Ratios in Simplest Form

Write a ratio in its simplest form: cancel common factors, change to the same units, clear decimals and fractions, and write it as 1 : n.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 32 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What Simplest Form Means

A ratio compares amounts. $2 : 3$ means for every 2 of the first there are 3 of the second.

Multiply or divide every part by the same number and the ratio stays the same. These are all one ratio:

Ratio	How you got it
$12 : 18$	the start
$6 : 9$	divide both by 2
$2 : 3$	divide both by 3

Rule

Simplest form means whole numbers that share no common factor. $2 : 3$ is the simplest form of $12 : 18$ .

Common mistake

Taking the same number away from both parts. $12 : 18$ take away 6 is $6 : 12$ , which is $1 : 2$ . That is a different ratio. Only multiply or divide.

Cancelling Two Numbers

The 1 or 2-mark question: "Write the ratio in its simplest form." A class has 36 boys and 60 girls. Find boys : girls.

Worked example

1
$36 : 60$ — boys first, as the question says
2
$6 : 10$ — both are in the 6 times table, so divide by 6
3
$3 : 5$ — 6 and 10 are still both even, so divide by 2
4
$3 : 5$ — 3 and 5 share no factor, so stop

Key idea

Quicker: divide by the highest common factor, 12, in one go. Any route ends at the same answer.

Common mistake

Stopping at $6 : 10$ . Always look again: is there a number that still goes into every part?

Three Parts

The 2-mark question gives three amounts, often from a table or chart. A shop sells 45 small, 30 medium and 75 large cups. Find small : medium : large.

Worked example

1
$45 : 30 : 75$ — in the order the question names them
2
$9 : 6 : 15$ — all end in 0 or 5, so divide every part by 5
3
$3 : 2 : 5$ — all in the 3 times table, so divide by 3

Rule

A common factor must go into every part, not just two of them.

Common mistake

Dividing $9 : 6 : 15$ by 2 because 6 is even. 2 does not go into 9 or 15, so you would get decimals.

Key idea

When the numbers come from a table, write them in the order the question asks for, not the order in the table.

Different Units: Change First

The 2-mark question mixes units. Write 350 ml : 2.1 litres in its simplest form.

Unit	Is the same as
1 kg	1000 grams
1 litre	1000 ml
1 km	1000 metres
1 metre	100 cm
1 hour	60 minutes

Worked example

1
$350 ml : 2.1 litres$ — different units, so you cannot compare yet
2
$2.1 \times 1000 = 2100$ — 1 litre is 1000 ml
3
$350 : 2100$ — both in ml now, so drop the units
4
$35 : 210$ — divide both by 10
5
$1 : 6$ — 35 goes into 210 six times, so divide by 35

Common mistake

Writing $350 : 2.1$ . The numbers are in different units, so the ratio is wrong by 1000 times.

Key idea

Change to the smaller unit. Then the numbers are usually whole.

Decimals and Fractions: Make Them Whole

The 2-mark question has decimals or fractions. Multiply every part by the same number until all are whole.

Worked example

1
$2.4 : 0.9$ — one decimal place each
2
$24 : 9$ — multiply both by 10
3
$8 : 3$ — divide both by 3

With fractions, multiply by a number that every bottom number goes into.

Worked example

1
$\frac{3}{4} : \frac{5}{6}$ — 4 and 6 both go into 12
2
$\frac{3}{4} \times 12 = 9$ — 12 ÷ 4 = 3, and 3 × 3 = 9
3
$\frac{5}{6} \times 12 = 10$ — 12 ÷ 6 = 2, and 2 × 5 = 10
4
$9 : 10$ — no common factor, so this is simplest

Common mistake

Multiplying $2.45 : 1.4$ by 10. You get 24.5, still a decimal. Two decimal places need $\times 100$ : $245 : 140$ , which is $7 : 4$ .

Key idea

"B costs one and a half times A" means A : B is $1 : 1.5$ . Double both: $2 : 3$ .

The Form 1 : n

The 1 or 2-mark question: "Write in the form $1 : n$ ". The first part must be exactly 1, so $n$ can be a decimal.

Worked example

1
$8 : 30$ — the first part must become 1
2
$\frac{8}{8} : \frac{30}{8}$ — so divide both parts by 8
3
$1 : 3.75$ — 30 ÷ 8 = 3.75. Leave it as a decimal

Rule

For $1 : n$ divide both parts by the first part. For $n : 1$ divide both by the second part.

Common mistake

Writing $4 : 15$ . That is the simplest form, but the question asked for $1 : n$ , so the first part must be 1.

Key idea

A sheet of A4 paper is 210 mm by 297 mm. In the form 1 : n that is about 1 : 1.41. Fold it in half and the A5 sheet you get has almost exactly the same 1 : 1.41 shape.

Which Method? Read the Question

Every question ends the same way: whole numbers, nothing left to cancel. What changes is the first step.

You see	You do
“in its simplest form”	divide every part by a common factor until none is left
Units like grams and kg, ml and litres	change to the smaller unit first
Decimals	× 10 or × 100 until all are whole
Fractions	× a number every bottom number goes into
“in the form 1 : n”	divide every part by the first part
“Share … in the ratio …”	See Dividing in a Given Ratio
A fraction or % first, then a ratio	See Dividing in a Given Ratio
Recipes, mixing, which runs out	See Ratio Mixture Problems
A map scale as 1 : n	See Scale Drawings

Key idea

Check your answer: whole numbers, the same order as the question, and no number except 1 goes into every part.

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Real past-paper questions on ratios in simplest form

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