Ratios

Ratio problems typically tell you the size of a group compared to the size of another group, or compared to the total. Often you’ll need to solve for a missing piece of information or figure out the relationship between the groups.

Cross multiplication

When you’re working with two groups and only one value is given, cross multiplication is easily the best strategy. For example:

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

To solve this, we can rewrite the problem as two fractions and cross multiply.

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

The variables $c$ and $p$ stand for the number of cups and the number of plates. We build the equation just like translating the ratio from numbers to words, with a ratio of $2:3$ (two to three) turning into $2/3$, and $c:p$ (cups to plates) turning into $c/p$. Then we can swap out the $p$ value with the actual number of plates we’re given, and solve for the missing $c$ value.

By cross multiplying, we get $24=3c$.

After dividing both sides by $3$, we find that the number of cups $c$ is $8$.

Making equations

If only the ratio and totals are given, it’s best to write out equations that represent the situation and solve for the variable needed. 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 $3c=2p$

We can isolate $c$ in the first equation:

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

And then we can use that result and plug it into the other equation. This will leave us 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 we’ve found there are a total of 12 plates.

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: