Direct proportionality: a multiplicative relation — Mathematics, 11–13
Two quantities are directly proportional when multiplying one by a number multiplies the other by the same number, so one known pair unlocks all the others.
Multiply one, multiply the other
Some quantities travel together: the more paint, the bigger the wall you can cover. They are directly proportional when multiplying one by a number multiplies the other by the same number. If 3 litres of paint cover 12 m², then 6 litres (3 × 2) cover 12 × 2 = 24 m², and 9 litres (3 × 3) cover 12 × 3 = 36 m². Dividing works the same way: 1 litre (3 ÷ 3) covers 12 ÷ 3 = 4 m². The relation is multiplicative: what counts is doing the same multiplying, or dividing, on both sides, not the same adding.
One pair opens them all, but not every pair is proportional
With direct proportionality, one known pair gives you every other pair, in a text, in a table or on a graph, where each point is a pair of values you can read and test. But not everything that grows together is proportional. Ana is 10 and her brother is 12; when Ana is 20 (twice as old), he is 22, not 24: their ages are linked by adding, not by multiplying. A laundry cycle lasts 45 minutes whether the machine holds 2 shirts or 6: doubling the shirts does not double the time.
Litres of paint and square metres
Table: 3 L → 12 m², 6 L → 24 m², 9 L → 36 m². How much do 15 L cover? Look at the litres: 15 = 3 × 5, so the area is 12 × 5 = 60 m². That compares two values of the same quantity. Another route goes through 1 litre: 12 ÷ 3 = 4 m² per litre, so 7 litres cover 7 × 4 = 28 m², and 15 litres cover 15 × 4 = 60 m², the same answer. Test the table: 6 L and 24 m² are double 3 L and 12 m², and 9 L and 36 m² are triple. It is directly proportional.
Adding the same amount is not proportional
A juice recipe uses 2 cups of concentrate and 3 cups of water. How much water for 6 cups of concentrate? Many people see that the concentrate went up by 4 (from 2 to 6) and add 4 to the water too: 3 + 4 = 7. It is a reasonable idea, but the taste changes. In a proportion you do the same multiplication on both sides: 6 = 2 × 3, so the water is 3 × 3 = 9 cups. Check: 2 cups of concentrate for 3 of water and 6 for 9 are the same mix, because both were multiplied by 3.
Recipes, steady pace and equal prices
Direct proportionality is at work when you scale a recipe for more people, when a cyclist keeps the same pace (if 3 km take 10 minutes, 9 km take 30 minutes) or when you buy several items that all cost the same. It only holds while the rate stays the same: if the cyclist gets tired, or a shop gives a discount for many items, the table stops being proportional. Before you multiply, ask whether the relation really is multiplicative.
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