Two quantities are linked multiplicatively when you get from one to the other by multiplying (or dividing) — not by adding.
• Multiplier: $A = 2.5 \times B$ (because $15 \div 6 = 2.5$)
• Fraction: $A$ is $\dfrac{5}{2}$ of $B$
• Ratio: $A : B = 15 : 6 = 5 : 2$
Learning to slide between these three forms is the whole of this subtopic.
- Decide which quantity you are starting from and which you are going to.
- Divide: multiplier $= \dfrac{\text{the quantity you want}}{\text{the quantity you start from}}$.
- State it as a decimal or as a fraction, whichever is neater.
- Check by multiplying back.
A shop's takings rose from £$800$ on Monday to £$1000$ on Tuesday.
So Tuesday's takings are $1.25$ times Monday's, and Monday's are $0.8$ of Tuesday's.
A pencil is $18$ cm long and a pen is $12$ cm long. Express the pen's length as a multiple of the pencil's.
| Given | To a ratio | To a fraction | To a multiplier |
|---|---|---|---|
| $A : B = 3 : 4$ | — | $A = \dfrac{3}{4}B$ | $A = 0.75B$ |
| $A = \dfrac{5}{2}B$ | $A : B = 5 : 2$ | — | $A = 2.5B$ |
| $A = 1.6B$ | $A : B = 8 : 5$ | $A = \dfrac{8}{5}B$ | — |
| $A$ is $40\%$ of $B$ | $A : B = 2 : 5$ | $A = \dfrac{2}{5}B$ | $A = 0.4B$ |
$A = 1.6B$. Write $A : B$ in its simplest form.
Answer: $A : B = 8 : 5$
The ratio of Ella's savings to Fred's is $7 : 4$. Fred has £$260$. Find Ella's savings using a multiplier.
Check with the parts method: one part $= 260 \div 4 = 65$, so Ella $= 7 \times 65 = £455$ ✓
If you apply one multiplier and then another, the overall multiplier is found by multiplying them together.
A price is increased by $20\%$ and then reduced by $15\%$. Find the overall multiplier and describe the overall change.
An overall increase of $2\%$.
Note that this is not the same as $20 - 15 = 5\%$. Percentages must be combined by multiplying, never by adding.
$P$ is $\dfrac{3}{5}$ of $Q$, and $Q$ is $\dfrac{10}{9}$ of $R$. Express $P$ as a fraction of $R$.
$P = \dfrac{2}{3}R$, so $P : R = 2 : 3$.
To undo a multiplication you divide — equivalently, you multiply by the reciprocal.
| Forward multiplier | Reverse multiplier |
|---|---|
| $\times 4$ | $\div 4$ (that is, $\times 0.25$) |
| $\times 1.15$ | $\div 1.15$ ($\approx \times 0.8696$) |
| $\times \dfrac{3}{7}$ | $\times \dfrac{7}{3}$ |
| $\times 0.8$ | $\times 1.25$ |
After a $35\%$ discount, a bike costs £$266.50$. What was the original price?
Check: $410 \times 0.65 = 266.50$ ✓
Exam questions increasingly ask you to write the link as an equation, because that is what you need for proportion questions later.
"There are $5$ times as many sheep as cows." Let $s$ be the number of sheep and $c$ the number of cows. Write an equation.
$a : b = 4 : 3$ and $b : c = 9 : 2$. Write $a$ in terms of $c$.
$a = 6c$, so $a : c = 6 : 1$.
Multiplier
$k = \dfrac{\text{quantity you want}}{\text{quantity you start from}}$.
Bigger or smaller?
$k > 1$ means growth; $0 < k < 1$ means shrinkage.
Ratio → fraction
$A : B = a : b$ means $A = \dfrac{a}{b}B$.
Decimal → ratio
Write $k : 1$, then clear the decimal and simplify.
Chains
Multiply the multipliers together; never add them.
Reversing
Divide by the multiplier — the reciprocal undoes it.
Algebra
"$A$ is $n$ times $B$" $\Rightarrow A = nB$. Always test with real numbers.
Percentages
$p\%$ of $B$ means $\dfrac{p}{100} \times B$ — just another multiplier.
$A = 24$ and $B = 15$. Write (a) the multiplier from $B$ to $A$, (b) $A$ as a fraction of $B$, (c) $A : B$ in its simplest form.
▶ Show solution
(a) $24 \div 15 = 1.6$
(b) $\dfrac{24}{15} = \dfrac{8}{5}$
(c) $24 : 15 = 8 : 5$ (divide both by $3$)
The number of visitors to a museum fell from $4500$ to $3600$. Find the multiplier, and state the percentage change.
▶ Show solution
Multiplier $= 3600 \div 4500 = 0.8$
$0.8 = 80\%$, so visitors are now $80\%$ of what they were.
That is a decrease of $100 - 80 = 20\%$.
$P = 0.35Q$. Write $P : Q$ in its simplest form.
▶ Show solution
$P : Q = 0.35 : 1$
Multiply both by $100$: $35 : 100$
Divide by the HCF $5$: $\;7 : 20$
The ratio of Jaz's height to Kim's height is $6 : 5$. Kim is $145$ cm tall. Find Jaz's height.
▶ Show solution
Jaz $= \dfrac{6}{5} \times$ Kim $= 1.2 \times 145$
$= 174$ cm
A quantity is multiplied by $1.5$ and then by $0.6$. Find the single multiplier that has the same effect, and say whether the quantity has grown or shrunk.
▶ Show solution
$1.5 \times 0.6 = 0.9$
Since $0.9 < 1$, the quantity has shrunk — it is now $90\%$ of the original, a $10\%$ decrease.
$X$ is $\dfrac{4}{7}$ of $Y$, and $Y$ is $\dfrac{7}{2}$ of $Z$. Express $X$ in terms of $Z$.
▶ Show solution
$X = \dfrac{4}{7} \times \dfrac{7}{2} \times Z = \dfrac{28}{14}Z = 2Z$
$X = 2Z$, so $X : Z = 2 : 1$.
After a $12\%$ increase, a salary is £$33\,600$. Find the salary before the increase.
▶ Show solution
Forward multiplier $= 1.12$.
Original $= 33\,600 \div 1.12 = £30\,000$
Check: $30\,000 \times 1.12 = 33\,600$ ✓
There are $\dfrac{3}{8}$ as many buses as cars in a depot. Write an equation linking the number of buses $b$ and the number of cars $c$, and find $c$ when $b = 21$.
▶ Show solution
$b = \dfrac{3}{8}c$
When $b = 21$: $\;21 = \dfrac{3}{8}c$
Multiply both sides by $8$: $168 = 3c$, so $c = 56$ cars.
A shop reduces all prices by $30\%$, then adds $20\%$ VAT to the reduced price. A jacket was originally £$95$ before VAT. Find the final price and the overall multiplier.
▶ Show solution
Overall multiplier $= 0.7 \times 1.2 = 0.84$
Final price $= 95 \times 0.84 = £79.80$
The overall effect is a $16\%$ decrease on the original.
$m : n = 5 : 2$ and $n : p = 6 : 7$.
(a) Write $m$ in terms of $p$. (b) If $p = 42$, find $m$ and $n$.
▶ Show solution
(a) $m = \dfrac{5}{2}n$ and $n = \dfrac{6}{7}p$.
$m = \dfrac{5}{2} \times \dfrac{6}{7} \times p = \dfrac{30}{14}p = \dfrac{15}{7}p$
(b) $n = \dfrac{6}{7} \times 42 = 36$
$m = \dfrac{15}{7} \times 42 = 90$
Check: $m : n = 90 : 36 = 5 : 2$ ✓ and $n : p = 36 : 42 = 6 : 7$ ✓