3. Operations On whole numbers

By studying this lesson you will be able to

• Identify the convention used in simplifying numerical expressions, and

• Simplify numerical expressions consisting of whole numbers.

3.1 Mathematical operations on two whole numbers

You have already learnt how to add and multiply two whole numbers.

Here each mathematical operation wasperformed only once.

3.2 The order in which mathematical operations in a numerical expression are performed

Consider the expression 3 + 7 × 5.

This is a numerical expression with three whole numbers and two operations.

Here + and × are defined as the operations of this expression.

The order in which the operations appear is + first and then ×.

If we consider the expression 15 ÷ 3 - 2, the order in which the operations appear is ÷ first and then -.

(1) For each of the following numerical expressions, write down the mathematical operations in the order in which they appear.

(i) 5 + 3 + 2 

(ii) 6 × 3 - 6 

(iii) 10 - 8 ÷ 2 × 3 

(iv) 11 × 2 + 5 - 2 

(v) 24 ÷ 6 + 6 ÷ 3

3.3 Simplifying numerical expressions

● Simplifying expressions involving only addition

Let us simplify the expression 8 + 7 + 2 in two different ways.

Let us add 8 and 7 first, and then add 2 to the result. This yields the answer 17.

8 + 7 + 2 = 15 + 2 = 17

Adding 7 and 2 first, followed by adding 8 to the result also yields the answer 17.

8 + 7 + 2 = 8 + 9 = 17

● Simplifying expressions involving only multiplication

Let us simplify the expression 5 × 2 × 3 in two different ways.

Multiplying 5 by 2 first, and then multiplying the result by 3 yields theanswer 30. 

5 × 2 × 3 = 10 × 3 = 30

Multiplying 2 by 3 first, and then multiplying the result by 5 also yields the answer 30.

5 × 2 × 3 = 5 × 6 = 30

Thus, if either addition or multiplication is the only operation in a numerical expression, then irrespective of the order in which the operations are performed, the result obtained is the same.

(1) Simplify each of the following expressions.

(i) 12 + 5 + 8 

(ii) 5 × 8 × 3 

(iii) 7 + 3 + 2 + 6 

(iv) 2 × 5 × 4 × 3

3.4 Further simplification of numerical expressions

Let us first add 10 to 2 and then multiply the answer by 3. 

10 + 2 × 3 = 12 × 3 = 36.

Now let us multiply 2 by 3, and then add 10 to it.

10 + 2 × 3 = 10 + 6 = 16

Therefore, it is clear that when numbers simplify such numerical expressions which involve more than two terms and several operations, we may end up with different answers, depending on the order in which we perform the operations.

This emphasises the need for a convention when simplifying expressions involving two or more operations.

Let us consider below the convention used when simplifying such expressions.

👉 First perform all divisions (÷) and multiplications (×), working from left to right.

👉 Then perform all additions (+) and subtractions (-), working from left to right.

Only the operations of addition and multiplication appear in the expression 10 + 2 × 3 . According to the above convention, multiplication should be performed first.

10 + 2 × 3 = 10 + 6 = 16

Also, if only subtraction (-) and addition (+), or only division (÷) and multiplication (×) appear in a numerical expression, simplification is done from left to right in the order that the operations appear.

>> Simplifying expressions involving only addition and subtraction

Let us simplify the expression 10 - 7 + 2'

Here the order in which the operations appear from left to right is – first and then +.

When simplifying 10 - 7 + 2, first 7 is subtracted from 10 and then 2 is added to the result.

∴ 10 - 7 + 2 = 3 + 2 = 5

Another example is: 6 + 7 - 2 = 13 - 2 = 11

>> Simplifying expressions involving only multiplication and division

Let us deal with the expression 36 ÷ 6 × 3 in a similar way.

Here the order in which the operations appear from left to right is ÷ first

and then ×.

Let us first divide 36 by 6, and then multiply the answer by 3.

We then obtain 36 ÷ 6 × 3 = 6 × 3 = 18

Another example is: 36 × 6 ÷ 3 = 216 ÷ 3 = 72

>> Simplifying expressions in which only subtraction (-) or division (÷) appears several times.

When simplifying expressions involving only subtraction (-) or division (÷), the order in which the operations are performed is from left to right.

Consider the expression 10 - 3 - 2, where the operation of subtraction is applied twice. When the expression 36 ÷ 6 ÷ 3 is considered, division is applied twice.

Let us simplify these expressions.

Now let us subtract 3 from 10, and then subtract 2 from the result.

10 - 3 - 2 = 7 - 2 = 5.

Let us divide 36 by 6, and then divide the answer by 3.We then obtain,

36 ÷ 6 ÷ 3 = 6 ÷ 3 = 2.

(1) Place a  next to the correct statements and a  next to the incorrect statements.

(i) 8 - 5 + 2 = 1 

(ii) 12 × 3 - 11 = 25

(iii) 7 + 18 ÷ 6 = 10 

(iv) 5 × 6 ÷ 3 + 7 = 3

(2) Simplify the following expressions.

(i) 10 × 4 + 17 

(ii) 8 × 3 + 5 

(iii) 14 ÷ 7 × 5

(iv) 448 + 12 ÷ 3 

(v) 7 × 200 + 108 (vi) 8 × 9 - 61

(vii) 100 - 7 × 8 

(viii) 195 - 12 × 10 ÷ 5 

(ix) 7 + 5 × 37 + 278

 >> Simplifying expressions with brackets

If we want to subtract 2 from 3 first, and then subtract the result from 10, we write it as 10 - (3 - 2), with 3 - 2 within brackets. This emphasises that the operation within brackets has to be done first.

That is, 10 - (3 - 2) = 10 - 1 = 9.

Consider the following example.

A practical examination in music is held over six days. Each day, twelve candidates participate in the morning session, while 8 participate in the afternoon session. Let us find the total number of candidates.

Number of candidates in each morning session = 12

Number of candidates in each afternoon session = 8

Total number of candidates during the six days = (12 + 8) × 6 = 20 × 6 = 120

Observe that the usage of brackets has been necessary in deriving the correct answer.

The convention followed when simplifying expressions involving whole numbers and the operations +" -" ×" ÷" and brackets is as follows.

👉 First perform any calculations inside brackets.

👉 Then perform all multiplications and divisions, working from left to right.

👉 Finally perform all additions and subtractions, working from left to right.

(1)Simplify the following expressions.

(i) (12 + 8) - 15 

(ii) 35 - (14 + 9) 

(iii) 7 (12 - 7)

(iv) 108 + 3 (27 - 13) 

(v) 24 ÷ (17 - 5) 

(vi) 3 (5 + 2) × 8

(vii) 31 + (16 ÷ 4) 

(viii) 73 - (8 × 9) 

(ix) (19 × 10) + 38

(x) 475 - (30 ÷ 6)

(4) Simplify the following expressions.

(i) 30 ÷ 10 × 5 

(ii) 40 × 10 ÷ 5 

(iii) 400 - 20 × 10

(iv) 30 ÷ (10 × 3) 

(v) (40 ÷ 10) × 8 

(vi) 3 + 7 × 5

(vii) 6 ÷ 2 + 7 

(viii) (24 × 3) ÷ 8 

(ix) 24 ÷ (3 × 4)

(x) 3 + 6 × (5 + 4) ÷ 3 - 7 

(xi) 10 + 8 (11 - 3) × 4 - 4