3 Ratios and Proportions

Understanding ratios

A ratio compares two quantities. For example, the ratio of $8$ apples to $12$ oranges can be written as $8:12$, “8 to 12,” or $\frac{8}{12}$. Keep the order clear: apples to oranges is not the same comparison as oranges to apples.

A ratio can compare two parts of a group, such as apples to oranges, or a part to the whole, such as apples to all the fruit. Ratios and rates are usually written as fractions when calculating.

Comparing and simplifying ratios

To compare ratios, first ensure that they describe quantities in the same order and use compatible units. Then simplify the ratios or convert them to fractions with a common denominator or to decimals.

To simplify a ratio, divide both terms by the same positive common factor. For example, dividing both terms of $18:24$ by $6$ gives $3:4$. The simplified ratio is equivalent to the original and expresses the same comparison. If quantities have different units, convert them to the same unit before simplifying: $1\text{ m}:50\text{ cm}=100\text{ cm}:50\text{ cm}=2:1$.

Equivalent ratios and proportions

Equivalent ratios have the same value. Multiply or divide both terms by the same nonzero number to make an equivalent ratio; for example, $3:5=6:10=9:15$.

A proportion is an equation stating that two ratios are equivalent. In $\frac{3}{5}=\frac{x}{20}$, the denominator $5$ is multiplied by $4$ to get $20$, so multiply the numerator $3$ by $4$ as well. This gives $x=12$.

Solving by cross-multiplication

A proportion can also be solved by cross-multiplying. If

then

For the example, $\frac{3}{5}=\frac{x}{20}$ gives $3(20)=5x$, so $x=12$. Check the result by confirming that the ratios are equal: $\frac{12}{20}=\frac{3}{5}$. Scaling equivalent ratios and cross-multiplication are both ways to solve proportions.

Rates and unit rates

A rate compares quantities with different units, such as miles and hours. A unit rate gives the amount for one unit of the second quantity. Divide the first quantity by the second and include the units: $\frac{150\text{ miles}}{3\text{ hours}}=50\text{ miles per hour}$.

Unit rates help compare rates. A $12$-ounce cereal box for $\$3$ costs $\$3\div12=\$0.25$ per ounce. A $20$-ounce box for $\$4$ costs $\$4\div20=\$0.20$ per ounce, so the second box costs less per ounce.

A unit rate can also be used to solve a proportion. If a recipe uses $4$ cups of flour for $10$ servings, the rate is $0.4$ cup per serving. For $25$ servings, calculate $0.4\times25=10$ cups. Keep units attached to the quantities so the calculation answers the question asked.