Lesson 2: Determine Unit Rates with Ratios of Fractions

Property

A rate is a ratio of two quantities.
The unit rate is the amount of one quantity that corresponds to 1 unit of the other quantity.
The designation of unit rate must be clear about the choice and order of the units.

For a ratio $a:b$ with $b \neq 0$, the unit rate is $\frac{a}{b}$ units of the first quantity per 1 unit of the second quantity.

Examples

• If you pay 9 dollars for 3 sandwiches, the unit rate is found by dividing: $9 \div 3 = 3$ dollars per sandwich.
• A cyclist travels 30 miles in 2 hours. The unit rate for her speed is $30 \div 2 = 15$ miles per hour.
• A team scores 45 points in 3 quarters. Their unit rate is $45 \div 3 = 15$ points per quarter.

Property

To divide a number by a fraction, multiply the number by the reciprocal of the fraction. For example, to divide by $\frac{2}{3}$, you multiply by its reciprocal, $\frac{3}{2}$.

First, convert any mixed numbers to improper fractions. Then, apply the rule:

Examples

• How many $\frac{3}{4}$-foot lengths of rope can be cut from a 6-foot rope? We calculate $6 \div \frac{3}{4} = 6 \times \frac{4}{3} = \frac{24}{3} = 8$ lengths.

Property

When working with rates, you may encounter complex fractions where the numerator or denominator contains a fraction. To simplify a complex fraction in a rate, treat the main fraction bar as division.
Step 1. Rewrite the complex fraction as a division problem.
Step 2. Multiply by the reciprocal of the divisor.
Step 3. Simplify to find the unit rate.

Examples