You already compare amounts with words like "three times as many" (see Times as Many). A ratio is another way to compare two amounts. It tells how much of one thing there is for every so much of another thing.
Key idea
The ratio $a : b$, read "$a$ to $b$", means: for every $a$ of the first thing, there are $b$ of the second thing. The order follows the words, so the thing named first is written first.
Why it works
Why not just say "there is 1 more cup of flour than milk"? Make a double batch: 6 cups of flour and 4 cups of milk. Now the difference is 2 cups, but the batter tastes exactly the same. What stayed the same is the mix: for every 3 cups of flour there are still 2 cups of milk. A ratio describes the mix, so it does not change when the batch grows.
Part to part, and part to whole
A bag holds 4 red marbles and 6 blue marbles. Some ratios compare one part with another part; others compare a part with the whole bag of 10.
| Comparison | Ratio | Kind |
|---|---|---|
| red to blue | $4 : 6$ | part to part |
| blue to red | $6 : 4$ | part to part |
| red to all marbles | $4 : 10$ | part to whole |
| blue to all marbles | $6 : 10$ | part to whole |
A part-to-whole ratio is the same as a fraction of the whole: $\frac{4}{10}$ of the marbles are red. You can also write a ratio in simplest form: $4 : 6$ and $2 : 3$ describe the same mix.
Worked example: Read the order from the words
A fruit bowl has 5 apples and 3 bananas. Write the ratio of bananas to apples.
- The words say bananas first, then apples.
- There are 3 bananas and 5 apples.
Answer: $3 : 5$, read "3 to 5".
Worked example: From a ratio to a fraction
In a club, the ratio of girls to boys is $4 : 5$. What fraction of the club are girls?
- The ratio says: for every 4 girls there are 5 boys.
- So each group of $4 + 5 = 9$ members has 4 girls.
- Girls are 4 out of every 9 members.
Answer: $\frac{4}{9}$ of the club are girls.
Common mistake
Two slips happen a lot. One is writing the numbers in the wrong order: "cats to dogs" must start with the cats. The other is mixing up part-to-part and part-to-whole: in the club above, $\frac{4}{5}$ compares girls with boys, not with the whole club. Quick check: the fractions for all the parts must add to 1. Here $\frac{4}{9} + \frac{5}{9} = 1$, but $\frac{4}{5} + \frac{5}{4}$ is much more than 1.