Proportional Relationships
Double a recipe and you double every ingredient; drive twice as long at the same speed and
you cover twice the distance; buy three times the petrol and you pay three times as much.
Whenever two amounts grow in lockstep like this, they share a proportional
relationship — one of the most common patterns in shopping, cooking, science and
money.
Two quantities have a proportional relationship when their ratio
stays the same no matter how big or small they get. Whatever pair of
values x and y you pick, dividing
one by the other gives the very same number every time:
\frac{y}{x} = k \quad\text{(always the same } k\text{)}
That fixed number k is the constant ratio. It
is exactly the unit
rate — the amount of y for one unit
of x. Once the ratio is locked, the two quantities move together:
triple x and y triples too, because the
ratio has to stay put.
A food stall charges the same for every taco. Check three different orders and work out
the ratio £\div tacos each time:
£6 · 4 = £12 · 6 = £18
6 \div 2 = 3, 12 \div 4 = 3,
18 \div 6 = 3. Every order gives the same ratio
k = 3 pounds per taco, so the price and the number of tacos
are proportional. That single number 3 prices
any order.
Two ways to spot it
A proportional relationship shows itself in two matching ways — one in a
table, one on a graph:
- Table test: divide y \div x in every row.
If you get the same answer each time, it is proportional.
- Graph test: plot the points. If they form a straight line
that passes through the origin (0, 0), it is
proportional.
Worked example. A table pairs hours worked with pay: 2 \to 16,
3 \to 24, 5 \to 40. Testing:
16 \div 2 = 8, 24 \div 3 = 8,
40 \div 5 = 8. The ratio is always 8,
so pay is proportional to hours, at £8 per hour.
Worked example. Another table: 1 \to 5,
2 \to 8, 3 \to 11. Testing:
5 \div 1 = 5 but 8 \div 2 = 4 —
the ratios disagree, so this is not proportional. (Look
closely: it climbs by 3 each step but starts at
2, so it is y = 3x + 2.)
A hot-air balloon rises a steady 3 metres every second — but it launches from a
2-metre platform. So "more seconds means more height", yet it is
not proportional: at 0 seconds the height is
2, not 0.
height = 3t + 2
The ratio height \div time keeps changing
(5 \div 1 = 5, then 8 \div 2 = 4,
then 11 \div 3 \approx 3.7), and its graph is a straight line
that misses the origin. Growing together is not enough — the ratio must be
constant.
Quantities x and y are proportional
when:
- the ratio y \div x = k is the same for
every pair — this constant k is the unit rate;
- they are linked by y = kx;
- the graph is a straight line through the origin
(0, 0).
See it: straight, and through the origin
Below is a random proportional line y = kx. Press
Play to reveal it step by step: first the line, then a couple of table
points marked on it. Every marked point has the same ratio
y \div x = k, and the line drives straight through the
origin. Press Refresh for a new
k.
Drive the ratio yourself
Pull the slider to change the constant ratio k. However you
set it, the line stays straight and stays pinned to the origin — a bigger
k just tilts it steeper. That is what "proportional" looks
like — the same idea as direct
proportion. Later, that multiplier k gets its
own name: the constant of
proportionality.
The classic trap
The most common mistake is thinking any "more means more" pairing is
proportional. A phone plan that charges £5 to join plus £2 per month costs more the longer
you stay — but it is not proportional, because month 0
already costs £5. Always run the ratio test and check the origin.
Two traps to avoid:
- "More of one means more of the other" is NOT enough. A straight line
with a starting amount (like y = x + 2) also grows together,
yet its ratio y \div x keeps changing — so it is
not proportional.
- Proportional means the graph is straight AND passes through
(0, 0). If there is a joining fee, flag-fall, or any
fixed starting amount, the line misses the origin and the relationship is not
proportional.
See it explained