Converting Units Using Dimensional Analysis

There are countless ways to convert units mathematically, most of which rely on the principal of proportions. The method we will be using here is called dimensional analysis; it can be used whether you are converting imperial units to imperial, metric units to metric or even when transitioning between imperial and metric systems.

To apply dimensional analysis, you start with the given quantity and set up a series of conversion factors in a way that cancels out the unwanted units, leaving you with the desired units. Each conversion factor is essentially a ratio of equivalent values expressed with appropriate units.

Dimensional analysis

Conversion factors

A conversion factor is a known measurement that expresses the relationship between two different units of the same quantity. It serves as a bridge between the original and desired units, allowing for the accurate conversion of measurements. Conversion factors are often written as simple equations, for example:

\[1\;\mathrm{in} = 25.4\;\mathrm{mm}\]

Every conversion factor can also be written as a ratio instead, as in:

Using a single conversion factor

Suppose we want to convert \(78\;\mathrm{mm}\) to inches. Using dimensional analysis, we must multiply this value by one of the two ratios written above such that the unit of "mm" cancels out in the multiplication process – remember, only one of these can work!

Stringing together conversion factors

While this method may seem too elaborate for a single conversion, its benefits really shine when you need to string together multiple conversions at once.

Suppose we want to convert \(78\;\mathrm{mm}\) to feet, but we aren't given a direct conversion factor that relates millimeters and feet. Rather, we only know the following:

• \(1\;\mathrm{in} = 25.4\;\mathrm{mm}\)
• \(1\;\mathrm{ft} = 12\;\mathrm{in}\)

In this situation, we could apply dimensional analysis twice: once to convert millimeters to inches, then again for what we found in inches to feet. However, it is better to simply string the conversion factors together in one large multiplication statement, as this avoids intermediate rounding errors.

\[78\;\mathrm{mm} \times \frac{1\;\mathrm{in}}{25.4\;\mathrm{mm}} \times \frac{1\;\mathrm{ft}}{12\;\mathrm{in}}\]

\[78\;\color{red}{\cancel{\mathrm{mm}}} \times \frac{1\;\color{blue}{\cancel{\mathrm{in}}}}{25.4\;\color{red}{\cancel{\mathrm{mm}}}} \times \frac{1\;\mathrm{ft}}{12\;\color{blue}{\cancel{\mathrm{in}}}}\]

\[\frac{78}{1} \times \frac{1}{25.4} \times \frac{1\;\mathrm{ft}}{12} =\frac{78\;\mathrm{ft}}{304.8}\]

\[= 0.2559...\;\mathrm{ft} \approx 0.26 \;\mathrm{ft}\]

Test your understanding

Convert 135 ounces (oz) to gallons (gal) given the following conversion factors:

• \(1\;\mathrm{gallon} = 4\;\mathrm{quarts}\)
• \(1\;\mathrm{quarts} = 2\;\mathrm{pints}\)
• \(1\;\mathrm{pint} = 16\;\mathrm{ounces}\)

Solution

To convert ounces to gallons given only the conversion factors provided, we must convert ounces → pints (pt) → quarts (qt) → gallons (gal). This means all three conversion factors will be used in the mathematical statement.

\[135\;\color{red}{\cancel{\mathrm{oz}}} \times \frac{1\;\color{blue}{\cancel{\mathrm{pt}}}}{16\;\color{red}{\cancel{\mathrm{oz}}}} \times \frac{1\;\color{purple}{\cancel{\mathrm{qt}}}}{2\;\color{blue}{\cancel{\mathrm{pt}}}} \times \frac{1\;\mathrm{gal}}{4\;\color{purple}{\cancel{\;\mathrm{qt}}}} \]

\[\frac{135\;\mathrm{gal}}{16 \times 2 \times 4} = 1.0546...\;\mathrm{gal}\]

Therefore, the 135 ounces is approximately 1.05 gallons.

Modifying conversion factors

Sometimes a conversion factor may not be given directly. For instance, you may be given:

\[2.54\;\mathrm{cm} = 1\;\mathrm{in}\]

But your units are squared or cubed. If you plan to convert, for example, \(20\;\mathrm{cm}^2\) to square inches, you cannot use the conversion factor the way it is. Instead, you will have to modify it by squaring both sides of the conversion factor before using it.

\[(2.54\;\mathrm{cm})^2 = (1\;\mathrm{in})^2\]

\[6.4516\;\mathrm{cm}^2 = 1\;\mathrm{in}^2\]

Only now can the conversion factor can be used in conjunction with the dimensional analysis technique.

\[20\;\cancel{\mathrm{cm}^2} \times \frac{1\;\mathrm{in}^2}{6.4516\;\cancel{\mathrm{cm}^2}}\]

\[\approx 3.1\;\mathrm{in}^2\]

Test your understanding

Convert 5.56 cubic inches to cubic centimeters.

Solution

Let's reuse the conversion factor from earlier: \(2.54\;\mathrm{cm} = 1\;\mathrm{in}\). We will cube both sides first:

\[(2.54\;\mathrm{cm})^3 = (1\;\mathrm{in})^3\]

\[\boxed{16.387064\;\mathrm{cm}^3 = 1\;\mathrm{in}^3}\]

The conversion:

\[5.56\;\cancel{\mathrm{in}^3} \times \frac{16.387064\;\mathrm{cm}^3}{1\;\cancel{\mathrm{in}^3}}\]

\[\approx 91.1\;\mathrm{cm}^3\]

Therefore, 5.56 cubic inches is approximately 91.1 cubic centimeters.

Supplementary Video

The following video shows dimensional analysis in action using various examples.