Ratio, proportion and constant of proportionality — Mathematics, 11–13
A ratio compares two quantities with a fraction, a proportion says two ratios are equal, and the constant of proportionality is the ratio that never changes.
A fraction that compares two parts
A ratio compares two quantities by division, and it is written as a fraction. In a class with 12 girls and 8 boys, the ratio of girls to boys is 12/8 = 3/2: for every 3 girls there are 2 boys. Both are parts of the same whole, the class. A proportion says that two ratios are equal, like 12/8 = 6/4. In a direct proportionality the ratio between corresponding values is always the same number, the constant of proportionality k, so y = k × x.
Comparing without being fooled by size
Ratios let you compare situations of different sizes. Luís puts 6 strawberries in 30 cl of milk, and Rosa puts 9 in 40 cl. Rosa has more strawberries, but also more milk, so counting them proves nothing. Compare the ratios of strawberries to milk: 6/30 = 1/5 = 8/40 for Luís and 9/40 for Rosa. With the same denominator, 9/40 is more than 8/40, so Rosa's smoothie tastes more of strawberry. Enlarging or reducing a ratio (1/5 to 8/40) keeps its value, which is what makes the comparison fair.
The constant and the expression
A tap runs at a steady rate. In 2 minutes it delivers 30 L, in 5 minutes 75 L, in 8 minutes 120 L. The ratios are 30/2 = 15, 75/5 = 15 and 120/8 = 15, so 30/2 = 75/5 is a proportion and the constant of proportionality is 15: every minute, 15 litres come out. So litres = 15 × minutes, or y = 15 × x. In 12 minutes: 15 × 12 = 180 L. How long for 270 L? 270 ÷ 15 = 18 minutes. In the expression, x is the minutes, y the litres and 15 the litres per minute.
Part to part is not part to whole
A basket has 4 apples and 6 pears. The ratio of apples to pears is 4/6 = 2/3: two parts of the same basket compared with each other. Many people use this fraction to say what share of the fruit is apples, but a share compares the apples with the whole basket: 4 out of 4 + 6 = 10 pieces, so 4/10 = 2/5. The order matters too: pears to apples is 6/4 = 3/2. Before you write a fraction, say what you are comparing with what.
Rates, speeds and circles
The constant of proportionality hides in everyday rates: litres per minute, kilometres per hour, price per kilogram. In geometry the perimeter of a circle is P = π × d: P and the diameter d depend on each other, and π (about 3.14) is the constant of proportionality, the same for every circle. A circle with d = 10 cm has a perimeter of about 3.14 × 10 = 31.4 cm. Whenever you see “per”, a constant of proportionality may be hiding there.
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