Conversion Factors

Many problems involve converting measurements from one unit to another. You can convert units by using an equation that shows how units are related. For example, 1 in. = 2.54 cm relates inches and centimeters.

To write a conversion factor, divide both sides of the equation by 1 in.

$\genfrac{}{}{0.1ex}{}{1in.}{1in.}=\genfrac{}{}{0.1ex}{}{2.54cm}{1in.}1=2.54cm/in.$

Because the conversion factor is equal to 1, you can multiply one side of an equation by it and preserve equality. You can make a second conversion factor by dividing both sides of the equation by 2.54 cm.

$\genfrac{}{}{0.1ex}{}{1in.}{2.54cm}=\genfrac{}{}{0.1ex}{}{2.54cm}{2.54cm}=1$

One conversion factor converts inches to centimeters and the other converts centimeters to inches. Choose the conversion factor that cancels out the unit that you have a measurement for.

Example

Convert 25 inches to centimeters. Use the conversion factor 2.54 cm/in. so that the inches units cancel.

$25in.\times \genfrac{}{}{0.1ex}{}{2.54cm}{1in.}=64cm$

Some conversions are more complicated and require multiple steps.

Example

Convert 23°F to a Celsius temperature. The conversion formula is ${}^{\circ }F=\left(\genfrac{}{}{0.1ex}{}{9}{5}{\times }^{\circ }C\right)+{32}^{\circ }F$.

First solve the equation for °C.

Now substitute in 23°F.

${}^{\circ }C=\genfrac{}{}{0.1ex}{}{5}{9}({23}^{\circ }F-{32}^{\circ }F)=\genfrac{}{}{0.1ex}{}{5}{9}(-9)=-5$

Thus, 23°F is equivalent to −5°C.

Example

A grocer is selling oranges at a price of 3 for $1.00. How much would 10 oranges cost?

Use the equality 3 oranges = $1.00 to write a conversion factor. The desired conversion factor should have dollars in the numerator so that the oranges units cancel.

$10oranges\times \genfrac{}{}{0.1ex}{}{\$1.00}{3oranges}=\$3.33$

Example

Water runs through a hose at a rate of 2.5 gallons per minute. What is the rate of water flow in gallons per day?

To convert gal/min to gal/d, you must use conversion factors based on the following equalities.

$60min=1h24h=1d\genfrac{}{}{0.1ex}{}{2.5gal}{min}\times \genfrac{}{}{0.1ex}{}{60min}{1h}\times \genfrac{}{}{0.1ex}{}{24h}{1d}=3600gal/d$

Example

The density of nitrogen gas is 1.17 g/L. What is the density of nitrogen expressed in micrograms per deciliter (μg/dL)?

Derive the needed conversion factors from the following equalities.

$$\begin{gathered} {10}^6\mu g=1g \\ 10dL=1L \\ \genfrac{}{}{0.1ex}{}{1.17g}{L}\times \genfrac{}{}{0.1ex}{}{{10}^6\mu g}{1g}\times \genfrac{}{}{0.1ex}{}{1L}{10dL}=1.17\times {10}^5\mu g/dL \end{gathered}$$

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