What is proportionality?
Two quantities are proportional when they increase (or decrease) always in the same ratio. Example: the price of a bag of sweets is proportional to the number of bags bought — 2 bags cost twice as much as 1 bag, 3 bags cost three times as much, and so on.
It's not always the case! A person's age isn't proportional to their height (we don't keep growing forever), and the price of a taxi ride isn't always proportional to the distance (there's often a fixed starting fee).
The ratio table
To check for proportionality, we organise the data in a two-row table: one row per quantity.
| Number of bags | 1 | 2 | 5 |
|---|---|---|---|
| Price in € | 3 | 6 | 15 |
The price is proportional to the number of bags: each column, divided, gives 3
To check that the table IS proportional, divide the second row by the first, column by column: if you get the same number every time, it's proportional. Here: 3÷1=3, 6÷2=3, 15÷5=3 → the table is indeed proportional.
The scale factor
This single number that keeps showing up (here, 3) is called the scale factor (or constant of proportionality). To find any missing value, simply multiply by this factor.
Cross-multiplication
When the scale factor isn't a nice round number (easy to work out in your head), we use cross-multiplication: multiply diagonally the two numbers that correspond to each other, then divide by the remaining number.
Example: 4 pens cost €6. How much do 10 pens cost?
| Pens | 4 | 10 |
|---|---|---|
| Price in € | 6 | ? |
- Multiply diagonally the two known numbers that don't line up: 6 × 10 = 60.
- Divide by the remaining number (the one that lines up with the unknown): 60 ÷ 4 = €15.
Mistakes to avoid
- Applying proportionality everywhere, even when the situation doesn't fit (a plan with a fixed starting price, for instance).
- Multiplying the wrong diagonal in cross-multiplication.
- Forgetting to check the result against the scale factor for consistency.
- Confusing proportionality with a simple increase: a quantity that goes up isn't necessarily proportional to another.
Worked exercises
3 kg of apples cost €6. How much do 5 kg cost?
A recipe for 4 people uses 200 g of flour. How much for 6 people?
Is this table proportional? 2→5, 4→9
Now it's your turn!
On EasyMaths, your child practises recognising and solving proportional situations, for free.
Start the exercises →Frequently asked questions
How do you recognise a proportional relationship?
Two quantities are proportional when multiplying one by a number multiplies the other by the same number. In a table, check that every column gives the same result when you divide the second row by the first.
What is cross-multiplication?
It's a method for finding a missing number in a 4-cell ratio table: multiply diagonally the two known numbers that don't line up, then divide by the fourth number.
What age do children learn proportionality?
The first proportional situations (doubling, tripling) are met from around age 9. The ratio table and scale factor are learned around age 10-11, and cross-multiplication is formalised around age 11-12.