Two situations are in proportion when their ratios are equal. That single idea is what lets you scale recipes, compare prices and solve for missing values.
The ratio apples : cost is $3 : 1.20$ in the first case and $6 : 2.40$ in the second.
Both simplify to $1 : 0.40$ — the ratios are equal, so the situations are in proportion.
When one of the four numbers is missing, the fastest reliable method is cross-multiplication.
- Write the proportion as two equal fractions, keeping like with like (e.g. cost on top of both sides, quantity underneath both).
- Multiply diagonally across the equals sign, both ways.
- Set the two products equal and solve the resulting equation.
- Sense-check the size of your answer.
Solve $\;4 : 7 = 20 : x$.
Check: $4 : 7 = 20 : 35$? Divide the second by $5$: $4 : 7$ ✓
A machine fills $150$ bottles in $8$ minutes. How long does it take to fill $525$ bottles?
Answer: $28$ minutes
Sense check: $525$ is $3.5$ times $150$, and $8 \times 3.5 = 28$ ✓
An alternative that many students find clearer: scale down to one, then scale up.
$7$ identical books weigh $1.96$ kg. What do $12$ such books weigh?
Answer: $3.36$ kg
A ratio table lays the information out so the multipliers are obvious. Whatever you do to one row, do to the other.
$5$ litres of paint cover $60\text{ m}^2$. How much paint is needed for $150\text{ m}^2$?
| Paint (litres) | $5$ | $1$ | $?$ |
|---|---|---|---|
| Area (m²) | $60$ | $12$ | $150$ |
Answer: $12.5$ litres
In practice you would buy $13$ litres!
"Which is better value?" questions are proportion questions. Make the comparison fair by reducing both options to the same base.
• Cost per unit — divide price by quantity. The smaller answer is better value.
• Quantity per pound — divide quantity by price. The larger answer is better value.
Either is fine, but say clearly which one you are using.
Which is better value: a $750$ g box of cereal for £$2.40$, or a $1.2$ kg box for £$3.66$?
The $1.2$ kg box is better value — by $1.5$p per $100$ g.
Shop A sells juice at £$1.80$ per carton with "buy 3 get 1 free". Shop B sells the same juice at £$1.50$ per carton with "buy 2 get the third half price". Which shop is cheaper?
Shop B is cheaper.
To test a table of values, work out $\dfrac{y}{x}$ for every pair. If the answer is the same each time, the quantities are in direct proportion.
Are $x$ and $y$ in direct proportion?
| $x$ | $2$ | $5$ | $8$ |
|---|---|---|---|
| $y$ | $7$ | $17.5$ | $28$ |
Yes — the ratio is constant, and $y = 3.5x$.
A taxi charges £$3$ plus £$2$ per mile. Are cost and distance in proportion?
| Miles | $1$ | $2$ | $5$ |
|---|---|---|---|
| Cost (£) | $5$ | $7$ | $13$ |
No, they are not in proportion — because of the fixed £$3$ charge. Doubling the distance does not double the cost.
The relationship is $C = 2m + 3$: a linear relationship, but not a proportional one (the graph does not pass through the origin).
Definition
In proportion means equal ratios: $a : b = c : d$.
Cross-multiplying
$\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc$. Keep like with like.
Unitary method
Divide to find one, then multiply up to the amount wanted.
Ratio tables
Do the same operation to both rows; work via $1$ if it helps.
Best buys
Compare cost per unit (smaller wins) or amount per £ (bigger wins).
Testing proportion
$\dfrac{y}{x}$ constant $\Rightarrow$ direct proportion. Any fixed charge breaks it.
Solve $\;3 : 8 = x : 40$.
▶ Show solution
$\dfrac{3}{8} = \dfrac{x}{40}$
Cross-multiply: $8x = 3 \times 40 = 120$
$x = 15$
Six identical chairs cost £$414$. How much do $11$ chairs cost?
▶ Show solution
One chair $= 414 \div 6 = £69$
Eleven chairs $= 69 \times 11 = £759$
A recipe for $9$ muffins uses $360$ g of flour. How many muffins can be made with $600$ g of flour?
▶ Show solution
$\dfrac{9}{360} = \dfrac{m}{600}$
Cross-multiply: $360m = 9 \times 600 = 5400$
$m = 15$ muffins
$4$ workers lay $180$ bricks in an hour. Working at the same rate, how many bricks would $7$ workers lay in an hour?
▶ Show solution
One worker lays $180 \div 4 = 45$ bricks per hour.
Seven workers lay $45 \times 7 = 315$ bricks per hour.
Which is better value: $12$ eggs for £$3.24$, or $18$ eggs for £$4.68$?
▶ Show solution
Cost per egg (pack of 12): $324 \div 12 = 27$p
Cost per egg (pack of 18): $468 \div 18 = 26$p
The pack of $18$ is better value (1p cheaper per egg).
Exchange rate: £$1$ = $$1.27$. Convert (a) £$350$ into dollars, (b) $$508$ into pounds.
▶ Show solution
(a) $350 \times 1.27 = 444.50$, so $$444.50$
(b) $508 \div 1.27 = 400$, so £$400$
A $2.5$ litre tin of paint covers $30\text{ m}^2$. A decorator must paint a wall $12$ m long and $2.8$ m high, giving it two coats. How many tins must she buy?
▶ Show solution
Area of wall $= 12 \times 2.8 = 33.6\text{ m}^2$
Two coats: $33.6 \times 2 = 67.2\text{ m}^2$
Tins needed $= 67.2 \div 30 = 2.24$
Round up: she must buy $3$ tins.
Decide whether $x$ and $y$ are in direct proportion. Justify your answer.
| $x$ | $3$ | $4$ | $9$ |
|---|---|---|---|
| $y$ | $10.5$ | $14$ | $31.5$ |
▶ Show solution
$10.5 \div 3 = 3.5$; $14 \div 4 = 3.5$; $31.5 \div 9 = 3.5$
Yes — the ratio $\dfrac{y}{x}$ is constant, so $y = 3.5x$.
Shop A sells $6$ tins of soup for £$4.20$. Shop B sells the same tins at $80$p each with an offer of "buy 3 get 1 free". Which shop gives the cheaper tin?
▶ Show solution
Shop A: $420 \div 6 = 70$p per tin.
Shop B: you pay for $3$ tins and take home $4$.
Cost of $4$ tins $= 3 \times 80 = 240$p.
Cost per tin $= 240 \div 4 = 60$p.
Shop B is cheaper at $60$p per tin, saving $10$p on every tin.
On a school trip the ratio of teachers to students must be at least $1 : 12$ (that is, no more than $12$ students per teacher).
(a) What is the smallest number of teachers needed for $150$ students?
(b) The trip costs £$18$ per student and £$0$ per teacher, plus a fixed coach hire of £$260$. Find the total cost.
▶ Show solution
(a) $150 \div 12 = 12.5$
You cannot have half a teacher, and $12$ teachers could only supervise $144$ students, so round up: $13$ teachers.
(b) Student cost $= 150 \times 18 = £2700$
Total $= 2700 + 260 = £2960$
Note: because of the fixed £$260$, the total cost is not in direct proportion to the number of students.