Ratio Notation and Simplifying

A ratio compares two or more quantities of the same kind. It is written using a colon.

Ratio notation

$a : b$ is read as "$a$ to $b$"

Order matters. If the ratio of cats to dogs is $3 : 5$, then the ratio of dogs to cats is $5 : 3$. Always write the quantities in the order the question names them.

Ratios can compare more than two things: $2 : 3 : 5$ is a perfectly good ratio (for example cement : sand : gravel in concrete).

A ratio is in its simplest form when the numbers are whole numbers with no common factor other than $1$. Simplifying works exactly like simplifying a fraction: divide every part by the same number.

Make sure both quantities are in the same unit.
Drop the units — a ratio has none.
Find the HCF (highest common factor) of all the parts.
Divide every part by the HCF.
Check that no common factor is left.

Worked Example 1 — Simple case

Simplify $24 : 36$.

①Factors of $24$: $1,2,3,4,6,8,12,24$. Factors of $36$: $1,2,3,4,6,9,12,18,36$.

②HCF $= 12$.

③$24 \div 12 = 2$ and $36 \div 12 = 3$.

Answer: $2 : 3$

If you can't spot the HCF, just keep halving or dividing by small numbers: $24:36 \to 12:18 \to 6:9 \to 2:3$. You get there in the end.

Worked Example 2 — Three parts

Simplify $18 : 30 : 42$.

①All three are even, and all are multiples of $3$, so try $6$.

②$18 \div 6 = 3$, $\;30 \div 6 = 5$, $\;42 \div 6 = 7$.

③$3, 5, 7$ have no common factor.

Answer: $3 : 5 : 7$

Worked Example 3 — Different units

Simplify $250$ g $: 2$ kg.

①Same unit: $2$ kg $= 2000$ g.

②$250 : 2000$

③Divide both by $250$: $\;1 : 8$

Answer: $1 : 8$

Never write $250 : 2$. That would say a quarter of a kilo is more than a kilo!

A ratio in simplest form should contain whole numbers. If you start with decimals or fractions, get rid of them first.

Decimals: multiply every part by $10$, $100$ or $1000$ until they are whole.
Fractions: multiply every part by the LCM of the denominators.

Worked Example 4 — Decimals

Simplify $1.2 : 0.9$.

①Both have one decimal place, so multiply both by $10$: $12 : 9$

②HCF of $12$ and $9$ is $3$: $\;4 : 3$

Answer: $4 : 3$

Worked Example 5 — Fractions

Simplify $\dfrac{2}{3} : \dfrac{4}{5}$.

①Denominators are $3$ and $5$; their LCM is $15$.

②Multiply both parts by $15$: $\;\dfrac{2}{3}\times 15 = 10$ and $\dfrac{4}{5}\times 15 = 12$.

③$10 : 12$, and dividing by $2$ gives $5 : 6$.

Answer: $5 : 6$

Sometimes a ratio is more useful when one side is exactly $1$ — for example, map scales are always written $1 : n$, and "best buy" comparisons often use $n : 1$.

Method

For $1 : n$ → divide both parts by the left number.
For $n : 1$ → divide both parts by the right number.

The answer often is not a whole number, and that is fine. Do not round unless asked.

Worked Example 6 — $1 : n$ form

Write $8 : 20$ in the form $1 : n$.

①Divide both parts by the left number, $8$.

②$8 \div 8 = 1$ and $20 \div 8 = 2.5$

Answer: $1 : 2.5$

Worked Example 7 — $n : 1$ form

Write $45 : 12$ in the form $n : 1$, giving $n$ to 2 decimal places.

①Divide both parts by the right number, $12$.

②$45 \div 12 = 3.75$ and $12 \div 12 = 1$

Answer: $3.75 : 1$

To compare two ratios, put them both in the same form — either $1 : n$, or with a matching first (or second) part.

Worked Example 8 — Which squash is stronger?

Drink A mixes concentrate to water in the ratio $2 : 9$. Drink B mixes them $3 : 13$. Which is stronger?

①Write both as $1 : n$ (concentrate : water).

②A: $2 : 9 \to 1 : 4.5$

③B: $3 : 13 \to 1 : 4.333\ldots$

④B has less water per unit of concentrate.

Drink B is stronger.

Worked Example 9 — Combining two ratios

$A : B = 3 : 4$ and $B : C = 6 : 5$. Find $A : B : C$.

①$B$ appears in both, as $4$ and as $6$. Make them match using the LCM of $4$ and $6$, which is $12$.

②$A : B = 3 : 4$, multiply by $3$ → $9 : 12$

③$B : C = 6 : 5$, multiply by $2$ → $12 : 10$

④Now line them up: $A : B : C = 9 : 12 : 10$

Answer: $9 : 12 : 10$ (no common factor, so this is simplest form).

Multiplying or dividing every part of a ratio by the same number produces an equivalent ratio — a different way of writing the same relationship.

Ratio	$2 : 3$	$4 : 6$	$10 : 15$	$1 : 1.5$	$24 : 36$
Equivalent?	✓	✓	✓	✓	✓

You must never add or subtract the same number from each part. $2 : 3$ is not equivalent to $3 : 4$ — the first says "$1.5$ times", the second says "$1.33$ times".

Worked Example 10 — Finding a missing part

Complete the equivalent ratio $\;5 : 8 = 35 : \;?$

①What turns $5$ into $35$? $35 \div 5 = 7$

②Apply the same multiplier to the other part: $8 \times 7 = 56$

Answer: $35 : 56$

Notation

$a : b$ means "$a$ to $b$". Order follows the wording of the question.

Same units first

Convert before simplifying; ratios themselves have no units.

Simplifying

Divide every part by the HCF. Repeated halving works too.

Decimals

Multiply all parts by $10$, $100$… until whole.

Fractions

Multiply all parts by the LCM of the denominators.

$1 : n$ form

Divide both parts by the left-hand number. Decimals are allowed.

Combining

Make the shared letter match using the LCM, then line up all three.

Golden rule

Multiply or divide every part — never add or subtract.

Question 1

Simplify these ratios: (a) $16 : 24$ (b) $45 : 27$ (c) $14 : 21 : 35$.

▶ Show solution

(a) HCF of $16$ and $24$ is $8$: $\;2 : 3$

(b) HCF of $45$ and $27$ is $9$: $\;5 : 3$

(c) HCF of $14$, $21$, $35$ is $7$: $\;2 : 3 : 5$

Question 2

Simplify $40$ cm $: 1.2$ m.

▶ Show solution

$1.2$ m $= 120$ cm.

$40 : 120$; divide both by $40$: $\;1 : 3$

Question 3

Simplify $2.5 : 4$.

▶ Show solution

Multiply both by $2$ to clear the decimal: $5 : 8$.

(Multiplying by $10$ first gives $25 : 40$, which also simplifies to $5 : 8$ ✓)

Question 4

Simplify $\dfrac{3}{4} : \dfrac{5}{6}$.

▶ Show solution

LCM of $4$ and $6$ is $12$. Multiply both parts by $12$:

$\dfrac{3}{4} \times 12 = 9$ and $\dfrac{5}{6} \times 12 = 10$

Answer: $9 : 10$

Question 5

Write $12 : 30$ in the form (a) $1 : n$ and (b) $n : 1$.

▶ Show solution

(a) Divide by $12$: $\;1 : 2.5$

(b) Divide by $30$: $\;0.4 : 1$

Question 6

Complete the equivalent ratios: (a) $3 : 7 = 21 : \;?$ (b) $? : 44 = 5 : 11$.

▶ Show solution

(a) $21 \div 3 = 7$, so multiply $7$ by $7$: the answer is $49$. Ratio $= 21 : 49$.

(b) $44 \div 11 = 4$, so multiply $5$ by $4$: the answer is $20$. Ratio $= 20 : 44$.

Question 7

Two paints are mixed. Paint P uses white : blue in the ratio $5 : 2$. Paint Q uses white : blue in the ratio $7 : 3$. Which paint is a darker blue?

▶ Show solution

Write both as blue as a fraction of the total.

P: blue $= \dfrac{2}{7} = 0.2857$

Q: blue $= \dfrac{3}{10} = 0.3$

$0.3 > 0.2857$, so paint Q is darker.

Question 8

$X : Y = 2 : 5$ and $Y : Z = 3 : 4$. Find $X : Y : Z$ in its simplest form.

▶ Show solution

$Y$ is $5$ in the first ratio and $3$ in the second. LCM of $5$ and $3$ is $15$.

$X : Y = 2 : 5$, $\times 3$ → $6 : 15$

$Y : Z = 3 : 4$, $\times 5$ → $15 : 20$

$X : Y : Z = 6 : 15 : 20$ (no common factor).

Question 9

In a class the ratio of left-handed to right-handed students is $1 : 6$. There are $28$ students. Explain why $28$ is a possible class size, and find how many are left-handed.

▶ Show solution

Total parts $= 1 + 6 = 7$. The class size must be a multiple of $7$.

$28 = 7 \times 4$ ✓, so $28$ is possible.

One part $= 28 \div 7 = 4$.

Left-handed $= 1 \times 4 = 4$ students (and $24$ are right-handed).

Question 10

A concrete mix uses cement, sand and gravel in the ratio $1 : 2\frac{1}{2} : 3\frac{1}{2}$. Write this ratio using whole numbers in its simplest form, and state what fraction of the mix is gravel.

▶ Show solution

Write as improper fractions: $1 : \dfrac{5}{2} : \dfrac{7}{2}$.

Multiply every part by $2$: $\;2 : 5 : 7$.

No common factor, so simplest form is $\mathbf{2 : 5 : 7}$.

Total parts $= 2 + 5 + 7 = 14$, so gravel is $\dfrac{7}{14} = \dfrac{1}{2}$ of the mix.