Convert units with dimensional analysis

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Difficulty: Intermediate
Estimated time: 40 minutes
Domains: Problem Solving and Data Analysis
Techniques: Dimensional-analysisConversion-factorsUnit-cancellationDerived-unitsSquared-and-cubed-units

What you’ll learn

See why a conversion factor equals $1$ .
Set up a factor so the unit you don’t want cancels.
Build one-step and multistep conversion chains.
Convert both parts of a derived unit like miles per hour.
Raise a complete conversion factor to the right power for square and cubic units.
Do quick arithmetic by hand, and type longer arithmetic into Desmos as one expression once the units are right.

Prerequisites

Why this matters on the SAT

Let the units check your setup

SAT unit-conversion questions might ask for a familiar measurement in a new unit, a speed with both of its units changed, or an area or volume where the conversion factor needs a power. The numbers can look like they have nothing to do with each other. Dimensional analysis, which means keeping track of the units as you go, gives you a built-in check: every unit you don’t want has to cancel, and the unit the question asks for has to be what’s left.

SAT example

A delivery vehicle travels at a constant speed of $54$ miles per hour. What is this speed, in feet per second?

Use $1$ mile $=5{,}280$ feet.

A
$15.0$
B
$79.2$
C
$237.6$
D
$285{,}120$

Solution to the example

Start with the speed you’re given. You need to change miles to feet and hours to seconds:

54\frac{\text{miles}}{\text{hour}} \left(\frac{5{,}280\text{ feet}}{1\text{ mile}}\right) \left(\frac{1\text{ hour}}{3{,}600\text{ seconds}}\right)

Miles cancel with miles and hours with hours, so feet per second is what’s left. Now type 54*5280/3600 into Desmos in one go. It gives $79.2$ , and the answer is B. Choice D is the trap: it changes miles to feet but never changes hours to seconds.

That’s the whole method in four words: units first, numbers second.

On paper, set up each factor and cancel units until only the unit the question asks for is left.
Then do the numbers. Some are quick by hand, like $3.4\times100$ . For a longer chain like this one, type every number into Desmos as one expression, instead of working out a piece such as $5{,}280\div3{,}600$ by hand first.

Desmos hands back a bare $79.2$ . It knows nothing about miles or seconds, so your written setup is what tells you the answer is $79.2$ feet per second.

Calculator loads as you approach

Miles and hours cancelled on paper, so the whole chain goes in at once: 79.2 feet per second.

Make unwanted units cancel

A conversion factor is a fraction made from two measurements of the same amount. Since the top and the bottom are equal, the fraction equals $1$ .

For example, $1$ meter and $100$ centimeters are the same length, so both of these fractions equal $1$ :

\frac{100\text{ centimeters}}{1\text{ meter}} \qquad\text{and}\qquad \frac{1\text{ meter}}{100\text{ centimeters}}.

Multiplying by $1$ doesn’t change the actual amount. Only the unit and the number change.

Say you want to change $3.4$ meters to centimeters. Start with what you’re given:

3.4\text{ meters}.

Meters have to go, so put meters on the bottom of the factor:

3.4\cancel{\text{ meters}} \left(\frac{100\text{ centimeters}}{1\cancel{\text{ meter}}}\right) =340\text{ centimeters}.

The cancellation told you which way up the factor goes. You didn’t need a memorized rule about which way the decimal moves. It moved because every meter became $100$ centimeters.

Check your understanding:

Why doesn’t multiplying by $\frac{12\text{ inches}}{1\text{ foot}}$ change the actual length of an object?

Common mistake:

It’s tempting to choose between multiplying and dividing by asking whether the new unit is smaller. That shortcut gets unreliable in a long chain or a rate like miles per hour. Instead, put the unit you want gone on the other side of the fraction bar, cancel it, and check that the unit the question asks for is what’s left.

Orient one factor, then build a chain

Every conversion comes down to three moves:

Write down the quantity you’re given, with its unit.
Multiply by a factor that cancels the unit you don’t want yet.
Keep going until only the unit the question asks for is left.

One factor

To change $7.5$ feet to inches, put feet on the bottom of the fraction:

7.5\cancel{\text{ feet}} \left(\frac{12\text{ inches}}{1\cancel{\text{ foot}}}\right) =90\text{ inches}.

Flip that factor over and you’d end up with feet squared over inches, which can’t answer a question about inches.

Several factors

You don’t need one fact that links your starting unit straight to your ending unit. Other units can act as a bridge.

For example, change $2.5$ hours to seconds, with minutes as the bridge:

2.5\cancel{\text{ hours}} \left(\frac{60\cancel{\text{ minutes}}}{1\cancel{\text{ hour}}}\right) \left(\frac{60\text{ seconds}}{1\cancel{\text{ minute}}}\right)

Only seconds are left, so all three numbers go into Desmos together: 2.5*60*60 gives $9{,}000$ . So $2.5$ hours is $9{,}000$ seconds.

Read the chain from left to right:

\text{hours}\longrightarrow\text{minutes}\longrightarrow\text{seconds}.

Each bridge unit shows up once on top and once on the bottom, so it cancels out. Keep every unit written in until the whole path works. The numbers can wait.

Check your understanding:

You need to change a length in yards to centimeters, using $1$ yard $=3$ feet and $1$ foot $=30.48$ centimeters. Write the two conversion factors in the right order, each the right way up. You don’t need to calculate.

Converting a rate or finding one?

Use dimensional analysis when the question gives you a quantity or a rate and asks for the same amount in different units. Watch for wording like “equivalent to,” “in feet per second,” “in square meters” or “use the following relationships.”

When the main job is to build or compare a rate from raw numbers, that’s a rates and unit rates question. Finding miles per gallon from the miles driven and the gallons used is a rate problem. Changing a miles-per-gallon value you already know into kilometers per liter is a conversion.

Some SAT questions need both. Find the rate first, then convert it with a chain. The units show you where one job ends and the next one starts.

Check your understanding:

What’s the main job in each question? (1) A cyclist travels $45$ miles in $3$ hours. Find miles per hour. (2) A cyclist travels at $15$ miles per hour. Convert the speed to feet per second.

Convert both parts of a derived unit

A derived unit is built from other units by multiplying or dividing. Speed, density, fuel efficiency and price per area all use one:

\frac{\text{miles}}{\text{hour}}, \qquad \frac{\text{grams}}{\text{cubic centimeter}}, \qquad \frac{\text{dollars}}{\text{square foot}}.

Units multiplied together work the same way. Changing kilowatt-hours to watt-hours, for example, changes the power unit and leaves the hours alone:

2.4\cancel{\text{ kilowatts}}\cdot\text{hours} \left(\frac{1{,}000\text{ watts}}{1\cancel{\text{ kilowatt}}}\right) =2{,}400\text{ watt-hours}.

When the question changes the unit on top and the unit on the bottom, convert both. Read “per” as a fraction bar, then check the top and the bottom separately.

Say a runner’s speed is $18$ meters per second, and you want kilometers per hour:

18\frac{\cancel{\text{meters}}}{\cancel{\text{second}}} \left(\frac{1\text{ kilometer}}{1{,}000\cancel{\text{ meters}}}\right) \left(\frac{3{,}600\cancel{\text{ seconds}}}{1\text{ hour}}\right)

That leaves kilometers per hour, and 18/1000*3600 in Desmos gives $64.8$ . The speed is $64.8$ kilometers per hour.

The time factor might look upside down, since an hour is longer than a second. Cancellation settles it: seconds start on the bottom, so they have to be on top in a factor to cancel.

Common mistake:

It’s easy to convert only the top of a rate. Changing $18$ meters per second to $0.018$ kilometer per second is a fine first step, but it isn’t kilometers per hour yet. Convert the bottom unit too, then check that each unit the question asks for sits where it should, on top or on the bottom.

Check your understanding:

To change gallons per minute to liters per hour, should minutes go on the top or the bottom of the time factor? Use cancellation to explain.

Match the power of square and cubic units

A linear conversion factor, like one between yards and feet, converts a length, which has one dimension. An area has two length dimensions, and a volume has three.

Picture a square yard. It’s $3$ feet long and $3$ feet wide, so it covers $9$ square feet, not $3$ . That’s why the yard-to-foot factor gets squared for area:

\left(\frac{1\text{ yard}}{3\text{ feet}}\right)^2 = \frac{1\text{ square yard}}{9\text{ square feet}}.

For cubic units, use three copies:

\left(\frac{1\text{ yard}}{3\text{ feet}}\right)^3 = \frac{1\text{ cubic yard}}{27\text{ cubic feet}}.

So raise the complete conversion factor, number and units together, to the same power as the starting unit:

\boxed{ \text{square units}\rightarrow(\text{linear factor})^2 \qquad \text{cubic units}\rightarrow(\text{linear factor})^3 }

For example, change $1{,}458$ square feet to square yards:

1{,}458\cancel{\text{ feet}^2} \left(\frac{1\text{ yard}}{3\text{ feet}}\right)^2

The square applies to the $3$ and to both unit labels. In Desmos, 1458*(1/3)^2 gives $162$ , so the area is $162$ square yards. Use the linear factor only once and one foot is left over that never cancels.

Check your understanding:

Given $1$ meter $=100$ centimeters, fill in the blank: $1\text{ cubic meter}=\ ?\text{ cubic centimeters}$ . Explain the exponent.

Common mistake:

A common slip is using the linear factor only once, or raising only one unit label to the power. Put the whole fraction in parentheses before you apply the exponent. Then write the units out once to check that every copy of the starting unit cancels.

Example: Convert a rainfall rate

Worked example

During a storm, rain accumulates at a rate of $6.8$ millimeters per hour. Which choice is closest to this rate in gallons per square yard per minute?

Use $1$ inch $=25.4$ millimeters, $1$ yard $=36$ inches, and $1$ gallon $=231$ cubic inches.

A
$0.0000193$
B
$0.0250$
C
$1.50$
D
$90.0$

Step 1

Compare the starting and ending units

Rainfall depth is a length per time:

\frac{\text{millimeters}}{\text{hour}}.

Now picture one square yard of ground in the storm. After an hour, it’s covered in water $6.8$ millimeters deep. That water has a volume: its depth times the square yard it covers. Gallons measure volume, so the question wants the gallons that land on each square yard in each minute:

\frac{\text{volume}}{\text{area}\cdot\text{time}}.

That gives the chain four jobs: change millimeters to inches, bring in square inches for each square yard, change cubic inches to gallons, and change hours to minutes.

Step 2

Build the unit path

Square yards belong on the bottom, so use the yard-to-inch fact twice by squaring it:

\begin{aligned} 6.8\frac{\cancel{\text{millimeters}}}{\cancel{\text{hour}}} &\left(\frac{1\text{ inch}}{25.4\cancel{\text{ millimeters}}}\right) \left(\frac{36\text{ inches}}{1\text{ yard}}\right)^2\\[1.4em] &\left(\frac{1\text{ gallon}}{231\text{ inches}^3}\right) \left(\frac{1\cancel{\text{ hour}}}{60\text{ minutes}}\right). \end{aligned}

This is the tricky part, so count the inches on top. One comes from the rainfall depth and two come from the squared area factor. Together they make cubic inches, which cancel with the cubic inches in $1$ gallon.

Step 3

Let the calculator handle the arithmetic

The units work, so type the numbers as one expression:

6.8/25.4*36^2/231/60

Desmos gives about

0.0250332.

The closest choice is $0.0250$ , so the answer is B.

Calculator loads as you approach

Your written cancellation settles the units. Desmos works out the whole chain of numbers in one expression.

Try it yourself:

Cover the calculator line and check only the units. Find the three inch factors that make cubic inches, then point to the factors that leave square yards and minutes on the bottom. If the units work, type the chain in one go.

Practice problems

Same routine on every problem: units first, numbers second.

Convert time in one step

Practice problem

How many seconds are equivalent to $2.75$ hours?

Use $1$ hour $=60$ minutes and $1$ minute $=60$ seconds.

Your answer

Calculator loads as you approach

Once hours and minutes cancel, type the whole product.

Link two length conversions

Practice problem

A support beam is $12.5$ feet long. What is its length, in centimeters?

Use $1$ foot $=12$ inches and $1$ inch $=2.54$ centimeters.

Calculator loads as you approach

Once feet and inches cancel, type the whole product.

Convert both parts of a speed

Practice problem

A drone travels at a constant speed of $22$ meters per second. What is this speed, in kilometers per hour?

Use $1$ kilometer $=1{,}000$ meters and $1$ hour $=3{,}600$ seconds.

Calculator loads as you approach

Once both parts of the rate cancel the right way, type the whole chain.

Convert area before applying a unit price

Practice problem

A floor has an area of $68$ square meters. A protective finish costs $3.15 per square foot. What is the total cost of the finish, to the nearest dollar?

Use $1$ meter $=3.28$ feet.

Calculator loads as you approach

Write the squared factor first, then type the area and the price together.

Build a cubic volume chain

Practice problem

A tank contains $32$ gallons of water. Approximately how many cubic inches of water does the tank contain?

Use the following relationships:

$1$ gallon $=3.785$ liters
$1$ liter $=1{,}000$ cubic centimeters
$1$ inch $=2.54$ centimeters

Calculator loads as you approach

Set up the whole unit chain, then type its numbers as one expression.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A conversion factor equals $1$ because its top and bottom are the same amount.
Start with what you’re given, put each unit you don’t want on the other side of a fraction bar, and keep going until the unit you want is left.
In a rate or other derived unit, convert every unit that changes, top and bottom.
Square a complete linear factor for area, and cube it for volume.
Finding a new rate and converting a rate you already have are different jobs, even when one SAT question asks for both.
Units first, numbers second: settle the units on paper, then do quick arithmetic by hand and type anything longer into Desmos as one expression.

Related lessons

Go back to Use rates and unit rates when the main job is building or comparing a rate. For more calculator practice with long chains you can check step by step, try Build ratio, rate, and unit chains in Desmos.

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