When two quantities are in a
proportional relationship,
the ratio between them never changes. Divide y by
x for any matching pair and you always get the
same number. That number is so important it has its own name: the
constant of proportionality, written k.
k = \dfrac{y}{x} \qquad\Longrightarrow\qquad y = kx
Once you know k, the whole relationship is captured in one tidy
equation, y = kx. Feed in any x and it
hands you the matching y. The constant
k is exactly the
unit rate
— the amount of y for one unit of
x.
A takeaway sells identical pizzas at a fixed price. The cost and the number of pizzas are
proportional, so cost divided by number is always the same:
= £9
Three pizzas cost £9, so k = 9 \div 3 = 3 pounds per pizza. That
3 is the constant of proportionality and the price of a single pizza — the unit
rate. The equation is y = 3x, so 7 pizzas cost
3 \times 7 = 21 pounds. One little number prices any order.
Find k, then predict
The constant of proportionality can be dug out of a table, a
graph, an equation, or a sentence of
words. In every case the recipe is the same: divide
y by x for one matching pair to get
k, then use y = kx for anything else.
From a table. A car uses fuel at a steady rate:
\begin{array}{c|c|c|c} \text{litres } (x) & 2 & 5 & 8 \\ \hline \text{km } (y) & 30 & 75 & 120 \end{array}
Check any column: 30 \div 2 = 15,
75 \div 5 = 15,
120 \div 8 = 15. The same k = 15 every
time confirms it is proportional, so y = 15x and 10 litres take the
car 15 \times 10 = 150 km.
From words. "A recipe needs 4 eggs for every 2 cakes." Here
k = 4 \div 2 = 2 eggs per cake, so
y = 2x and 9 cakes need
2 \times 9 = 18 eggs.
From an equation. If you are simply handed
y = 6x, the constant is staring at you: k = 6.
The number multiplying x is the constant of proportionality.
Whatever the story — pizzas at £3 each, a car going 15 km per litre, 2 eggs per cake —
the shape is always y = kx. Learn to spot the constant and every
"how much for that many?" question becomes a single multiplication.
