6 + 4)
a. 3 × (5 + 7)
d. 12 + (9 × 2)
g. 18 + (3 × 2)

2. Simplify:
a. 40 + (20 ÷ 5 + 6) - 30 b. 21 ÷ 7 - 9 + (12 ÷ 3) c. 5 (6 + 2) + 3 - 3 (6 + 2)
d. 6 (5 - 4) - 3 (4 - 6) ÷ 6 e. (24 ÷ 4 × 2) ÷ (6 ÷ 3 × 3) f. 25 + (5 + 2 × 2 - 3)

3. Simplify the following: b. 18 -[20 - {7 + (15 - 12)}]
a. 3 × [15 - {3 + (15 ÷ 3)}] d. 26 - 5 × {15 - (45 ÷ 5)}
c. 40 - {(30 ÷ 10) +10} f. [{84 - (16 × 4 )} ÷ 5] ÷ 4
e. 9 - [7 + {4 - (5 - 2)}] h. 3{24 + 8 ÷ (6 - 4)} - 60
g. 32 ÷ [7 + {4 - (5 - 2)}] j. 8 + [3 × {14 - (16 × 2 + 3) ÷ 7}]
i. 39 - 7 × {28 ÷ (17 - 10) + 1} l. 60 ÷ [150 ÷ 2 - {6 + 3 × (17 - 4)}]
k. 5 × [10 + {6 + 2 × (6 - 3)}]
m. 60 - [ 18 -{ 30 - (8 × 4 - 8)}]

72 Oasis School Mathematics Book-5

4. Convert the following statements into mathematical expressions and
simplify:

a. 24 is divided by the sum of 2 and 4.
b. 3 multiplied by the difference between 15 and 7.
c. 27 is subtracted from the sum of 12 and 6.
d. 7 times the sum of 5 and 8 is added to the quotient of 40 divided by 5.
e. The difference of 35 and 7 is divided by the product of 7 and 2.
f. 14 is added to 4 times the difference of 10 and 6.
g. 3 times the sum of 4 and 2 is added to the quotient of 36 divided by 3.
h. 5 times the difference of 25 and 20 is added to the quotient of 64 divided by 4.

Answers :
1. a) 36 b) 2 c) 15 d) 30 e) 3 f) 9 g) 24 h) 2 2. a) 20 b) -2 c) -39 d) 5
e) 2 f) 31 3. a) 21 b) 8 c) 27 d) -4 e) 1 f) 1 g) 4 h) 24 i) 4 j) 35
k) 110 l) 2 m) 48 4. a) 4 b) 24 c) -9 d) 99 e) 2 f) 30 g) 30 h) 41

Worksheet 4 5

Cross number puzzle.

12

3

7 9 11
6 10

8

Across: Down:
1. 123 × 9 + 4
1. 20 × 8 – 48 ÷ 3 2. 100 ÷ 2 – 1
3. 1000 – (25 × 4 - 8) 3. 200 × 5 - 1
4. (4 × 100) + 7 4. 8 × 5
7. (20 × 3) – (18 ÷ 3) 5. 11445 ÷ 15
8. 265 + 332 – 90 6. product of 40 and 4 is increased by 4
10. 15 is subtracted from the 7. (40 + 10) × 100
product of 16 and 3 8. 14 multiplied by 4
9. 125 × 35
11. (123 × 9 + 4) × 3

Oasis School Mathematics Book-5 73

Unit Test b. 3 4 8 7 4 5 Full marks: 20
+ 2045 1.5 × 2 = 3
1. Add: +17346
a. 6357871 + 246897 + 38942

2. Subtract: 1.5 × 2 = 3
a. 5 6 3 4 7 5 7 b. 6 3 7 4 5 8 3 2 – 2 7 6 4 8 2 3 7

– 2 4 6 8 9 6 3

3. Simplify: 2
28372456+4684357–34827320

4. Multiply: 1.5 × 2 = 3
a. 3 7 6 2 8 × 2 7 6
b. 5 8 3 × 3000

5. Divide: 1.5 × 2 = 3
a. 18468 ÷ 22
b. 540000 ÷ 1800
6. Simplify:
a. 18 × 3 – 5 × 6 – 35 ÷ 7 + 10 1.5 × 2 = 3
b. 3 × [15 - {3 + (15 ÷ 3)}]

7. Solve the following problems: 1.5 × 2 = 3

a. One truck can carry 950 bags of cement. How many trucks are

needed to carry 85500 bags of cement?

b. The population of a city is 843567 and the population of another city
is 232872. Find the total population of the two cities.

74 Oasis School Mathematics Book-5

UNIT Properties of
whole number
5

12 Estimated Teaching Hours: 9
93

6

Contents • Even and odd number
• Prime and composite number
• Test of divisibility
• Factors and multiples
• Prime factorisation
• Highest common factor (HCF)
• Lowest common multiple (LCM)
• Square and square roots
• Cube and cube roots

Expected Learning Outcomes

Upon completion of the unit, students will be able
to develop the following competencies:

• To identify odd and even numbers
• To identify prime and composite numbers
• To identify whether the given numbers are divisible by the

number 2 to 10
• To find the factors and multiples of the given numbers
• To factorise the given number
• To find the H.C.F. and L.C.M. of the given numbers by

factorisation method
• To find the square, square root, cube and cube root of the

given numbers

Materials Required : A4 size paper, chart paper, scissor, Glue, etc.

Oasis School Mathematics Book-5 75

Number system

We have already learnt about whole numbers, natural numbers, odd numbers,
even numbers, prime numbers and composite numbers. Let's review.
Natural numbers:
The numbers 1, 2, 3, 4, 5, ...... which are used for counting are the natural
numbers or counting numbers.

1, 2, 3, 4, ..... are the natural numbers.
Whole numbers
The set of natural numbers including zero are the whole numbers:

0, 1, 2, 3, 4....... are the whole numbers.

The smallest natural number is 1 and
the smallest whole number is 0.

The greatest natural number and
whole number is not defined.

Even and odd numbers Even numbers are the paired
Numbers which are exactly divisible by 2 numbers.
are called even numbers.
Even numbers are the multiple of 2.
2, 4, 6, 8, 10 ..... etc are the even numbers

Numbers which are not exactly divisible by 2 Odd numbers are the unpaired
are the odd numbers. Odd numbers are not numbers.
the multiple of 2.

1, 3, 5, 7, 9 ....... are the odd numbers.

How to determine whether the given
number is odd or even if the number is

very large?

If the digit in the ones place is odd then the
number is odd. If the digit in the ones place is

even or zero, then the number is even.

76 Oasis School Mathematics Book-5

427590 is an even number. Note:
237835 is an odd number.
126524 is an even number. • The sum and difference of two even numbers
is even.

• The sum of two odd numbers is even.

• The sum and difference of an even and
odd number is odd.

Exercise 5.1

1. Answer the following questions:

a. Which are the smallest and the greatest natural numbers?

b. Which are the smallest and the greatest whole numbers?

c. Are all natural numbers whole numbers?

d. Name the only one whole number which is not a natural number.

e. What is the number called which is divisible by 2?

f. A number is not a multiple of 2. What is the number called?

2. Identify whether the given numbers are odd or even:

a. 2460 b. 82693 c. 45857 d. 94282 e. 56740

f. 79934 g. 81647 h. 25810 i. 56219

3. Identify whether the given numbers are odd or even:

a. Sum of two even numbers
b. Sum of two odd numbers
c. Sum of an even and an odd numbers
d. Difference of two even numbers
e. Difference of two odd numbers
f. Difference of an odd and an even number

4. Identify whether the given numbers are odd or even (identify without
addition or subtraction):

a. 15 + 7 b. 12 - 6 c. 18 + 8

d. 234 + 825 e. 23 - 17 f. 265 - 134

Oasis School Mathematics Book-5 77

Prime and composite number

Prime number

A number which is divisible by 1 and by itself but not divisible by other

number is called a prime number. • 3 is divisible by 1 and 3 itself
So, 3 is also a prime number
• 2 is divisible by 1 and 2 itself. Similarly, 5, 7, 11, 13, etc. are
So, 2 is a prime number.

Composite number prime numbers.

Let's take a number 4.

It is divisible by 1 and 4 itself.

Again, it is divisible by 2 also.

Therefore, 4 is a composite number.

Hence, a number which is exactly divisible by other numbers also except by 1
or by itself is called a composite number.

6, 8, 9, 10, 12, etc. are composite numbers.

Prime and composite numbers from 1 to 50 10
123456789 20
11 12 13 14 15 16 17 18 19 30
21 22 23 24 25 26 27 28 29 40
31 32 33 34 35 36 37 38 39 50
41 42 43 44 45 46 47 48 49

Steps:
• Circle 1 which is neither prime nor composite.
• Circle all the even numbers except 2.
• Circle the numbers divisible by 3, except 3.
• Circle all the numbers which are divisible by 5, except 5.
• Circle all the numbers which are divisible by 7, except 7.

All the circled numbers are composite numbers except 1.
All the remaining numbers are prime numbers.
\ Prime numbers from 1 to 50 are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

78 Oasis School Mathematics Book-5

Note: Do you know!

• 1 is neither a prime nor a composite Two prime numbers which are
number. close to each other with a gap of
only one number are twin prime
• 2 is the only even number which is numbers. 17 and 19 are twin
prime. prime numbers.

• The smallest prime number is 2.

Exercise 5.2

1. a. Which even number is prime?
b. What are the two factors of prime number?
c. A number has two factors 1 and the number itself. What is the
number called?
d. If a number has more than two factors what is it called?
e. Which number is neither prime nor composite?
f. Give an example of twin prime numbers.

2. Write all the prime number from 1 to 10.

3. Write all the prime number from 10 to 20.

4. Write the number from 1 to 30
- circle 1
- circle all the numbers divisible by 2
- circle all the numbers divisible by 3
- circle all the numbers divisible by 4, 5, 6, 7, 8 and 9
a. Make the list of remaining numbers.
b. What are the numbers called?

Oasis School Mathematics Book-5 79

Test of divisibility

A number is said to be divisible by another number if the remainder is zero while
dividing the dividend by the divisor.

To find out without actual division whether the given number is divisible by other
numbers or not there are certain tests. Here are some rules of divisibility of the
numbers from 2 to 1.

Divisibility by 2

A number is divisible by 2 if the digit in the place of ones is zero or an even number.

Test:

Number Digit in the ones place Divisible by 2

278 8 (even) Yes

590 0 Yes

3217 7 (odd) No

5369 9 (odd) No

Divisibility by 3

A number is divisible by 3 if the sum of the digits of the given number is a multiple
of 3.

Test: Sum of the digits Divisible by 3
Number 1 + 2 + 3 = 6 (a multiple of 3) Yes
123

946 9 + 4 + 6 = 19 (not a multiple of 3) No

2769 2 + 7 + 6 + 9 = 24 (a multiple of 3) Yes

1039 1 + 0 + 3 + 9 = 13 (not a multiple of 3) No

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible
by 4.

Test:

Number Number formed by last two digits Divisible by 4

336 36 (divisible by 4) Yes

1352 52 (divisible by 4) Yes

1681 81 (not divisible 4) No

2574 74 (not divisible 4) No

80 Oasis School Mathematics Book-5

Divisibility by 5

A number is divisible by 5 if the digit in the place of ones is 0 or 5.

Test: Digits in the ones place Divisible by 5
Number 0 Yes
120

2785 5 Yes

3789 9 No

1487 2 No

Divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.

Test:

Number Digit in the Divisible Sum of the digits Divisible Divisible
by 3 by 6
place of ones by 2 7 + 3 + 6 = 16
(not divisible by 3) No No
736 6 (even) Yes

3336 6 (even) Yes 3 + 3 + 3 + 6 = 15 Yes Yes
(divisible by 3)

1503 3 (odd) No 1 + 5 + 0 + 3 = 9 Yes No
1397 7 (odd) (divisible by 3) No

No 1 + 3 + 9 + 7 = 20 No
(not divisible by 3)

Divisibility by 7

A number is divisible by 7 if the difference between twice the last digit and the
number formed by two other digits is divisible by 7.

Test:

Number Twice the Number Difference Divisible by 7
224 Yes
Number formed 22 - 8 = 14
(divisible by 7)
2 × 4 = 8 22

434 2 × 4 = 8 43 43 - 8 = 35 Yes

(divisible by 7)

Oasis School Mathematics Book-5 81

1035 2 × 5 = 10 103 103 - 10 = 93 No
363 2 × 3 = 6 36 (not divisible by 7 No

36 - 6 = 30
(not divisible by 7)

Divisibility by 8

A number is divisible by 8 if the number formed by last three digits is divisible
by 8 or last three digits are zero.

Test: Number formed by last 3 digits Divisible by 8
Number 088 (divisible by 8) Yes
4088 000 Yes
25000 No
3614 614 (not divisible by 8) No
3124 124 (not divisible by 8)

Divisibility by 9

A number is divisible by 9 if the sum of the digits of the number is divisible
by 9.

Test: Sum of the digits Divisible by 9
Number 4 + 0 + 8 + 8 + = 20 (not divisible by 9) No
4088 Yes
1827 1 + 8 + 2 + 7 = 18 (divisible by 9) No
728 7 + 2 + 8 = 17 (not divisible by 9) Yes
1737 1 + 7 + 3 + 7 = 18 (divisible by 9)

Divisibility by 10
A number is divisible by 10 if the digits in the place of ones is zero.

Test: Digit in ones place Divisible by 10
Number 0 Yes
300 3 No
563 0 Yes
3750

82 Oasis School Mathematics Book-5

Exercise 5.3

1. a. On which condition a number is divisible by 2?
b. Write the condition in which a number is divisible by 3.
c. What digits should be in the ones place for a number to be divisible

by 5?

d. Write the condition of a number to be divisible by 6.

2. Which of the following numbers are divisible by 2.

11, 42, 47, 30, 39, 37, 56

3. Which of the following numbers are divisible by 3.

42, 47, 56, 128, 162, 214, 261

4. Which of the following numbers are divisible by 4.

63, 72, 122, 242, 244, 167, 164

5. Which of the following numbers are divisible by 5.

23, 37, 94, 125, 246, 330, 574, 825

6. Test whether the given numbers are divisible by 6 or not.

a. 162 b. 274 c. 532 d. 673 e. 522

7. Test whether the given numbers are divisible by 7 or not.

a. 224 b. 372 c. 371 d. 434 e. 716

8. Test whether the given numbers are divisible by 8 or not.

a. 2154 b. 2352 c. 3416 d. 6000 e. 7603

9. Test whether the given numbers are divisible by 9 or not.

a. 1827 b. 824 c. 1674 d. 2511 c. 3251

10. Test whether the given numbers are divisible by 10 or not.

a. 6450 b. 1723 c. 2486 d. 5760 e. 8364

Answers: Consult your teacher

Oasis School Mathematics Book-5 83

Factors and Multiples

Factors:
Look and learn:

12 = 12 × 1, 12 = 6 × 2, 12 = 4 × 3
12 is divisible by 1, 2, 3, 4, 6, and 12.

Here, 1, 2, 3, 4, 6 and 12 are the factors of 12.
Again,

20 = 20 × 1

20 = 10 × 2

20 = 5 × 4

Here, 1, 2, 4, 5, 10 and 20 are the factors of 20.

Hence, a number is said to be a factor of another if it divides that number
exactly.

Remember !

• 1 is a factor of every number.
• Every natural number is a factor of itself.
• A factor of a number is either less than or equal to the number.

Multiples:

Look and learn:

5×1=5 5, 10, 15, 20 ... etc are the multiples of 5.
5 × 2 = 10

5 × 3 = 15

5 × 4 = 20

Again,

6×1=6

6 × 2 = 12 6, 12, 18, 24, 30 ... etc are the multiples of 6.

6 × 3 = 18

6 × 4 = 24

6 × 5 = 30

Hence, the multiples of a number are the product of the number by any number.

3 × 5 = 15

Remember ! 15 is a multiple of 3 and 5 are any two
3 and 5. factors of 15.

• Every number is a multiple of 1.

• Every number is a multiple of itself.

• 0 is a multiple of any number.

84 Oasis School Mathematics Book-5

Exercise 5.4

1. Answer the following questions:

a. What are the two factors of a prime number?

b. Which number is the factor of each number?

c. Which number is the multiple of each number?

2. Find the possible factors of:

a. 12 b. 10 c. 18 d. 30 e. 45 f. 50

g. 96 h. 144 i. 150 j. 180 k. 256

3. Write down the first five multiples of:

a. 5 b. 9 c. 12 d. 18 e. 25 f. 28
k. 96 l. 108
g. 32 h. 52 i. 56 j. 76

4. Circle the multiples of given numbers: 35
5 10 14 20 25 28 32 79
7 7 11 21 23 28 40 55
11 11 20 22 30 44 45

5. Write all the prime numbers from:

a. 10 to 20 b. 20 to 30 c. 30 to 40

d. 40 to 50 e. 50 to 70 f. 80 to 100

Answers: Consult your teacher

Prime factors:
Let's find the factors of composite numbers:
24 = 24 × 1 24 = 12 × 2
24 = 8 × 3 24 = 6 × 4
\ The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Among these, 2 and 3 are prime
numbers. So they are called prime factors.
Hence, the smallest possible factors of any number which are the prime
numbers are called the prime factors of the number.

Prime factorisation:
There are two methods of finding prime factors.
They are,
a. by making factor tree
b. by successive division method

Oasis School Mathematics Book-5 85

Factorisation by making

factor tree:

Let's take a number

56. Observe the given

factorisation properly

and get the idea of prime

factorisation, using factor

tree.

Factorisation by successive division method:
Let's take a composite number and factorise it by successive division method.

Example :

Factorise: 144 Apply test of divisibility rule to 144 ÷ 2 = 72
2 144 factorise a number. 72 ÷ 2 = 36
2 72 36 ÷ 2 = 18
2 36 The successive division is carried 18 ÷ 2 = 9
2 18 out by dividing the given number 9÷3=3
3 9 with respective prime number
until the last dividend is prime.

3

\ 144 = 2 × 2 × 2 × 2 × 3 × 3

Example :

Factorise 150 by using

a. factor tree method b. Successive division method

Solution: Solution:

2 150 150 ÷ = 75
75 ÷ 3 = 25

3 75

5 25

5

\ 150 = 2 × 3 × 5 × 5 \ 150 = 2 × 3 × 5 × 5

86 Oasis School Mathematics Book-5

Exercise 5.5

1. Write all the possible factors of the given numbers and make the list of
prime factors:

a. 24 b. 36 c. 48 d. 56 e. 80 f. 84

g. 96 h. 112 i. 128 j. 148 k. 150 l. 156

m. 180 n. 196 o. 198 p. 200

2. Complete the given factor tree and write the given number as the
product of prime factors:

3. Find the prime factors of the following numbers by factor tree method
and express the given number as the product of prime factors:

a. 28 b. 44 c. 36 d. 60 e. 100

f. 150 g. 180 h. 128 i. 220

4. Find the prime factors of the following numbers by successive division
method:

a. 24 b. 32 c. 40 d. 64 e. 72

f. 84 g. 98 h. 110 i. 256 j. 196

k. 175 l. 216 m. 243 n. 343 o. 625

Answers: Consult your teacher

Oasis School Mathematics Book-5 87

Highest Common Factor (H.C.F.)

Let's take any two numbers 18 and 24. The greatest common factor is the
highest common factor.

The possible factors of 18 are 1, 2, 3, 6, 9, and 18.

The possible factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

The common factor of 18 and 24 are 1, 2, 3 and 6.

Here, the greatest common factors of 18 and 24 is 6.

Hence, the highest common factor (H.C.F.) of two or more than two numbers is the
greatest number that is the factor of all the numbers.

Finding H.C.F. by prime factorisation method:

Let's find the H.C.F. of two numbers using prime factorisation method.

Example :

Find the H.C.F. of 24 and 54. Steps:
• Break each of the given numbers into
Solution: 2 54
3 27 their prime factors.
2 24
2 12 • Take out the common prime factors.

2 6 3 9 • Multiply the common prime factors
3 3 obtained from the second step.
Now,
• The product so obtained is H.C.F.

24 = 2 × 2 × 2 × 3 H.C.F. of 5 and 7
54 = 2 × 3 × 3 × 3 is 1.

H.C.F. = 2 × 3 (Taking common factors)

=6

Note:
H.C.F. of two prime numbers = 1.

Lowest Common Multiple (L.C.M.)

Let’s take any two numbers 6 and 8.
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48 etc.
The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64 etc.
The common multiples are 24, 48, .... etc., smallest common multiple is 24.
\ L.C.M. of 6 and 8 is 24.

88 Oasis School Mathematics Book-5

Finding of L.C.M. by prime factorisation method:

Let's take any two numbers 12 ans 18.

Here, 2 12 2 18 Steps:
2 6 3 9 • Resolve each of the given numbers into
Now, 3 3
its factors.
12 = 2 × 2 × 3
18 = 2 × 3 × 3 • Select all the common factors.

• Select rest of the factors which are not
common.

• Multiply these factors.

\ L.C.M. = 2 × 3 × 2 × 3 = 36 • The product so obtained is L.C.M.

2 and 3 are common factors and rest
of the factors are 2 and 3.
L.C.M. = 2 × 3 × 2 × 3

Finding L.C.M. by division method:

If the given numbers are large, we use division method to find the L.C.M. Find

the L.C.M. of 12, 16 and 24: Steps:
• Write all the numbers in a line.

2 12, 16, 24 • Divide the given numbers by a prime
2 6, 8, 12 factors if it divides at least two of them.
2 3, 4, 6
• Write down the quotient and the undivided
3 3, 2, 3 number side by side in the next line.
1, 2, 1
• Proceed in this way until you get all the
\ L.C.M = 2 × 2 × 2 × 3 × 1 × 2 × 1 numbers prime in a row.

• Find the product of divisors and the
numbers in the last row.

= 48 • The product so obtained is L.C.M.

Note: L.C.M. of 3 and 11 is
L.C.M. of two prime numbers = their product 3 × 11 = 33.
L.C.M. of 5 and 7 = 5 × 7
= 35

Oasis School Mathematics Book-5 89

Exercise 5.6

1. Write all the possible factors of the following numbers. List the
common factors and then find H.C.F.

a. 9, 12 b. 15, 18 c. 15, 20 d. 16, 24 e. 20, 25

f. 36, 60 g. 48, 64

2. Find the twelve multiples of the following numbers. List the common
multiples and then find L.C.M.

a. 2, 5 b. 4, 6 c. 3, 5 d. 6, 8 e. 4, 8

f. 6, 12 g. 6, 10

3. Using prime factorisation method, find the H.C.F. of:

a. 6, 8 b. 8, 12 c. 6, 16 d. 9, 12 e. 12, 16

f. 18, 27 g. 12, 15 h. 27, 45 i. 24, 28 j. 36, 48

k. 10, 15, 20 l. 18, 24, 36 m. 12, 15, 18 n. 28, 42, 56 o. 36, 18, 12

4. Using division method, find the L.C.M. of:

a. 6, 16, 24 b. 9, 12, 18 c. 15, 25, 35 d. 20, 30, 40
h. 28, 42, 84
e. 36, 48, 60 f. 40, 45, 60 g. 48, 72, 96

i. 26, 52, 65 j. 28, 36, 42

5. a. Find the greatest number which exactly divides 12 and 21?
b. Find the greatest number which exactly divides 15, 20 and 25?

6. a. Find the smallest number which is exactly divisible by 8, 12 and 18.
b. Find the smallest number which is exactly divisible by 12, 18 and 24.

7. a. If ‘a’ and ‘b’ are the prime numbers, find their H.C.F. and L.C.M.
b. What is the H.C.F. of 5 and 11?
c. What is the L.C.M. of 2 and 3?

Answers :
1. a) 3 b) 3 c) 5 d) 4 e) 5 f) 12 g) 16 2. a) 10 b) 12 c) 15 d) 24 e) 8 f) 12

g) 30 3. a) 2 b) 4 c) 2 d) 3 e) 4 f) 9 g) 3 h) 9 i) 4 j) 12 k) 5 l) 6 m) 3 n) 14
o) 6 4. a) 48 b) 36 c) 525 d) 120 e) 720 f) 360 g) 288 h) 84 i) 260 j) 252
5. a) 3 b) 5 6. a) 72 b) 72 7. a) H.C.F.=1, L.C.M. = ab, b) H.C.F. = 1,
L.C.M.=55, c) H.C.F. = 1, L.C.M. = 6.

90 Oasis School Mathematics Book-5

Activity

To explore the concept of L.C.M. Steps:
• Fill the letters of word FOUR in the first
Take any two words having 4 and
6 letters row and NUMBER in the second row.

Words Number of letters • Keep writing them until at the same
position.
FOUR 4
• Count the number of letters in one row
NUMBER 6 of the poster, which represents the
L.C.M. of 4 and 6.

F OUR a. 3 and 5 letters b. 3 and 4 letters
NUMB E R
Try this with any two words having:

Square and Square Roots To form a square number, we have
Square number: to multiply a number by itself.

1 × 1 = 1, 1 is square of 1

2 × 2 = 4, 4 is square of 2

3 × 3 = 9, 9 is square of 3

4 × 4 = 16, 16 is square of 4

From the above, we understand that a square number is the product of two
same numbers.

The square numbers can be shown as follows:

1 9 I got the idea, if the number of
4 dots can be shown in square
shape, then it represents a square
16
number.

25

All these patterns are square shaped.

Remember !
• Square of 3 is written as 3².

Oasis School Mathematics Book-5 91

Example : 15
Find the square of 15. × 15
Solution: 75
Given number = 15
Square of 15 = (15)2 15
= 15 × 15 225
= 225
Square root:

1 × 1 = 1, 1 is the square root of 1

2 × 2 = 4, 2 is the square root of 4

3 × 3 = 9, 3 is the square root of 9

4 × 4 = 16 4 is the square root of 16

In each of the above,
1, 4, 9, 16 all have two equal factors. One of the equal factors is the square root

of the product.

5² = 25

5 × 5 = 25 5 is the square root of 25 = 5
25 is the square of 5 25

The symbol of square root is denoted by a radical sign ‘ ’.
‘5 is the square root 25’ can be written as 25 = 5

Method of finding square root:

Square root of a number can be obtained by prime factorisation method. Let's
see an example and get the idea about the method of finding square root.

Example : Steps:

Find the square root of 144. • Express the given number as the product of its
Solution: prime factors.

2 144 • Make the pairs of equal factors.
2 72
2 36 • Take one factor from each pair and multiply
3 18 them.

3 6 • The product is the square root of given
number.

2

92 Oasis School Mathematics Book-5

Here, 144 = 2 × 2 × 2 × 2 × 3 × 3
= 22 × 22 × 32
\ Square root of 144 = 2 × 2 × 3
= 12

Shortcut method of finding square of the numbers like 10, 20, 40, 200, 300, etc:

Square of 20 = 20² = 20 × 20 Multiply 2 and 2: 2 × 2 = 4
= 400 Write 2 zeros after 4: 400.

Square of 500 = 500² = 500 × 500 Multiply the numbers except
= 250000 zeros: 5 × 5 = 25

Write four zeros after the product:
250000.

Shortcut method of finding square roots of the numbers like 400, 900, 1600, 2500 etc:

Square root of 400 = 400 4 = 2, 400 = 20
= 20 One zero is taken as squares root of
Square root of 2500 = 2500 2 zeros.

= 50

Example :

Identify whether 250 is a square number or not?

Here, 2 250 Here, I understand! To be a square
number, each of its factor should
5 125 250 = 2 × 5 × 5 × 5
have a pair.
5 2 5
5 = 2 × 5 × 5²

Since 2 and 5 have no pair, 250 is not a square number.

Cube and Cube Roots If a number is multiplied with
itself three times, we can get a cube
Cube number:
1 × 1 × 1 = 1 1 is the cube of 1 number.
2 × 2 × 2 = 8 8 is the cube of 2
3 × 3 × 3 = 27 27 is the cube of 3
4 × 4 × 4 = 64 64 is the cube of 4

Oasis School Mathematics Book-5 93

From the above, we understand that a cube number is the product of three same
numbers.

Cube of 4 is denoted by 4³

Cube of 5 is denoted by 5³

Example:

Find the cube of 8. 64
Solution:

Cube of 8 = 8³ ×8
=8×8×8 512

= 64 × 8

= 512

Cube roots:

1 × 1 × 1 = 1, 1 is cube root of 1

2 × 2 × 2 = 8, 2 is cube root of 8 3 × 3 × 3 = 27
3 × 3 × 3 = 27, 3 is cube root of 27 27 is the cube of 3
4 × 4 × 4 = 64, 4 is cube root of 64 3 is the cube root of 27.

In each of the above cases,

1, 8, 27and 64 have three equal factors. One of the equal
factors is the cube root of the product.

Cube root is denoted by a radical sign 3
2 is the cube root of 8 can be written as, 3 8 = 2

Method of finding cube roots

Cube root of a number can be obtained by prime factorisation method. Let's
see an example and get the idea of finding the cube of given numbers.
Example:

Find the cube root of 216 Steps:
Solution: 2 216
• Express the given number as the product of its
2 108 prime factors.
2 54
3 27 • Make the groups of three similar factors.

3 9 • Take one factor from each group and multiply
3 them.

• The product is the cube root of given number.

94 Oasis School Mathematics Book-5

Here,

216 = 2 × 2 × 2 × 3 × 3 × 3

= 2³ × 3³

\ Cube root of 216 = 2 × 3

=6

Short cut method of finding cubes of the number like 20, 50, 300, 400 etc.

Cube of 20 = 20³ Multiply the numbers except zeros

= 20 × 20 × 20 2 × 2 × 2 = 8, Write 3 zeros after the product.

= 800 3 × 3 × 3 = 27
Cube of 300 = (300)³ 300 × 300 × 300 = 27000000

= 300 × 300 × 300

= 27000000

Shortcut method of finding cube roots of the number like 1000, 8000, 27000
etc.

Cube root of 1000 = 3 1000 One zero is taken as the cube root
= 10 of 3 zeros.
Cube root of 8000 = 3 8000
= 20 Cube root of 8 = 2
Cube root of 8000 = 20

Exercise 5.7

1. Find the squares of the following numbers:

a. 24 b. 18 c. 42 d. 64 e. 86 f. 96

g. 112 h. 124 i. 156

2. Find the squares of the following numbers:

a. 20 b. 40 c. 60 d. 90 e. 200 f. 300

g. 500 h. 700 i. 900 j 5000 k. 7000 l. 8000

3. Determine whether the given numbers are square numbers or not?

a. 144 b. 128 c. 392 d. 1200 e. 625 f. 2304

Oasis School Mathematics Book-5 95

4. Find the square root of the following numbers:

a. 81 b. 324 c. 441 d. 256 e. 576 f.625
f. 24
g. 676 h. 729 i. 1296 j. 1444 k. 1156

5. Find the cube of the following numbers:

a. 8 b. 9 c. 12 d. 16 e. 18

g. 27 h. 45 i. 48

6. Find the cube of the following numbers:

a. 10 b. 20 c. 30 d. 70 e. 90
j. 900
f. 200 g. 300 h. 400 i. 800

7. Find the cube root of the given numbers:

a. 8 b. 27 c. 64 d. 125 e. 216

f. 512 g. 2744 h. 5832 i. 3375

8. a. Which number should be multiplied to 125 to make it a perfect
square?

b. Which number should be multiplied to 108 to make it a perfect
square?

c. Which number should be multiplied to 98 to make it a perfect
square?

9. a. 18 flowers are planted in a row and column of a square garden.
Find the total number of flowers in the garden.

b. 35 students are kept in each row and column of a square ground.
Find the number of students in the ground.

10. a. 196 students are arranged in a square ground. How many students
are there in a row?

b. What number multiplied by itself gives the product 324?

Answers :
Q. 1 to 7 : Consult your teacher, 8. a) 5 b) 3 c) 2 9. a) 324 b) 1235

10. a) 14 b) 18

96 Oasis School Mathematics Book-5

Objective Questions

Colour the correct alternatives:
1. The only one even number which is prime is

246

2. If the digit in the ones place is 0 or 5, the number is divisible by

10 3 5

3. If the sum of the digits of a number is the multiple of 9, then the number
is divisible by
692

4. Which of the following statements is not true?

1 is a factor of every number

Every number is a factor of itself

2 is a factor of every number

5. H.C.F. of two prime numbers is

1 their product their sum
their sum
6. L.C.M. of two prime numbers is 9

1 their product
7. Cube root of 216 is 6

8

Oasis School Mathematics Book-5 97

Worksheet
Factors of the numbers 1 to 20

20
19
18
17
16
15
14
13
12
11
10 10
99
88
77
666
5555
44444
333333
2222222222
11111111111111111111

• Make the table of 20 × 20 grid. • In the bottom row write 1 in all the box.
• Write 2 in the gap of 1 boxes. • Write 3 in the gap of 2 boxes.
• Write 4 in the gap of 4 boxes.

Similarly, complete the table.

To find the factor 4, see the number 4 on the grid. Under 4 in the same column
there are 4, 2, 1.
\ Factors of 4 are 4, 2, 1.

98 Oasis School Mathematics Book-5

From the given table the factors of the number are: 16
6 11 17
7 12 18
8 13 19
9 14 20
10 15

The prime numbers from 1 to 20 are ......................................................................

Unit Test Full marks: 35

1. Add or subtract: 2x2=4
a. 6357871 + 246897 + 38942
b. 37564245 - 2867433

2. Multiply: 2x2=4
a. 37628 × 276
b. 583 × 3000

3. Divide: 2x2=4
a. 18468 ÷ 22
b. 540000 ÷ 900

4. Simplify: 2×2=4
a. 18×3-5×6-35 ÷ 7+10
b. 3×[15-{3+(15 ÷ 3)}]

5. Test whether the given numbers are divisible by 3 or not. 2x1x=2
a. 432 b. 863

6. Find the possible factors of 18. 2

7. Find the prime factors of. 2x2=4
a. 60 b. 84
2
8. Find the H.C.F. of 15 and 18. 2
2
9. Find the L.C.M. of 18 and 27. 1
4x1=4
10. Find the square root of 225.

11. Find the cube of 15.

12. Answer the following questions:
a. Which are the smallest and the greatest natural numbers?
b. What is a number called which is divisible by 2?
c. Which number is the factor of each number?
d. If 'a' and 'b' are two prime numbers,
find their H.C.F. and L.C.M.

Oasis School Mathematics Book-5 99

UNIT Fractions and
Decimals
6

12 Estimated Teaching Hours: 10
93

6

Contents • Conversion of unlike fractions into like fractions
• Addition and subtraction of fractions
• Multiplication of a fraction by a fraction and by

a whole number
• Division of a fraction by a fraction and by a

whole number
• Simplification of fractions
• Conversion of a decimal into fraction and

fraction into a decimal
• Addition and subtraction of a decimal
• Multiplication of decimal by decimal and by

whole number
• Division of a decimal by a decimal and by a

whole number
• Rounding off the decimal number

Expected Learning Outcomes

Upon completion of the unit, students will be able
to develop the following competencies:

• To convert the unlike fractions into like fractions
• To add and subtract the fractions
• To multiply and divide a fraction by a fraction and a fraction

by whole number
• To simplify the fractions
• To convert a decimal into a fraction
• To add and subtract decimal numbers
• To multiply a decimal number by a decimal number and by a

whole number
• Rounding off the decimal number to the nearest tenths and to

the nearest hundredths

Materials Required : A4 size paper, chart paper, cryons, clue, scissors, etc.
100 Oasis School Mathematics Book-5