Unit Rates with Fractions and Different Units | Grade 7 Math | LevelKite

A rate compares two positive quantities, each with its own unit, such as miles and hours. A unit rate tells you how much of the first quantity goes with exactly 1 unit of the second. You already know how to find one with whole numbers: 150 miles in 3 hours is $150 \div 3 = 50$ miles per hour.

This lesson adds two twists. First, the amounts can be fractions, such as $\frac{3}{4}$ mile in $\frac{1}{4}$ hour. Second, a measurement can be given in one unit, such as minutes, when the question asks about another unit, such as hours. The big idea stays the same: find the amount that goes with exactly 1.

Start with what a rate made of fractions means. Suppose you walk $\frac{3}{4}$ mile in $\frac{1}{4}$ hour. How far do you walk in 1 whole hour at the same pace?

Four quarter-hours make 1 hour, because $\frac{1}{4} \times 4 = 1$ . So you multiply the time by 4. To keep the same pace, you multiply the distance by 4 too: $\frac{3}{4} \times 4 = \frac{12}{4} = 3$ miles. Your unit rate is 3 miles per hour.

Dividing the miles by the hours gives the same answer: $\frac{3}{4} \div \frac{1}{4} = 3$ , because $\frac{1}{4}$ fits into $\frac{3}{4}$ exactly 3 times. Written as one fraction, that division is a complex fraction: a fraction that has a fraction on top (in the numerator), on the bottom (in the denominator), or both.

\frac{\dfrac{3}{4}\text{ mile}}{\dfrac{1}{4}\text{ hour}} = \left(\frac{3}{4} \div \frac{1}{4}\right)\text{ miles per hour} = 3\text{ miles per hour}

Key concept

Divide by the quantity you want exactly 1 of

To find a unit rate, divide the first quantity by the second quantity, the one you want exactly 1 of. The number you divide by is called the divisor. Keep each number with its unit, and put each measurement in the unit the question asks for. This works when both quantities are positive, so the divisor is never zero.

To divide by a fraction that is not zero, multiply by its reciprocal. In the formula, $a$ , $b$ , $c$ and $d$ stand for numbers, and $b$ , $c$ and $d$ are not zero. The rule works for any divisor except zero, not only for fractions with 1 on top.

Why it works: multiplying both quantities by the same positive number keeps the rate the same, and multiplying by the reciprocal of the second quantity turns it into exactly 1.

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Worked example

A fraction divided by a fraction

A hiker walks $\frac{5}{6}$ mile in $\frac{2}{3}$ hour at a steady pace. What is the hiker's unit rate in miles per hour?

Miles per hour means miles for each 1 hour, so the time in hours is the quantity you want exactly 1 of. Write the miles over the hours as a complex fraction. Both amounts are positive, so the divisor, 2/3 hour, is not zero. The number part is 5/6 divided by 2/3, and the unit is miles per hour.
$\frac{\dfrac{5}{6}\text{ mile}}{\dfrac{2}{3}\text{ hour}} = \left(\frac{5}{6} \div \frac{2}{3}\right)\text{ miles per hour}$
Dividing by a number that is not zero is the same as multiplying by its reciprocal. The reciprocal of 2/3 is 3/2, because 2/3 times 3/2 is 6/6, which equals 1. Multiplying the time by 3/2 turns it into 1 hour, so you multiply the distance by 3/2 as well to keep the same pace.
$\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2}$
Multiply the two fractions. To multiply fractions, multiply the numerators together and the denominators together.
$\frac{5}{6} \times \frac{3}{2} = \frac{5 \times 3}{6 \times 2} = \frac{15}{12}$
Simplify. Dividing the numerator and the denominator by the same number, as long as it is not zero, does not change the value of the fraction. Here that number is 3. Then write 5/4 as a mixed number, which is a whole number plus a fraction: 5/4 is 4/4 plus 1/4, or 1 and 1/4.
$\frac{15}{12} = \frac{15 \div 3}{12 \div 3} = \frac{5}{4} = 1\frac{1}{4}$
Check by multiplying the unit rate by the time. Multiplying by 2/3 undoes dividing by 2/3, so a correct rate gives back the original distance, 5/6 mile. The product 10/12 simplifies to 5/6 when you divide the numerator and the denominator by 2.
$\frac{5}{4} \times \frac{2}{3} = \frac{5 \times 2}{4 \times 3} = \frac{10}{12} = \frac{5}{6}$

Answer

The hiker's unit rate is $\frac{5}{4}$ miles per hour, which is $1\frac{1}{4}$ miles per hour.

The divisor does not have to be written as a fraction. It can be a mixed number such as $1\frac{1}{2}$ , or a decimal such as $0.4$ . Write it as a fraction first. Then find its reciprocal and multiply, just as before. The table shows the same steps for four divisors: a fraction with 1 on top, another fraction, a mixed number and a decimal.

The reciprocal rule works for every divisor except zero: a fraction, a mixed number, a decimal, even a negative number. In this lesson, the amounts in a rate are measurements such as distances and times, so they are always positive.

Same steps, different kinds of divisors

Situation	Divisor as a fraction	Reciprocal	Multiply by the reciprocal	Unit rate
$\frac{3}{4}$ mile in $\frac{1}{4}$ hour	$\frac{1}{4}$	$4$	$\frac{3}{4} \times 4 = \frac{12}{4} = 3$	3 miles per hour
$\frac{5}{6}$ mile in $\frac{2}{3}$ hour	$\frac{2}{3}$	$\frac{3}{2}$	$\frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4}$	$1\frac{1}{4}$ miles per hour
6 pages in $1\frac{1}{2}$ minutes	$1\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}$	$\frac{2}{3}$	$6 \times \frac{2}{3} = \frac{12}{3} = 4$	4 pages per minute
6 miles in $0.4$ hour	$0.4 = \frac{4}{10} = \frac{2}{5}$	$\frac{5}{2}$	$6 \times \frac{5}{2} = \frac{30}{2} = 15$	15 miles per hour

Now look closely at the units. A unit rate carries a label made from both units, such as miles per hour. When you find a unit rate by dividing two positive amounts, the unit after per is the unit of the divisor.

Sometimes a measurement is given in a different unit from the one in the question. Suppose you jog for 9 minutes, and the question asks for miles per hour. If you divided by 9, you would get the miles for each 1 minute, because the 9 counts minutes. So you first rewrite the time in hours.

For that you need a conversion fact: a statement that two measurements are equal, such as 1 hour = 60 minutes. Since 60 minutes make 1 hour, each minute is $\frac{1}{60}$ hour, and 9 minutes is $\frac{9}{60}$ hour. Rewriting a measurement in another unit does not change the amount of time, so the rate stays the same.

Worked example

Changing minutes to hours first

You jog $\frac{3}{4}$ mile in 9 minutes at a steady pace. Use the conversion fact 1 hour = 60 minutes to find your unit rate in miles per hour.

The question asks for miles per hour, so the time must be in hours. Because 60 minutes make 1 hour, each minute is 1/60 hour, so 9 minutes is 9/60 hour. Simplify by dividing the numerator and the denominator by 3. The amount of time has not changed, only its unit.
$9\text{ minutes} = 9 \times \frac{1}{60}\text{ hour} = \frac{9}{60}\text{ hour} = \frac{3}{20}\text{ hour}$
Now the units match the question: miles and hours. Write the miles over the hours. The divisor, 3/20 hour, is positive, so it is not zero. The number part is 3/4 divided by 3/20, and the unit is miles per hour.
$\frac{\dfrac{3}{4}\text{ mile}}{\dfrac{3}{20}\text{ hour}} = \left(\frac{3}{4} \div \frac{3}{20}\right)\text{ miles per hour}$
Multiply by the reciprocal of the divisor. The reciprocal of 3/20 is 20/3, because 3/20 times 20/3 is 60/60, which equals 1. Then simplify: 60 divided by 12 is 5.
$\frac{3}{4} \div \frac{3}{20} = \frac{3}{4} \times \frac{20}{3} = \frac{3 \times 20}{4 \times 3} = \frac{60}{12} = 5$
Check with a different route. First find a different unit rate, the miles for each 1 minute, by dividing the miles by the 9 minutes. The divisor 9 is not zero, and its reciprocal is 1/9, because 9 times 1/9 is 1. Dividing the numerator and the denominator of 3/36 by 3 gives 1/12 mile per minute.
$\frac{3}{4} \div 9 = \frac{3}{4} \times \frac{1}{9} = \frac{3 \times 1}{4 \times 9} = \frac{3}{36} = \frac{1}{12}$
One hour has 60 minutes, so in 1 hour you jog 60 times as far as in 1 minute. Multiply the miles per minute by 60. Both routes give 5 miles per hour, so the answer checks.
$\frac{1}{12} \times 60 = \frac{60}{12} = 5$

Answer

Your unit rate is 5 miles per hour.

Worked example

A unit rate that uses area

A worker tiles a rectangular floor that is 12 feet long and $7\frac{1}{2}$ feet wide in $2\frac{1}{4}$ hours, working at a steady pace. The area of a rectangle is its length times its width. What is the worker's unit rate in square feet per hour?

Write each mixed number as a fraction so that you can multiply and find a reciprocal. For the width, 7 wholes are 14 halves, and 14/2 plus 1/2 is 15/2. For the time, 2 wholes are 8 fourths, and 8/4 plus 1/4 is 9/4.
$7\frac{1}{2} = \frac{14}{2} + \frac{1}{2} = \frac{15}{2} \quad\text{and}\quad 2\frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}$
The rate compares area with time, so find the area next. Use the given formula: area equals length times width. Both lengths are in feet, so the area is in square feet.
$12\text{ feet} \times \frac{15}{2}\text{ feet} = \frac{12 \times 15}{2}\text{ square feet} = \frac{180}{2}\text{ square feet} = 90\text{ square feet}$
Square feet per hour means square feet for each 1 hour, so divide the area by the time. The number part is 90 divided by 9/4, and the unit is square feet per hour. The divisor, 9/4 hours, is positive, so it is not zero. Multiply by its reciprocal, 4/9.
$90 \div \frac{9}{4} = 90 \times \frac{4}{9} = \frac{90 \times 4}{9} = \frac{360}{9} = 40$
Check by multiplying the unit rate by the time. Multiplying by 9/4 undoes dividing by 9/4, so a correct rate gives back the area, 90 square feet.
$40 \times \frac{9}{4} = \frac{40 \times 9}{4} = \frac{360}{4} = 90$

Answer

The worker's unit rate is 40 square feet per hour.

Takeaway: A unit rate is the amount that goes with exactly 1 of another quantity. You find it by dividing by that quantity, which is never zero because the amounts in these rates are positive. When the divisor is a fraction, a mixed number or a decimal, write it as a fraction and multiply by its reciprocal. Before you divide, put each measurement in the unit the question asks for, using a conversion fact or a formula you are given. Then label your answer with both units.

Next, you will use unit rates to solve word problems and to compare ratios and rates.