Ratios

Ratio problems compare the size of one group to another group, or to the total. You’ll often use the ratio to find a missing value or to describe how the groups are related.

Cross multiplication

When you’re comparing two groups and you know one actual value, cross multiplication is usually the most direct strategy. For example:

If the ratio of cups to plates is 2 to 3, and there are 12 plates, how many cups are there?

Start by writing the ratio as two fractions and then cross multiply.

$$\begin{array}{rl}2/3& =c/p\\ 2/3& =c/12\\ 2\times 12& =c\times 3\\ 24& =3c\\ 8& =c\end{array}$$

Here, $c$ and $p$ stand for the number of cups and the number of plates. You set up the equation by matching the order in the ratio:

• $2:3$ (cups to plates) becomes $2/3$
• $c:p$ (cups to plates) becomes $c/p$

Then you replace $p$ with the given number of plates ($12$) and solve for $c$.

Cross multiplying gives $24=3c$.

Dividing both sides by $3$ shows that the number of cups $c$ is $8$.

Making equations

If you’re given a ratio and a total, it’s often best to write equations that represent the situation and solve for the variable you need. For example:

If there are 2 cups for every 3 plates, and there are a total of 20 items, how many plates are there?

The equations are:

• Total of 20 items means $c+p=20$
• 2 cups for every 3 plates means $3c=2p$

First, isolate $c$ in the total equation:

$$\begin{array}{rl}c+p& =20\\ c& =20-p\end{array}$$

Now substitute that expression for $c$ into the ratio equation. This leaves you with only $p$ to solve for:

$$\begin{array}{rl}3c& =2p\\ 3\times (20-p)& =2p\\ 60-3p& =2p\\ 60& =5p\\ 12& =p\end{array}$$

So there are a total of 12 plates.

Bringing it all together: question walkthrough video

Here’s a video that walks through one of the practice questions and shows these ideas in action: