Lesson 6: Ratios and Rate

New Concept

This lesson introduces ratios and rates—fractions for comparing quantities. You'll learn to write ratios for same-unit comparisons and rates for different-unit comparisons, then use them to find unit rates and compare prices.

What’s next

Next, you’ll tackle interactive examples for writing ratios and rates. Then, practice cards will challenge you to find unit prices and spot the best buy.

Property

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$, $\frac{a}{b}$, or $a:b$. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number.

Examples

• The ratio 20 to 36 is written as the fraction $\frac{20}{36}$, which simplifies to $\frac{5}{9}$.

• The ratio 45 to 18 is written as $\frac{45}{18}$, which simplifies to the improper fraction $\frac{5}{2}$. We keep it this way to see the two parts of the ratio.

Property

To handle ratios with decimals, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

Examples

• The ratio 4.8 to 11.2 becomes the fraction $\frac{4.8}{11.2}$. Move the decimal one place in both to get $\frac{48}{112}$, which simplifies to $\frac{3}{7}$.

• For the ratio 2.7 to 0.54, write $\frac{2.7}{0.54}$. Move the decimal two places to the right (adding a zero to 2.7) to get $\frac{270}{54}$, which simplifies to $\frac{5}{1}$.

Property

A rate compares two quantities of different units. A rate is usually written as a fraction. When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain.

Examples

• If Bob drove his car 525 miles in 9 hours, the rate is written as $\frac{525 \text{ miles}}{9 \text{ hours}}$, which simplifies to $\frac{175 \text{ miles}}{3 \text{ hours}}$.

• A price of 595 dollars for 40 hours of work is the rate $\frac{595 \text{ dollars}}{40 \text{ hours}}$.

Property

A unit rate is a rate with a denominator of 1 unit. To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of 1.

Examples

• If Anita was paid 384 dollars for 32 hours, her hourly (unit) pay rate is $384 \div 32 = 12$ dollars per hour.

• If a car travels 455 miles using 14 gallons of gasoline, its unit rate is $455 \div 14 = 32.5$ miles per gallon (mpg).

Property

A unit price is a unit rate that gives the price of one item. To find the unit price, divide the total price by the number of items. The better buy is the item with the lower unit price.

Examples

• If a case of 24 bottles of water costs 3.99 dollars, the unit price is $3.99 \div 24 \approx 0.17$ dollars per bottle.

• To find the better buy for detergent, compare a 64-load container at 14.99 dollars ($\approx 0.23$ dollars/load) to an 80-load container at 15.99 dollars ($\approx 0.20$ dollars/load). The 80-load container is the better buy.