Converting Rates and Measurements | Grade 8 Math | LevelKite

A speed of $60$ miles per hour and a speed of $88$ feet per second look very different. They are the same speed, written with different units. In this lesson you will learn how to show that.

A rate compares two quantities that have different units, such as miles and hours. You can write a rate as a fraction with units: $\frac{60\text{ mi}}{1\text{ h}}$ means $60$ miles for every $1$ hour, where mi stands for miles and h for hours. The quantity in the denominator cannot be zero, because division by zero is undefined.

To convert a rate is to rewrite it with different units so that it still describes the same relationship. This matters because you can compare the numbers in two rates only when both rates use matching units.

Start with a fact you already know: 1 mile is 5280 feet, written $1\text{ mi} = 5280\text{ ft}$ . Both sides name the same distance, so the fraction $\frac{5280\text{ ft}}{1\text{ mi}}$ has equal amounts on top and bottom, and its value is $1$ . A fraction built from two equal nonzero measurements like this is called a conversion factor.

Multiplying by $1$ never changes an amount. So when you multiply a measurement by a conversion factor, the amount stays the same and only its units change. A unit that appears in both a numerator and a denominator cancels, just as a common factor does.

In the line below, miles cancel and feet remain, so $3.5$ miles is the same distance as $18{,}480$ feet.

3.5\text{ mi} \times \frac{5280\text{ ft}}{1\text{ mi}} = 3.5 \times 5280\text{ ft} = 18{,}480\text{ ft}

Key concept

Multiply by 1 to change the units, not the amount

A conversion factor is a fraction whose numerator and denominator are equal nonzero measurements, so its value is $1$ . To convert a rate or a measurement, multiply it by one conversion factor for each unit you want to change. Turn each factor so that the unit you want to remove sits opposite its match, one in a numerator and one in a denominator, and cancels. Because you multiplied only by $1$ , the result describes exactly the same relationship as before. If a measurement fact is rounded, its factor is only about $1$ and the result is approximate. No rate and no conversion factor may have a zero denominator, because division by zero is undefined.

\frac{5280\text{ ft}}{1\text{ mi}} = 1 \quad \text{and} \quad \frac{1\text{ mi}}{5280\text{ ft}} = 1

Measurement facts used in this lesson and their conversion factors

Fact in words	Fact in symbols	Conversion factors	Value of each factor
1 minute is 60 seconds	$1\text{ min} = 60\text{ s}$	$\frac{60\text{ s}}{1\text{ min}}$ or $\frac{1\text{ min}}{60\text{ s}}$	Exactly $1$
1 hour is 60 minutes, or 3600 seconds	$1\text{ h} = 60\text{ min} = 3600\text{ s}$	$\frac{3600\text{ s}}{1\text{ h}}$ or $\frac{1\text{ h}}{3600\text{ s}}$	Exactly $1$
1 mile is 5280 feet	$1\text{ mi} = 5280\text{ ft}$	$\frac{5280\text{ ft}}{1\text{ mi}}$ or $\frac{1\text{ mi}}{5280\text{ ft}}$	Exactly $1$
1 inch is 2.54 centimeters	$1\text{ in} = 2.54\text{ cm}$	$\frac{2.54\text{ cm}}{1\text{ in}}$ or $\frac{1\text{ in}}{2.54\text{ cm}}$	Exactly $1$
1 meter is 100 centimeters	$1\text{ m} = 100\text{ cm}$	$\frac{100\text{ cm}}{1\text{ m}}$ or $\frac{1\text{ m}}{100\text{ cm}}$	Exactly $1$
1 mile is about 1.609 kilometers	$1\text{ mi} \approx 1.609\text{ km}$	$\frac{1.609\text{ km}}{1\text{ mi}}$ or $\frac{1\text{ mi}}{1.609\text{ km}}$	About $1$ , because $1.609$ is rounded

Each fact in the table gives two conversion factors, because you can put either measurement on top. Both are allowed because neither measurement is zero. Which one you use depends on which unit has to cancel.

You can follow the same routine for every rate conversion in this lesson:

Write the rate as a fraction with its units.
For each unit you want to change, pick the conversion factor that has the old unit on the other side of the fraction bar: in its denominator if the old unit is in a numerator, and in its numerator if the old unit is in a denominator.
Cancel each unit that appears in both a numerator and a denominator. Check that only the units you want are left.
Multiply the numerators and multiply the denominators. Then divide to find the amount for $1$ unit. The number you divide by is a product of nonzero numbers, so it is not zero.

Worked example

Changing one unit of a rate

A pump moves water at a steady rate of $90$ liters per minute. Write this rate in liters per second. Use $1\text{ min} = 60\text{ s}$ , and write L for liters.

Write the rate as a fraction with units so you can see where each unit sits. Liters are already the unit you want. Only minutes must change, and minutes are in the denominator.
$\frac{90\text{ L}}{1\text{ min}}$
Pick the conversion factor with minutes in its numerator, so that it sits opposite the minutes in the rate. This factor equals 1 because 1 minute and 60 seconds are the same nonzero amount of time, so multiplying by it keeps the rate the same.
$\frac{90\text{ L}}{1\text{ min}} \times \frac{1\text{ min}}{60\text{ s}}$
Cancel minutes, which now appear once in a numerator and once in a denominator. Then multiply the numerators and multiply the denominators. Liters over seconds remain, and those are the units you want.
$\frac{90 \times 1\text{ L}}{1 \times 60\text{ s}} = \frac{90\text{ L}}{60\text{ s}}$
Divide the numerator and the denominator by 60 to find the liters for 1 second. Dividing both by the same nonzero number keeps the fraction equal.
$\frac{90\text{ L}}{60\text{ s}} = \frac{1.5\text{ L}}{1\text{ s}}$

Answer

The pump moves $1.5$ liters per second, which is the same rate as $90$ liters per minute.

Worked example

Changing both units of a rate

A car travels at a steady $60$ miles per hour. Write this speed in feet per second. Use $1\text{ mi} = 5280\text{ ft}$ and $1\text{ h} = 3600\text{ s}$ .

Write the rate as a fraction with units. Two units must change: miles in the numerator and hours in the denominator.
$\frac{60\text{ mi}}{1\text{ h}}$
Use one conversion factor for each unit. Miles are in the rate's numerator, so the distance factor needs miles in its denominator. Hours are in the rate's denominator, so the time factor needs hours in its numerator. Each factor equals 1 because it compares two equal nonzero measurements, so the speed stays the same.
$\frac{60\text{ mi}}{1\text{ h}} \times \frac{5280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ h}}{3600\text{ s}}$
Cancel miles with miles and hours with hours, because each appears once in a numerator and once in a denominator. Feet over seconds remain. Then multiply 60 by 5280.
$\frac{60 \times 5280\text{ ft}}{3600\text{ s}} = \frac{316{,}800\text{ ft}}{3600\text{ s}}$
Divide the numerator and the denominator by 3600, which is not zero, to find the feet for 1 second.
$\frac{316{,}800\text{ ft}}{3600\text{ s}} = \frac{88\text{ ft}}{1\text{ s}}$

Answer

A speed of $60$ miles per hour is exactly $88$ feet per second, so the two speeds from the start of the lesson really are the same.

Worked example

A chain of three conversion factors

A conveyor belt moves at a steady $15$ inches per second. Write this speed in meters per minute. Use $1\text{ in} = 2.54\text{ cm}$ , $1\text{ m} = 100\text{ cm}$ and $1\text{ min} = 60\text{ s}$ .

Plan the path before you write anything. Inches become centimeters and then meters, and seconds become minutes. That takes three conversion factors. Each equals 1 because its two measurements are equal and nonzero.
$\text{in} \rightarrow \text{cm} \rightarrow \text{m} \qquad \text{s} \rightarrow \text{min}$
Turn each factor so its old unit sits opposite its match. Inches and then centimeters are in numerators, so the factors that remove them have those units in their denominators. Seconds are in a denominator, so the time factor has seconds in its numerator, the other way up from the pump example.
$\frac{15\text{ in}}{1\text{ s}} \times \frac{2.54\text{ cm}}{1\text{ in}} \times \frac{1\text{ m}}{100\text{ cm}} \times \frac{60\text{ s}}{1\text{ min}}$
Cancel inches, centimeters and seconds, because each appears once in a numerator and once in a denominator. Meters over minutes remain.
$\frac{15 \times 2.54 \times 60\text{ m}}{100\text{ min}}$
Multiply the numbers in the numerator in two stages so that each product is easy to check.
$15 \times 2.54 = 38.1 \qquad 38.1 \times 60 = 2286$
Divide the numerator and the denominator by 100, which is not zero, to find the meters for 1 minute.
$\frac{2286\text{ m}}{100\text{ min}} = \frac{22.86\text{ m}}{1\text{ min}}$

Answer

The belt moves $22.86$ meters per minute. Every measurement fact used is exact, so this value is exact, not rounded.

Exact or rounded? Facts such as $1\text{ mi} = 5280\text{ ft}$ and $1\text{ in} = 2.54\text{ cm}$ are exact, so an answer built only from them is exact unless you round it yourself. The fact $1\text{ mi} \approx 1.609\text{ km}$ is rounded, so its conversion factor is only about $1$ and any answer that uses it is an approximation. For example, multiplying $\frac{10\text{ mi}}{1\text{ h}}$ by $\frac{1.609\text{ km}}{1\text{ mi}}$ gives $10 \times 1.609 = 16.09$ kilometers per hour, so you write $\frac{10\text{ mi}}{1\text{ h}} \approx \frac{16.09\text{ km}}{1\text{ h}}$ and say about $16.09$ kilometers per hour.

Here is the idea to keep: converting a rate or a measurement changes its units, not the relationship it describes. You multiply by conversion factors that equal $1$ , or about $1$ when a fact is rounded, turned so that each unwanted unit cancels, and you divide only by nonzero numbers. Once two rates are written in matching units, their numbers can be compared. Next, you will compare unit rates, which are rates given for $1$ unit of the second quantity, and use them to solve word problems.