Level 3 · Ages 11–13
Read it again any time, including while you work through the exercises.
You already know how to spot a proportional situation: when you buy twice as many croissants, you pay twice as much. In a proportional table, one single cell is often missing: this mystery number is called the fourth proportional, because it completes the three numbers already known.
To find it, there is a powerful tool: cross multiplication. Here is the idea: in a proportional table with 4 cells, the diagonal products are equal. If the table holds a and b on the first row, c and d on the second, then a × d = b × c. Mentally draw the two diagonals: they form a cross, hence the name!
The method takes two moves: I multiply the two numbers on the complete diagonal, then I divide by the remaining number. For example, if 4 croissants cost 6 euros, how much do 10 croissants cost? I compute 10 × 6 = 60, then 60 ÷ 4 = 15 euros. Fast, reliable, elegant.
One final pro warning: cross multiplication ONLY works when the situation is truly proportional. A person's age and height, for instance, are not proportional (at age 20, you are not twice as tall as at age 10!). Before pulling out your cross product, always check that doubling one quantity really doubles the other.
4 croissants cost 6 euros. How much do 10 croissants cost?
4 × x = 10 × 6
I write the cross product: the diagonal products of the table are equal.
4 × x = 60
I compute the complete diagonal: 10 × 6 = 60.
x = 60 ÷ 4 = 15
I divide by the remaining number, 4, to isolate x.
Result: 10 croissants cost 15 euros.
Find the fourth proportional x such that 3/5 = x/20.
5 × x = 3 × 20
Two equal fractions mean a cross product: I cross the terms.
5 × x = 60
I compute 3 × 20 = 60.
x = 60 ÷ 5 = 12
I divide by 5. Check: 12/20 = 3/5, consistent since 20 = 5 × 4 and 12 = 3 × 4.
Result: x = 12, because 3/5 = 12/20.