Compare and Apply Unit Rates | Grade 8 Math | LevelKite

Printer A prints 96 pages in 8 minutes. Printer B prints 90 pages in 6 minutes. Each printer works at a steady pace. Which one is faster? The totals alone cannot tell you, because the printers ran for different lengths of time. To compare them fairly, you need to know what each one does in the same amount of time.

A rate compares two quantities measured in different units, such as 96 pages in 8 minutes. A unit rate is a rate whose second quantity is exactly 1 unit, such as 12 pages per 1 minute. The word per means for each.

You already know how to convert measurements and rates. In this lesson, you will use that skill to compare unit rates and to solve word problems.

To find a unit rate, divide the first quantity by the second quantity. Here is why that works. When you divide both quantities of a rate by the same nonzero number, you get an equivalent rate: a rate that describes the same relationship with different numbers. Divide 96 pages and 8 minutes both by 8, and the second quantity becomes exactly 1 minute.

The number you divide by is the divisor, and it must not be 0. Division by zero is undefined, so a rate per 0 minutes has no meaning.

\frac{96 \text{ pages}}{8 \text{ minutes}} = \frac{(96 \div 8) \text{ pages}}{(8 \div 8) \text{ minutes}} = \frac{12 \text{ pages}}{1 \text{ minute}} = 12 \text{ pages per minute}

Key concept

Unit rates make comparisons fair

A unit rate tells you how much of one quantity goes with exactly 1 unit of another. If $a$ units of a first quantity go with $b$ units of a second quantity, and $b \neq 0$ , the unit rate is $a \div b$ of the first unit per 1 of the second unit.

To compare two rates, write both as unit rates with the same units in the same order, such as dollars per ounce for both. Only then can you be sure that the greater number is the greater rate. The question tells you which rate you want. For speed, the faster runner has the greater rate. For cost per ounce, the cheaper snack has the lesser rate.

\text{unit rate} = \frac{a}{b}, \quad b \neq 0

Three printers, each printing at a steady rate

Printer	Pages printed	Time (minutes)	Unit rate (pages per minute)
A	$96$	$8$	$96 \div 8 = 12$
B	$90$	$6$	$90 \div 6 = 15$
C	$52$	$4$	$52 \div 4 = 13$

Read across each row of the table. Every unit rate is in pages per minute, so the units match and you can compare the numbers directly: $12 \lt 13 \lt 15$ . Printer B is the fastest at 15 pages per minute, and Printer A is the slowest at 12 pages per minute.

Notice that Printer A printed the most pages in total, yet it has the lowest unit rate. It simply ran for the longest time. A total tells you how much a printer did. A unit rate tells you how quickly it worked.

Worked example

Which brand costs less per ounce?

Brand A sells 12 ounces (oz) of trail mix for 3 dollars. Brand B sells 20 ounces of the same trail mix for 4 dollars. Which brand costs less per ounce?

Choose the unit rate that answers the question. Cost per ounce means dollars for 1 ounce, so divide dollars by ounces for both brands. The divisors, 12 and 20, are not 0, so you can divide.
Find Brand A's unit rate. Divide both quantities by 12 so the amount becomes 1 ounce. Dividing both by the same nonzero number keeps the rate equivalent.
$\frac{3 \text{ dollars}}{12 \text{ oz}} = \frac{0.25 \text{ dollars}}{1 \text{ oz}} = 0.25 \text{ dollars per oz}$
Find Brand B's unit rate the same way, this time dividing both quantities by 20.
$\frac{4 \text{ dollars}}{20 \text{ oz}} = \frac{0.20 \text{ dollars}}{1 \text{ oz}} = 0.20 \text{ dollars per oz}$
Compare. Both unit rates are in dollars per ounce, so the units match and you can compare the numbers. The question asks for the lower cost, so choose the lesser rate.
$0.20 < 0.25$

Answer

Brand B costs less per ounce: $0.20$ dollars per ounce, compared with $0.25$ dollars per ounce for Brand A.

Worked example

Which one is faster?

A cyclist rides 400 meters (m) in 50 seconds (s) at a steady speed. A scooter travels at a steady 27 kilometers per hour (km/h). Which one is faster?

Find the cyclist's unit rate. Divide both quantities by 50 so the time becomes 1 second. The divisor, 50, is not 0, so you can divide.
$\frac{400 \text{ m}}{50 \text{ s}} = \frac{8 \text{ m}}{1 \text{ s}} = 8 \text{ m/s}$
Check the units before you compare. The cyclist's speed is in meters per second, and the scooter's speed is in kilometers per hour. The units do not match, so you cannot compare 8 and 27 yet.
Convert the cyclist's speed to kilometers per hour. Because 3600 seconds make 1 hour (h) and 1000 meters make 1 kilometer (km), each fraction you multiply by equals 1. The units change, but the speed does not. The result, 28.8, is exact.
$\frac{8 \text{ m}}{1 \text{ s}} \times \frac{3600 \text{ s}}{1 \text{ h}} \times \frac{1 \text{ km}}{1000 \text{ m}} = \frac{28{,}800 \text{ km}}{1000 \text{ h}} = 28.8 \text{ km/h}$
Now both speeds are in kilometers per hour, so compare the numbers. The faster one has the greater speed.
$28.8 \text{ km/h} > 27 \text{ km/h}$
Check with the other unit. Write the scooter's speed in meters per second: 27 kilometers is 27,000 meters, and 1 hour is 3600 seconds, a nonzero divisor. Both speeds are now in meters per second, and the cyclist's 8 is still greater. The conclusion does not depend on which matching unit you choose.
$\frac{27{,}000 \text{ m}}{3600 \text{ s}} = 7.5 \text{ m/s} < 8 \text{ m/s}$

Answer

The cyclist is faster: $28.8$ kilometers per hour, compared with $27$ kilometers per hour for the scooter.

Now put unit rates to work. A unit rate also lets you find amounts that a problem does not give you. This works when the rate is constant (steady), which means every unit goes with the same amount.

Printer A prints 12 pages per minute at a constant rate. In 5 minutes it prints $12 \cdot 5 = 60$ pages, because each of the 5 minutes adds 12 pages. To print 84 pages it needs $84 \div 12 = 7$ minutes, because 84 pages is 7 groups of 12 pages.

Here is the same idea with variables. Let $r$ be the constant unit rate. Let $n$ be the number of units of the second quantity, with $n \ge 0$ . Let $T$ be the matching total of the first quantity. The units of $n$ must match the per unit of $r$ . Multiply to find a total. To find the number of units, divide the total by the unit rate, which must not be 0 because division by zero is undefined.

T = r \cdot n \qquad \text{and} \qquad n = \frac{T}{r} \quad (r \neq 0)

Worked example

Filling a pool at a constant rate

A garden hose fills a 15-liter (L) bucket in 1.25 minutes (min). The water flows at a constant rate. (a) How many liters flow out in half an hour? (b) How many minutes does the hose take to fill a 540-liter pool?

Name the quantities. Let t be the time in minutes, with t at least 0. Let V be the volume of water in liters. Let r be the constant flow rate in liters per minute (L/min). Because the rate is constant, the volume is the rate times the time.
$V = r \cdot t$
Find the unit rate from the bucket by dividing liters by minutes. The divisor, 1.25, is not 0, so you can divide. Multiplying both quantities by the nonzero number 4 clears the decimal and keeps the rate equivalent.
$r = \frac{15 \text{ L}}{1.25 \text{ min}} = \frac{60 \text{ L}}{5 \text{ min}} = 12 \text{ L/min}$
Part (a): match the time unit to the rate. The rate is per minute, so write half an hour as 30 minutes. Multiply the rate by the time. The minutes cancel, and liters remain.
$V = \frac{12 \text{ L}}{1 \text{ min}} \times 30 \text{ min} = 360 \text{ L}$
Part (b): now you know the volume and need the time. Substitute 540 for V and 12 for r in the relationship from step 1.
$540 = 12 \cdot t$
Divide both sides by 12. Dividing both sides of an equation by the same nonzero number keeps the two sides equal. Because t is in minutes, the time is 45 minutes.
$t = \frac{540}{12} = 45$
Check part (b) by multiplying the rate by the time. The time is in minutes, which matches the rate, and the product equals the pool's volume, so 45 minutes is correct.
$\frac{12 \text{ L}}{1 \text{ min}} \times 45 \text{ min} = 540 \text{ L}$

Answer

At a constant $12$ liters per minute, $360$ liters flow out in half an hour, and the hose takes $45$ minutes to fill the $540$ -liter pool.

Takeaway. A unit rate tells you how much goes with exactly 1 unit, and you find it by dividing by an amount that is not 0. To compare rates, write both as unit rates with the same units in the same order, then let the question tell you whether you want the greater rate or the lesser one. When a rate is constant, multiply the unit rate by the number of units to find a total, or divide a total by the nonzero unit rate to find the number of units.