Conversion Factors

Intuition

In the Rates lesson you turned gallons per second into gallons per minute by scaling both parts of the rate together. That works, but it asks you to invent the right multiplier each time and to keep the units straight in your head. There is a cleaner tool that never changes: the conversion factor. It rests on one plain observation. Since $1$ foot and $12$ inches are the same length, the fraction $\frac{12 \text{ in}}{1 \text{ ft}}$ has a top and a bottom that measure the same thing, so it equals $1$ . Multiply any length by it and the amount cannot change, because you multiplied by $1$ , yet the units flip from feet to inches. Setting up these factors so that the units you do not want cancel is called dimensional analysis. Dimensional analysis is the workhorse that scientists and engineers use to move between units without ever second-guessing the arithmetic.

Learning goals

Build a factor from two equal amounts, so it equals one
Orient the factor so the unwanted unit cancels
Chain several factors when one hop is not enough
Convert a rate with two factors, one per unit
Raise the factor to a power for area and volume

A conversion factor is the number one in disguise

Every unit conversion rests on an equality between two amounts. A foot is twelve inches. A kilometer is a thousand meters. An hour is sixty minutes. Each such fact, read as an equation, lets you build a fraction that equals exactly $1$ , and that fraction is a conversion factor.

Take $1 \text{ ft} = 12 \text{ in}$ . Because the two sides are equal, you may write the ratio of one to the other, and a ratio of equal quantities is $1$ :

\frac{12 \text{ in}}{1 \text{ ft}} = 1 \qquad \text{and} \qquad \frac{1 \text{ ft}}{12 \text{ in}} = 1.

Both fractions equal $1$ ; they are the same fact written two ways. Which one you reach for depends on the unit you want to cancel, and that choice is the only real decision in the whole method. First, though, here is why multiplying by such a factor is always allowed.

Why multiplying by a conversion factor keeps the amount the same#

Start from a fact about equal quantities. The statement $12 \text{ in} = 1 \text{ ft}$ says that twelve inches and one foot are two names for the very same length. Divide both sides of that equation by $1 \text{ ft}$ , and the right side becomes $1$ :

\frac{12 \text{ in}}{1 \text{ ft}} = \frac{1 \text{ ft}}{1 \text{ ft}} = 1.

So the conversion factor $\frac{12 \text{ in}}{1 \text{ ft}}$ is not close to $1$ ; it is exactly $1$ , because its top and bottom measure the same length. Dividing instead by $12 \text{ in}$ gives the other orientation, $\frac{1 \text{ ft}}{12 \text{ in}} = 1$ . Every conversion factor is a fraction whose numerator and denominator name equal amounts, so every conversion factor equals $1$ .

Now recall the multiplicative identity: multiplying any quantity by $1$ leaves it unchanged. When you multiply a length by a conversion factor, you are multiplying it by $1$ , so the amount of length cannot change. What changes is only the label. That is the whole justification: a conversion factor is the number $1$ in disguise, dressed up to trade one unit for another.

The last piece is why the units cancel. A unit behaves inside a fraction exactly as a number does. Just as $\frac{5}{5} = 1$ and a shared factor of $5$ cancels from a numerator and denominator, a shared factor of $\text{ft}$ cancels the same way. In $3 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}}$ the $\text{ft}$ on top of the first quantity meets the $\text{ft}$ on the bottom of the factor. Those two form $\frac{\text{ft}}{\text{ft}} = 1$ and disappear, leaving $3 \times 12 = 36$ carrying the unit $\text{in}$ . You did not delete the foot by decree; it cancelled because a quantity over itself is $1$ , the same rule that cancels numbers. This is why lining up the factor so the unwanted unit sits opposite itself is the entire technique.

Writing a factor both ways and choosing the orientation

A conversion factor comes in two orientations, and only one of them cancels the unit you are trying to remove. The rule is short: put the unwanted unit on the opposite side of the fraction from where it sits now. A plain measurement like $7 \text{ ft}$ carries its unit on top, so the factor needs that unit on the bottom; then the two line up as $\frac{\text{ft}}{\text{ft}}$ and cancel. Pick the wrong orientation and nothing cancels, which is your signal to flip the factor over.

Worked example 1 Convert $7$ feet to inches

Start with the quantity $7 \text{ ft}$ and the fact $1 \text{ ft} = 12 \text{ in}$ . You want feet to disappear and inches to appear, so choose the orientation with feet in the denominator, $\frac{12 \text{ in}}{1 \text{ ft}}$ . Multiply, and cancel the $\text{ft}$ that shows up on top and bottom:

7 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} = 7 \times 12 \text{ in} = 84 \text{ in}.

The foot cancels because $\frac{\text{ft}}{\text{ft}} = 1$ , leaving inches, so $7$ feet is $84$ inches. Had you picked $\frac{1 \text{ ft}}{12 \text{ in}}$ instead, nothing would cancel and you would be left with the meaningless $\text{ft}^2/\text{in}$ , the built-in warning that the factor is upside down.

Check your understanding

You want to convert $48$ inches into feet using $1 \text{ ft} = 12 \text{ in}$ . Which conversion factor should you multiply by?

\dfrac{1 \text{ ft}}{12 \text{ in}}

\dfrac{12 \text{ in}}{1 \text{ ft}}

\dfrac{12 \text{ ft}}{1 \text{ in}}

\dfrac{48 \text{ in}}{1 \text{ ft}}

Chaining several factors

When no single factor connects your starting unit to your target unit, string several together. Each factor still equals $1$ , so multiplying by all of them at once is still multiplying by $1$ , and the amount is preserved through the whole chain. You arrange the factors so that every intermediate unit appears once on top and once on the bottom. That arrangement lets each intermediate unit cancel and hands you the next unit in line.

Worked example 2 How many seconds are in one day?

No single conversion factor turns days straight into seconds, but a chain of familiar ones does the job. Use $1 \text{ day} = 24 \text{ h}$ , then $1 \text{ h} = 60 \text{ min}$ , then $1 \text{ min} = 60 \text{ s}$ , and line the factors up so each unwanted unit cancels the one before it:

1 \text{ day} \times \frac{24 \text{ h}}{1 \text{ day}} \times \frac{60 \text{ min}}{1 \text{ h}} \times \frac{60 \text{ s}}{1 \text{ min}}.

Read the units down the line: $\text{day}$ cancels the starting day, and $\text{h}$ cancels between the first and second factors. Then $\text{min}$ cancels between the second and third, leaving only $\text{s}$ . Now multiply the numbers:

24 \times 60 \times 60 = 86{,}400 \text{ s}.

So a day holds $86{,}400$ seconds. The chain never needed a day-to-second constant; each factor was one you already knew, and the intermediate units bookkept themselves by cancelling.

Check your understanding

How many minutes are in $3$ days? Use $1 \text{ day} = 24 \text{ h}$ and $1 \text{ h} = 60 \text{ min}$ .

72

minutes B

180

minutes C

4320

minutes D

1440

minutes

Converting rates

A rate such as miles per hour is a fraction with a unit above and a unit below, so it has two units to convert, not one. Handle each with its own factor: one factor cancels the top unit and installs the new one, and a second factor does the same for the bottom unit. Because a bottom unit sits in the denominator, its factor goes in the orientation that cancels a denominator, which usually means flipping it relative to the top factor.

Worked example 3 Convert $30$ miles per hour to feet per second

A rate carries two units, one on top and one on the bottom. Converting a rate therefore takes two factors: one to fix the distance unit and one to fix the time unit. Write the speed as a fraction and attach both factors, using $1 \text{ mi} = 5280 \text{ ft}$ and $1 \text{ h} = 3600 \text{ s}$ :

\frac{30 \text{ mi}}{1 \text{ h}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}}.

The $\text{mi}$ on top cancels the $\text{mi}$ in the second factor’s denominator, and the $\text{h}$ in the first denominator cancels the $\text{h}$ in the third numerator. What remains is $\text{ft}$ over $\text{s}$ , exactly the unit you want. Multiply the numbers, remembering the $3600$ divides:

\frac{30 \times 5280}{3600} = \frac{158{,}400}{3600} = 44 \text{ ft/s}.

So $30$ miles per hour is $44$ feet per second. Notice the time factor is flipped compared with the distance factor: hours needed to leave the denominator, so hours went on top of its factor. Miles, in contrast, needed to leave the numerator, so miles went on the bottom of its factor.

Converting 30 miles per hour to feet per second uses two factors. Line them up so mi meets mi and h meets h on opposite sides of the bar; each cancels as a quantity over itself, leaving feet on top and seconds on the bottom. Then 30 times 5280 divided by 3600 is 44.

Squared and cubed units

Area and volume carry powers of a unit, like square feet or cubic centimeters, and a power changes how many times you must convert. A squared unit hides two lengths and a cubed unit hides three. The conversion factor therefore has to be applied once for each hidden length, and that repetition raises the factor to that power.

Worked example 4 Convert $3$ square feet to square inches

It is tempting to multiply by $\frac{12 \text{ in}}{1 \text{ ft}}$ just once, but that leaves a stray unit and the wrong number. Square feet means feet multiplied by feet, $1 \text{ ft}^2 = 1 \text{ ft} \times 1 \text{ ft}$ , so there are two lengths to convert and the factor must appear twice:

3 \text{ ft}^2 \times \frac{12 \text{ in}}{1 \text{ ft}} \times \frac{12 \text{ in}}{1 \text{ ft}} = 3 \text{ ft}^2 \times \left(\frac{12 \text{ in}}{1 \text{ ft}}\right)^2.

Squaring the factor squares both the number and the unit, giving $\frac{144 \text{ in}^2}{1 \text{ ft}^2}$ , and the $\text{ft}^2$ now cancels the $\text{ft}^2$ you started with:

3 \text{ ft}^2 \times \frac{144 \text{ in}^2}{1 \text{ ft}^2} = 3 \times 144 = 432 \text{ in}^2.

So $3$ square feet is $432$ square inches. The factor is still equal to $1$ , since $1^2 = 1$ , so squaring it changes nothing about the amount, only how many times the unit gets converted.

Worked example 5 Convert $2$ cubic yards to cubic feet

Volume raises the same idea one step higher. A cubic yard is three lengths multiplied together, $1 \text{ yd}^3 = 1 \text{ yd} \times 1 \text{ yd} \times 1 \text{ yd}$ , so the yard-to-foot factor must appear three times, which cubes it. With $1 \text{ yd} = 3 \text{ ft}$ ,

2 \text{ yd}^3 \times \left(\frac{3 \text{ ft}}{1 \text{ yd}}\right)^3 = 2 \text{ yd}^3 \times \frac{27 \text{ ft}^3}{1 \text{ yd}^3}.

Cubing the factor turns $3$ into $3^3 = 27$ and turns the unit into $\text{ft}^3$ , which cancels the $\text{yd}^3$ you began with:

2 \times 27 = 54 \text{ ft}^3.

So $2$ cubic yards is $54$ cubic feet. The rule generalizes cleanly: a squared unit needs the factor squared, and a cubed unit needs it cubed. In the same way, an $n$ th power needs the factor to the $n$ th, because that is how many separate lengths are hidden inside the unit.

Check your understanding

A tile covers $2$ square feet. How many square inches is that? Use $1 \text{ ft} = 12 \text{ in}$ .

24

square inches B

288

square inches C

144

square inches D

48

square inches

Common mistakes

Writing the factor upside down. If the unwanted unit does not cancel, the factor is flipped. To convert $60$ inches to feet, inches must be in the denominator ( $\frac{1 \text{ ft}}{12 \text{ in}}$ ); using $\frac{12 \text{ in}}{1 \text{ ft}}$ leaves $\text{in}^2/\text{ft}$ and a wrong number. Let the units, not guesswork, choose the orientation.
Converting a squared or cubed unit only once. Square feet to square inches needs the factor twice ( $12^2 = 144$ ), and cubic units need it three times ( $12^3 = 1728$ ). Multiplying by $12$ a single time is the most common area-and-volume error.
Multiplying when you should divide the time part of a rate. Turning miles per hour into feet per second means dividing by the $3600$ seconds in an hour, because hours sit in the denominator. Multiplying by $3600$ instead makes the answer far too big.
Dropping the units and trusting the numbers. The units are the check. Keep them attached through every step. If they cancel to the unit you wanted, the setup is right, and if a strange unit like $\text{ft}^2/\text{in}$ survives, the setup is wrong.
Using a ratio whose two sides are not truly equal. $\frac{12 \text{ in}}{1 \text{ ft}}$ works because a foot really is twelve inches. A made-up ratio like $\frac{10 \text{ in}}{1 \text{ ft}}$ is not $1$ , so it corrupts the amount instead of preserving it.

Key takeaways

A conversion factor is a fraction of two equal amounts (like $\frac{12 \text{ in}}{1 \text{ ft}}$ ), so it equals $1$ . Multiplying by that factor changes a quantity’s units but not how much it is.
Units cancel like numbers. A unit on top and the same unit on the bottom form $\frac{\text{unit}}{\text{unit}} = 1$ and disappear, so orient each factor to cancel the unit you do not want.
Every factor has two orientations; choose the one that puts the unwanted unit opposite itself.
Chain factors when one hop is not enough, arranging them so each intermediate unit cancels; the whole chain is still a product of $1$ s.
Rates carry two units, so they need two factors, one for the top and one for the usually flipped bottom, as in $30$ mph $= 44$ ft/s.
Squared and cubed units need the factor raised to that power: $1 \text{ ft}^2 = 144 \text{ in}^2$ and $1 \text{ ft}^3 = 1728 \text{ in}^3$ , because area hides two lengths and volume hides three.

Conversion Factors

Learning goals

A conversion factor is the number one in disguise

Writing a factor both ways and choosing the orientation

Chaining several factors

Converting rates

Squared and cubed units

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Free Response Questions (FRQ)