Adding, Subtracting, Multiplying and Dividing Decimals

Add, subtract, multiply and divide decimals without a calculator: line up the points, count the decimal places, and make the number you divide by a whole number.

In the exam for Cambridge O Level Maths 4024, Cambridge IGCSE Maths 0580 · 14 real past-paper questions below

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What Decimal Arithmetic Is

On a non-calculator paper you add, subtract, multiply and divide decimals by hand. Each operation has one rule about the decimal point.

To	Do this
Add or subtract	line up the points, fill gaps with 0
Multiply	ignore the points, then count decimal places
Divide by a decimal	make the number you divide by whole
Divide by a whole number	the point stays where it is

Key idea

Decimals reached most of Europe through a small 1585 booklet by Simon Stevin, a Flemish engineer. It was called The Tenth. He wanted merchants to add and multiply fractions as easily as whole numbers.

Multiplying Decimals

The 1-mark question: work out something like $0.05 \times 0.3$ . It is the most common decimal question on the paper.

Rule

Multiply the digits as whole numbers. Then the answer has as many decimal places as the two numbers together.

Worked example

1
$5 \times 3 = 15$ — forget the points for now
2
$0.05 : 2 places$ — count the digits after the point
3
$0.3 : 1 place$ — and in the other number
4
$2 + 1 = 3 places$ — three places in the answer
5
$0.05 \times 0.3 = 0.015$ — 15 with three decimal places

Common mistake

Writing 0.15. Two numbers smaller than 1 multiply to something smaller than both. 0.15 is bigger than 0.05, so it cannot be right.

Key idea

A whole number works the same way. $40 \times 0.3$ : $40 \times 3 = 120$ , one decimal place gives 12.0, so the answer is 12.

Dividing by a Decimal

The 1-mark question: work out something like $72 \div 0.3$ . You cannot divide by 0.3 on paper, so change it into a division by a whole number.

Rule

Multiply BOTH numbers by 10, or by 100, until the number you divide by is whole. The answer does not change.

Worked example

1
$72 \div 0.3$ — 0.3 has one decimal place
2
$720 \div 3$ — both numbers × 10
3
$720 \div 3 = 240$ — an easy whole number division
4
$72 \div 0.3 = 240$ — same answer as before

Common mistake

Changing only the 0.3. $72 \div 3 = 24$ is ten times too small. Whatever you do to one number, do to the other.

Key idea

Why the answer is bigger: $1.5 \div 0.5$ asks how many halves fit in 1.5. Three fit, and $15 \div 5 = 3$ too.

Dividing a Decimal by a Whole Number

The 1-mark question: work out something like $25.8 \div 6$ . Use short division. The point in the answer goes straight above the point in the number.

Worked example

1
$25 \div 6 = 4 r 1$ — 6 into 25 goes 4 times, 1 left
2
$4. \dots$ — write the point straight above the point
3
$18 \div 6 = 3$ — carry the 1 to the 8 to make 18
4
$25.8 \div 6 = 4.3$ — the point stayed in line

Common mistake

Stopping with a remainder. Add zeros after the point and keep going: $7.3 \div 4 = 7.300 \div 4 = 1.825$ .

Adding and Subtracting Decimals

The 1-mark question: work out something like $4.2 - 1.75$ . Write one number above the other with the points in a line.

Worked example

1
$4.2 = 4. 20$ — fill the gap with 0 so both have two places
2
$4.20 - 1.75$ — points in a line, digits in their columns
3
$= 2.45$ — subtract column by column, as with whole numbers

Common mistake

Lining up the last digits. $0.86 - 0.3$ is $0.56$ , not 0.83. The 3 is three tenths, so it goes under the 8.

Key idea

Adding works the same way: $12.40 + 3.75 = 16.15$ .

Is the Answer Sensible?

A slip with the point makes the answer 10 or 100 times too big or too small. A quick estimate catches it.

Rule

Round each number to 1 significant figure and do the easy sum in your head. Your answer should be close.

Worked example

1
$4.8 \div 0.6 \approx 5 \div 0.6$ — round the 4.8
2
$\approx 8$ — 0.6 fits into 5 about 8 times
3
$4.8 \div 0.6 = 8$ — the answer 8 fits; 0.8 or 80 does not

Key idea

Multiply by a number smaller than 1: the answer gets smaller. Divide by it: the answer gets bigger.

Which Method? Read the Question

Look at the operation and at the number after it.

You see	You do
0.05 × 0.3	multiply digits, count decimal places
72 ÷ 0.3	× 10 both numbers, then divide
25.8 ÷ 6	short division, point stays in line
4.2 − 1.75	line up the points, fill with 0
× or ÷ by 10, 100, 1000	Understanding Decimal Place Value
Given 36 × 52 = 1872, find 3.6 × 5.2	Understanding Decimal Place Value
Estimate to 1 significant figure	Estimation
Give the answer to 2 decimal places	Rounding to Decimal Places

Key idea

Before you write the answer, ask: should it be bigger or smaller than the numbers in the question?

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