A *ratio* is a relationship between two numbers (or two objects) that defines the quantity of the first in comparison to the second. For example, for most mammals the ratio of legs to noses is $4:1$, but for humans, the ratio of legs to noses is $2:1$. Ratios can also be written in fractional form, so comparing three boys with five girls could be written $3:5$ or $\frac{3}{5}$.

Be careful, however; we often think of fractions in the form $\frac{\text{part}}{\text{whole}}$, but a ratio is not necessarily comparing a part to a whole. In the example above, the denominator, 5, represents girls, and if we wanted to compare boys to the **total** number of people, we would need the ratio $3:8 = \frac{3}{8}$.

When two objects are *proportional*, it means that their ratios are equal. Specifically, two ratios, $a:b$ and $c:d$, are "in the same proportion" or "proportional" if and only if:

$a:b :: c:d \implies \frac{a}{b}=\frac{c}{d} \implies ad = bc$

## The ratio of Alice's pay to Bob's pay is $\frac{5}{4}$. The ratio of Bob's pay to Charlie's pay is $10:9$. If Alice is paid $75, how much is Charlie paid?

Since the ratio of Alice's pay to Bob's pay is $5:4$, Bob's pay must be $b$, where $\frac{5}{4}=\frac{75}{b}$. Cross-multiplying by the denominators, $5b = 4(75)$, so $b = 60$.

Continuing in the same way, we compare Bob to Charlie: $\frac{10}{9}=\frac{60}{c} \implies 10c = 9(60) \implies c = 54$. Thus, Charlie is paid $54. $_\square$

## The total number of vegetables is 158. If the ratio of cucumbers to carrots is $4:7$, and the ratio of cucumbers to radishes is $\frac{16}{35}$, how many more radishes are there than carrots?

The ratio of cucumbers to carrots is $4:7 = 16:28$. The ratio of cucumbers to radishes is $16:35$. Thus the ratio of cucumbers to carrots to radishes is $16: 28: 35$.

Let $16x$, $28x$, and $35x$ be the number of cucumbers, carrots, and radishes, respectively. Then, since the total is 158:

$16x + 28x + 35x = 79x = 158$

Thus, $x = 2$ and there are $2(35) = 70$ radishes and $2(28) = 56$ carrots. Therefore, there are 14 more radishes than carrots. $_\square$

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