Writing and Using Ratios
A ratio is a comparison of quantities. For example, for most mammals, the ratio of legs to noses is \(4:1,\) but for humans, the ratio of legs to noses is \(2:1.\)
Writing Ratios
Ratios can be written using the word "to," a colon, or a fraction.
For example, in a group of 3 girls and 5 boys, the ratio of girls to boys can be written as \(3 \text{ to } 5, \) \(3:5,\) or \(\dfrac{3}{5}.\)
A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of kiwi to apples?
There are 9 kiwi and 3 apples, so the ratio of kiwi to apples is \(9:3,\) or in simplified form, \(3:1.\)
A bowl contains 3 apples, 5 oranges, and 9 kiwi. What is the ratio of oranges to total pieces of fruit?
There are 5 oranges and \(3+5+9=17\) total pieces of fruit, so the ratio of oranges to total pieces of fruit is \(5:17.\)
Using Ratios to Problem Solve
Ratios are often easiest to use if written in fraction form. To find an unknown value in a ratio, we can use two equivalent ratios.
For example, if the ratio \(3:4\) is equivalent to the ratio \(18:x,\) then \(\frac{3}{4}=\frac{18}{x}\) and \(x=24\) because the values in the fraction on the right are six times greater than the corresponding values in the fraction on the left.
If \( 6: 15 = 10 : x, \) what is \(x\,?\)
Expressing the ratios as fractions, we get \( \frac{6}{15} = \frac{10}{x} .\)
The fraction \(\frac{6}{15}\) simplifies to \(\frac{2}{5},\) so now we have \( \frac{2}{5} = \frac{10}{x} .\) The values in the ratio on the right are five times greater than the values in the ratio on the left, so \(x=25.\)
We can also rewrite the ratios as fractions and cross multiply to solve. \[\begin{align} \frac{6}{15} &= \frac{10}{ x} \\ 6x &= (10)(15) \\ 6x &= 150 \\ x &= 25.\end{align}\]
If Calvin paid $5 for 7 pencils, how much would he pay for 56 pencils?
Let \(x\) be the price of \(56\) pencils. Since the price of a single pencil does not change, we have \[\begin{align} \frac{\$5}{7} &= \frac{x}{56} \\ 7x &= \$5 \times 56 \\ x &= \$40. \end{align}\] Hence, Calvin would pay \(\$40\) for \(56\) pencils.
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, we get \( 5b = 4(75) \), 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. \( _\square \)