Math 6 · 6.RP.A.3D

Changing the units of a rate

A sloth’s crawl is timed in feet per minute, but speeds are usually given in miles per hour. How do you rewrite a rate in other units without changing how fast it is?

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When you reach for it

Tilda takes her pulse at rest and counts 72 beats in a minute. How many times does her heart beat in an hour, or in a whole day? Stavros times a sloth crawling along a branch and finds it covers 11 feet a minute, but animal speeds are usually given in miles per hour. Luzia’s class reads that a glacier flows 0.48 meters a day, and wants to know how far it creeps in one hour. Each time the rate is right, but its units do not fit the question.

A rate compares two amounts of different kinds, as in Rates and unit rates. Written as a fraction it has two units: the one on top names what is counted or measured (beats, feet, meters) and the one on the bottom names the “per” (per minute, per day). Either unit can be changed, or both. Reach for this move whenever a rate has to answer a question asked in other units, and whenever two rates must be compared but arrive in different units, such as two speeds, one in feet per minute and one in miles per hour. Like two unit prices, two rates can be compared only once their units match.

The tool is the conversion factor: a fraction with the same amount on top and bottom, such as \displaystyle \frac{60\text{ min}}{1\text{ h}}, built from a customary or metric fact. Part 1 (day one) changes one unit at a time and explains why the bottom unit takes its factor the other way up. Part 2 (day two) changes both units, goes through middle units, compares rates, and checks each answer by its size.

The procedure

Part 1: one unit at a time

Write the rate as a fraction with its units. Tilda’s 72 beats per minute is

\frac{72\text{ beats}}{1\text{ min}}

Multiply by a conversion factor. A factor such as \displaystyle \frac{60\text{ min}}{1\text{ h}} has one amount of time on top and the same amount on the bottom, so it is equal to 1. Multiplying by 1 changes nothing (the identity property of multiplication), so the heart beats exactly as fast as before. Only the name of the rate changes.

Let the old unit decide which way up the factor goes. The unit being replaced has to cancel, so it must appear once on a top and once on a bottom.

Changing the top unit. The old unit sits on top of the rate, so it goes on the bottom of the factor. This is the same way up as converting a plain amount. A glacier flowing 0.3 meters per day flows 0.3 \times 100 = 30 centimeters per day, using \displaystyle \frac{100\text{ cm}}{1\text{ m}}.
Changing the bottom unit. The old unit sits on the bottom of the rate, so it goes on the top of the factor. This is the factor the other way up from converting a plain amount.

That second case is the one that needs thought. To change 120 minutes of time into hours, minutes sit on top, the factor is \displaystyle \frac{1\text{ h}}{60\text{ min}}, and the number is divided by 60: 120 minutes is 2 hours. But to change beats per minute into beats per hour, minutes sit on the bottom, so the factor is \displaystyle \frac{60\text{ min}}{1\text{ h}}, and the number is multiplied by 60:

\frac{72\text{ beats}}{1\,\cancel{\text{min}}} \times \frac{60\,\cancel{\text{min}}}{1\text{ h}} = \frac{4{,}320\text{ beats}}{1\text{ h}}

The figure shows why the number grows. An hour is not a smaller amount of beating than a minute; it is 60 minutes of it, and every one of those minutes holds 72 beats.

Figure 1Every minute adds 72 beats. An hour holds 60 minutes, so it holds 60 \times 72 = 4{,}320 beats: the rate per hour is the bigger number.

Check the size. Each change pushes the number one way, and the direction can be said before any arithmetic:

The change	For example	The number
top unit to a smaller unit	meters to centimeters	grows
top unit to a bigger unit	feet to miles	shrinks
bottom unit to a bigger unit	per minute to per hour	grows
bottom unit to a smaller unit	per day to per hour	shrinks

Each row has a reason, not just a direction. A bigger “per” holds more of the thing counted (an hour holds 60 minutes of beating), so its number is bigger; a smaller “per” holds only a slice of it, so its number is smaller. On top it runs the other way: a smaller unit takes more of itself to cover the same amount (one meter is 100 centimeters), so its number is bigger, and a bigger unit takes fewer.

Part 2: both units, and comparing

One factor for each unit that changes. To turn feet per minute into miles per hour, one factor cancels feet and brings in miles, and another cancels minutes and brings in hours. Multiplication can be reordered and regrouped (the commutative and associative properties), so it does not matter which unit is changed first.

Go through a middle unit when there is no single fact. No common fact links hours to years directly, but hours to days and days to years are both known. One factor per step, each equal to 1, and the chain is still equal to 1.

Compare two rates in one set of units. Change one rate into the other’s units, usually whichever is easier, and then compare the two numbers.

Check the size, one factor at a time. Feet per minute to miles per hour divides by 5{,}280 (miles are far bigger than feet) and multiplies by 60 (an hour holds 60 minutes). Together that divides by \displaystyle \frac{5{,}280}{60} = 88, so the number in miles per hour is much smaller.

Worked examples

Examples 1 to 3 belong to Part 1, changing one unit at a time. Examples 4 to 6 belong to Part 2, changing both units and comparing rates.

Example 1

A resting heart per hour and per day (Part 1)

Willa’s heart beats 65 times a minute at rest. Write this rate in beats per hour, and then in beats per day, as if it stayed steady all day.

Solution

Minutes sit on the bottom of the rate and must cancel, so the factor has minutes on top:

\frac{65\text{ beats}}{1\,\cancel{\text{min}}} \times \frac{60\,\cancel{\text{min}}}{1\text{ h}} = \frac{3{,}900\text{ beats}}{1\text{ h}}

Now hours sit on the bottom. To turn them into days, the factor has hours on top:

\frac{3{,}900\text{ beats}}{1\,\cancel{\text{h}}} \times \frac{24\,\cancel{\text{h}}}{1\text{ day}} = \frac{93{,}600\text{ beats}}{1\text{ day}}

Turned the other way up, the first factor would leave units that do not cancel:

\frac{65\text{ beats}}{1\text{ min}} \times \frac{1\text{ h}}{60\text{ min}} = \frac{65}{60}\ \frac{\text{beats}\cdot\text{h}}{\text{min}\cdot\text{min}}

Minutes twice on the bottom and hours on top is not beats per hour, so the units show the mistake before any number is checked.

Check. A bigger “per” holds more beats, and 93{,}600 is more than 3{,}900, which is more than 65. As a single step, a day is 60 \times 24 = 1{,}440 minutes, and 65 \times 1{,}440 = 93{,}600.

Example 2

A pulse counted for fifteen seconds (Part 1)

Vivek counts 18 beats in 15 seconds. What is his heart rate in beats per minute?

Solution

The rate is not yet per one: it is \displaystyle \frac{18\text{ beats}}{15\text{ s}}. That does not stop a factor from changing its bottom unit. Seconds sit on the bottom, so the factor has seconds on top:

\begin{aligned} &\frac{18\text{ beats}}{15\,\cancel{\text{s}}} \times \frac{60\,\cancel{\text{s}}}{1\text{ min}} \\ &\quad = \frac{18 \times 60}{15}\ \frac{\text{beats}}{\text{min}} = \frac{1{,}080}{15}\ \frac{\text{beats}}{\text{min}} \\ &\quad = 72\ \frac{\text{beats}}{\text{min}} \end{aligned}

Vivek’s heart beats 72 times a minute, the same rate as Tilda’s in Figure 1.

Check. A minute is four stretches of 15 seconds, each with 18 beats, and 4 \times 18 = 72.

Example 3

A glacier per day, then per hour (Part 1)

Luzia’s class follows a glacier that flows 0.48 meters per day. (a) Write the rate in centimeters per day. (b) Then write it in centimeters per hour.

Solution

(a) Meters sit on top of the rate, so they go on the bottom of the factor, the same way up as converting a plain length:

\frac{0.48\,\cancel{\text{m}}}{1\text{ day}} \times \frac{100\text{ cm}}{1\,\cancel{\text{m}}} = \frac{48\text{ cm}}{1\text{ day}}

A centimeter is smaller than a meter, so it takes more of them: the number grew from 0.48 to 48.

(b) Days sit on the bottom of the rate, so they go on the top of the factor:

\frac{48\text{ cm}}{1\,\cancel{\text{day}}} \times \frac{1\,\cancel{\text{day}}}{24\text{ h}} = \frac{2\text{ cm}}{1\text{ h}}

This time the number shrank, because an hour is only one twenty-fourth of a day, and a day’s 48 centimeters are shared over its 24 hours.

Figure 2A day’s 48 cm shared over its 24 hours is 2 cm in each hour: the rate per hour is the smaller number.

Check. 2 centimeters in each of 24 hours is 2 \times 24 = 48 centimeters, one day’s flow.

Example 4

A sloth’s crawl in miles per hour (Part 2)

Stavros times a sloth crawling along a branch at 11 feet per minute. Write its speed in miles per hour.

Solution

Both units change. Feet sit on top, so the factor that brings in miles has feet on the bottom, \displaystyle \frac{1\text{ mi}}{5{,}280\text{ ft}}. Minutes sit on the bottom, so the factor that brings in hours has minutes on top, \displaystyle \frac{60\text{ min}}{1\text{ h}}:

\begin{aligned} &\frac{11\,\cancel{\text{ft}}}{1\,\cancel{\text{min}}} \times \frac{60\,\cancel{\text{min}}}{1\text{ h}} \times \frac{1\text{ mi}}{5{,}280\,\cancel{\text{ft}}} \\ &\quad = \frac{11 \times 60}{5{,}280}\ \frac{\text{mi}}{\text{h}} = \frac{660}{5{,}280}\ \frac{\text{mi}}{\text{h}} \\ &\quad = 0.125\ \frac{\text{mi}}{\text{h}} \end{aligned}

Since \displaystyle \frac{660}{5{,}280} = \frac{1}{8}, the sloth crawls an eighth of a mile per hour.

Check. Feet to miles divides by 5{,}280 and per minute to per hour multiplies by 60, so the two together divide by 88, and 11 \div 88 = 0.125. Another way: the sloth crawls 660 feet in an hour, and 8 \times 660 = 5{,}280, so eight such hours make one mile.

Example 5

Which sloth is faster? (Part 2)

Willa reads that a second sloth crawled at 0.1 miles per hour. Is it faster or slower than the sloth in Example 4, at 11 feet per minute, and by how much?

Solution

The two rates can be compared only in one set of units. Change the second sloth’s rate into feet per minute. Miles sit on top, so the factor has miles on the bottom; hours sit on the bottom, so the factor has hours on top:

\begin{aligned} &\frac{0.1\,\cancel{\text{mi}}}{1\,\cancel{\text{h}}} \times \frac{5{,}280\text{ ft}}{1\,\cancel{\text{mi}}} \times \frac{1\,\cancel{\text{h}}}{60\text{ min}} \\ &\quad = \frac{528}{60}\ \frac{\text{ft}}{\text{min}} = 8.8\ \frac{\text{ft}}{\text{min}} \end{aligned}

Each factor is the one from Example 4 turned upside down, because the units are being changed the other way. Now 11 is more than 8.8, so Stavros’s sloth is the faster one, by 11 - 8.8 = 2.2, that is 2.2 feet per minute.

Check. Compare in miles per hour instead: 0.125 against 0.1, a difference of 0.025 miles per hour. Multiplying by 88 turns miles per hour into feet per minute, and 0.025 \times 88 = 2.2. Either set of units gives the same verdict.

Example 6

Two glaciers, per hour and per year (Part 2)

Irina compares two glaciers. Glacier A flows 150 meters per year. Glacier B flows 1.5 centimeters per hour. Which flows faster, and by how much? Take a year as 365 days.

Solution

Change glacier B’s rate into meters per year. The top unit goes from centimeters to meters with \displaystyle \frac{1\text{ m}}{100\text{ cm}}. No single everyday fact links hours to years, so the bottom goes through a middle unit, days: \displaystyle \frac{24\text{ h}}{1\text{ day}} cancels hours, then \displaystyle \frac{365\text{ days}}{1\text{ year}} cancels days. (Day and days are the same unit, so they cancel.)

\begin{aligned} &\frac{1.5\,\cancel{\text{cm}}}{1\,\cancel{\text{h}}} \times \frac{24\,\cancel{\text{h}}}{1\,\cancel{\text{day}}} \\ &\quad \times \frac{365\,\cancel{\text{days}}}{1\text{ year}} \times \frac{1\text{ m}}{100\,\cancel{\text{cm}}} \\ &\quad = \frac{1.5 \times 24 \times 365}{100}\ \frac{\text{m}}{\text{year}} \\ &\quad = 131.4\ \frac{\text{m}}{\text{year}} \end{aligned}

Working left to right: 1.5 \times 24 = 36 centimeters per day, 36 \times 365 = 13{,}140 centimeters per year, and 13{,}140 \div 100 = 131.4 meters per year. Glacier A flows faster, by 150 - 131.4 = 18.6, that is 18.6 meters per year.

Check. Glacier B’s 36 centimeters a day is a little more than a third of a meter a day, and \displaystyle \frac{1}{3} \times 365 is about 122. The answer 131.4 should be a little more than that, and it is.

Boundaries

The rate is taken as steady. Changing 65 beats per minute into 93{,}600 beats per day says what a day at that same pace would hold. A real heart beats slower in sleep and faster in a race, and a glacier can flow faster in summer than in winter, so a converted rate describes a steady pace, not a count that was made.

Every factor must be a true name for 1. \displaystyle \frac{60\text{ min}}{1\text{ h}} is 1 because the two amounts of time are equal. A year is 365 days only in an ordinary year; a leap year has 366, so a rate per year built on 365 is a rate for an ordinary year. A rate itself, such as 72 beats per minute, is not a name for 1: multiplying by it changes what is being measured.

Each unit changes only into a unit of the same kind. The top unit of a speed is a length and stays a length; the bottom unit is a time and stays a time. Feet per minute can become miles per hour, but never beats per hour.

A rounded fact gives a rounded answer. Every fact in this lesson is exact: 5{,}280 feet in a mile, 60 minutes in an hour, 24 hours in a day, 100 centimeters in a meter. A factor built from a rounded fact, such as 1\text{ mi} \approx 1.609\text{ km}, gives an answer that is only about right.

Where this goes next

Once a rate can wear any units, rates measured in different ways can be set side by side. Water running in liters per minute is rewritten in milliliters per second the same way in flow rate. Reasoning with units takes the move further, to kilometers per hour and meters per second, to fuel use per kilometer, and to reading a set-up’s units to see whether it can possibly give the quantity asked for. When both amounts of a rate are fractions, such as \displaystyle \frac{2}{3} of a mile in \displaystyle \frac{1}{4} of an hour, unit rates with fractions finds the amount per one by dividing one fraction by the other; the unit rate it gives can then be converted exactly as here. And a converted rate is still the The constant of proportionality and proportional tables between the two amounts, only measured in new units.

Exercises

Exercises 1 to 10 go with Part 1 (Examples 1 to 3), Exercises 11 to 19 with Part 2 (Examples 4 to 6), and the reasoning group draws on both. Take a year as 365 days wherever one is needed.

Part 1: changing one unit

After a longer rest, Tilda’s heart beats 64 times a minute. How many beats is that per hour?
At the rate in Exercise 1, how many beats is that per day?
Vivek counts 21 beats in 15 seconds. What is his heart rate in beats per minute?
Irina counts 13 beats in 10 seconds. Find her heart rate in beats per minute and in beats per hour.
Write Tilda’s 72 beats per minute as beats per second.
A heart beats 108{,}000 times in a day at a steady rate. Find its rate in beats per hour and in beats per minute.
A glacier flows 0.35 meters per day. Write this in centimeters per day.
A glacier flows 0.6 meters per day. How many meters per year is that?
A glacier flows 36 centimeters per day. Write this in centimeters per hour.
To write 90 beats per minute as beats per hour, is the factor \displaystyle \frac{60\text{ min}}{1\text{ h}} or \displaystyle \frac{1\text{ h}}{60\text{ min}}? Say why, and give the rate per hour.

Part 2: changing both units

A sloth crawls 6.6 feet per minute. Write this in miles per hour.
A sloth crawls 0.22 feet per second. Write this in miles per hour.
A sloth crawls at 0.25 miles per hour. Write this in feet per minute.
A glacier flows 0.96 meters per day. Write this in centimeters per hour.
A glacier flows 1.2 centimeters per hour. Write this in meters per day.
A glacier flows 73 meters per year. Write this in centimeters per day.

Part 2: comparing rates

One sloth crawls 36 feet in 4 minutes. Another crawls at 0.11 miles per hour. Which is faster, and by how many feet per minute?
Glacier C flows 0.5 meters per day and glacier D flows 2.5 centimeters per hour. Which flows faster, and by how many centimeters per day?
While she walks, Tilda’s heart beats 4{,}560 times an hour. Vivek’s beats 72 times a minute. Whose heart beats faster, and by how many beats per minute?

Reasoning

Changing 120 minutes into hours divides by 60, but changing 72 beats per minute into beats per hour multiplies by 60. Explain why the one fact 1\text{ h} = 60\text{ min} is used in opposite directions.
Without working out any answer, say whether a glacier’s rate is a bigger or a smaller number in centimeters per hour than in meters per day. Explain.
Show that 1 mile per hour is exactly 88 feet per minute. Use this to check Example 4 and Exercise 11.
In Example 3 the top unit was changed first and the bottom unit second. Would changing the bottom unit first give the same rate? Explain.

Answers

3{,}840 beats per hour. \displaystyle \frac{64\text{ beats}}{1\text{ min}} \times \frac{60\text{ min}}{1\text{ h}}, and 64 \times 60 = 3{,}840.
92{,}160 beats per day. \displaystyle \frac{3{,}840\text{ beats}}{1\text{ h}} \times \frac{24\text{ h}}{1\text{ day}}, and 3{,}840 \times 24 = 92{,}160.
84 beats per minute. \displaystyle \frac{21\text{ beats}}{15\text{ s}} \times \frac{60\text{ s}}{1\text{ min}} comes to \displaystyle \frac{21 \times 60}{15} = 84. Or: a minute is four lots of 15 seconds, and 4 \times 21 = 84.
78 beats per minute and 4{,}680 beats per hour. \displaystyle \frac{13 \times 60}{10} = 78, then 78 \times 60 = 4{,}680.
1.2 beats per second. Minutes sit on the bottom and must become seconds, so the factor has minutes on top: \displaystyle \frac{72\text{ beats}}{1\text{ min}} \times \frac{1\text{ min}}{60\text{ s}}, and \displaystyle \frac{72}{60} = 1.2. A second is a small slice of a minute, so the number shrinks.
4{,}500 beats per hour and 75 beats per minute. The factor \displaystyle \frac{1\text{ day}}{24\text{ h}} gives 108{,}000 \div 24 = 4{,}500, and \displaystyle \frac{1\text{ h}}{60\text{ min}} gives 4{,}500 \div 60 = 75.
35 centimeters per day. \displaystyle \frac{0.35\text{ m}}{1\text{ day}} \times \frac{100\text{ cm}}{1\text{ m}}, and 0.35 \times 100 = 35.
219 meters per year. \displaystyle \frac{0.6\text{ m}}{1\text{ day}} \times \frac{365\text{ days}}{1\text{ year}}, and 0.6 \times 365 = 219.
1.5 centimeters per hour. \displaystyle \frac{36\text{ cm}}{1\text{ day}} \times \frac{1\text{ day}}{24\text{ h}}, and 36 \div 24 = 1.5.
\displaystyle \frac{60\text{ min}}{1\text{ h}}, giving 5{,}400 beats per hour. Minutes sit on the bottom of the rate, so to cancel they must sit on top of the factor. 90 \times 60 = 5{,}400.
0.075 miles per hour. \displaystyle \frac{6.6\text{ ft}}{1\text{ min}} \times \frac{60\text{ min}}{1\text{ h}} \times \frac{1\text{ mi}}{5{,}280\text{ ft}} comes to \displaystyle \frac{396}{5{,}280} = 0.075.
0.15 miles per hour. Per second to per minute to per hour multiplies by 60 \times 60 = 3{,}600, so 0.22 \times 3{,}600 = 792 feet per hour, and \displaystyle \frac{792}{5{,}280} = 0.15.
22 feet per minute. \displaystyle \frac{0.25\text{ mi}}{1\text{ h}} \times \frac{5{,}280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ h}}{60\text{ min}} comes to \displaystyle \frac{1{,}320}{60} = 22.
4 centimeters per hour. 0.96 \times 100 = 96 centimeters per day, and 96 \div 24 = 4.
0.288 meters per day. 1.2 \times 24 = 28.8 centimeters per day, and 28.8 \div 100 = 0.288.
20 centimeters per day. 73 \times 100 = 7{,}300 centimeters per year, and 7{,}300 \div 365 = 20.
The second sloth, by 0.68 feet per minute. The first crawls 36 \div 4 = 9 feet per minute. The second crawls 0.11 \times 5{,}280 \div 60 = 580.8 \div 60 = 9.68 feet per minute, and 9.68 - 9 = 0.68.
Glacier D, by 10 centimeters per day. Glacier C flows 0.5 \times 100 = 50 centimeters per day, and glacier D flows 2.5 \times 24 = 60 centimeters per day. 60 - 50 = 10.
Tilda’s, by 4 beats per minute. 4{,}560 \div 60 = 76 beats per minute, against Vivek’s 72. In beats per hour the gap is 4{,}560 - 4{,}320 = 240, which is 4 \times 60.
120 minutes is an amount of time with minutes on top, so it takes \displaystyle \frac{1\text{ h}}{60\text{ min}}, and since an hour is bigger than a minute there are fewer of them. In a rate, minutes are on the bottom: they are what the beats are shared over. They cancel only against \displaystyle \frac{60\text{ min}}{1\text{ h}}. In plain words, an hour holds 60 minutes and each holds 72 beats, so an hour holds 60 times as many beats.
A bigger number. Meters to centimeters multiplies by 100, and per day to per hour divides by 24. Multiplying by 100 outweighs dividing by 24, so the number ends up multiplied by \displaystyle \frac{100}{24}, a little more than 4. In Example 3, 0.48 meters per day became 2 centimeters per hour.
\displaystyle \frac{1\text{ mi}}{1\text{ h}} \times \frac{5{,}280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ h}}{60\text{ min}} comes to \displaystyle \frac{5{,}280}{60} = 88 feet per minute, exactly, since both facts are exact. So feet per minute divided by 88 gives miles per hour: 11 \div 88 = 0.125 (Example 4) and 6.6 \div 88 = 0.075 (Exercise 11).
Yes. Either way the rate is multiplied by the same two factors, which multiply by 100 and divide by 24, and multiplication may be reordered and regrouped (the commutative and associative properties). Bottom first: 0.48 \div 24 = 0.02 meters per hour, then 0.02 \times 100 = 2 centimeters per hour, the same rate.