Reducing and expanding a ratio

A ratio can also be expanded: multiply both numbers by the same number. So $\displaystyle {3:4}={6:8}$, just as $\displaystyle {\frac{3}{4}}={\frac{6}{8}}$. Here are two strips – one of $\displaystyle 3:4$, and one of $\displaystyle 6:8$:

The two strips are the same length, and the blue part is the same length in both. In the second strip each square was cut in two – but the parts did not change. So $\displaystyle 3:4$ and $\displaystyle 6:8$ are equal ratios, and $\displaystyle 9:12$ and $\displaystyle 30:40$ are equal to them too.

• Expanding a ratio: multiply both numbers by the same number – $\displaystyle {3:4}={9:12}$.
• Reducing a ratio: divide both numbers by the same number – $\displaystyle {30:40}={3:4}$.
• A ratio that cannot be reduced any further is a reduced ratio, like $\displaystyle 3:4$.

A ratio with decimals is first expanded until it holds whole numbers only:

• $\displaystyle {1.5:2}={3:4}$ – expanded by 2.
• $\displaystyle {0.6:0.9}={6:9}={2:3}$ – expanded by 10, then reduced by 3.

Write as a reduced ratio.

Masada School painted its classrooms in a colour mixed from 10 cans of blue paint and 6 cans of white paint.
Negohot School wants the same colour, and has 20 cans of blue paint.
How many cans of white paint does it need?

For the same colour, the ratio of blue to white must stay $\displaystyle 10:6$. 20 is $\displaystyle 10\times2$, so we expand the ratio by 2: $\displaystyle {10:6}={20:12}$. It needs 12 cans of white paint.

Alon School also wants its classrooms in the same colour as Masada’s.
It has 9 cans of white paint.
How many cans of blue paint does it need?

An onion soup recipe for 8 people:

• 8 onions;
• 2 litres of water;
• 4 soup cubes;
• 20 grams of butter;
• half a tablespoon of salt.

Change the amounts in the recipe for 12 people.

In a chess club the ratio of boys to girls is $\displaystyle 2:3$.
The club has 12 girls.
How many boys are in the club?