Converting Rates Using Dimensional Analysis

To convert one rate to another, we will be using the same technique introduced in the previous lesson called dimensional analysis. We will explore examples involving the conversion of a single unit in the rate, the conversion of both units, and some unusual cases as well.

Example 1: Single unit

Convert 45 kilometers per hour to kilometers per minute.

Note:

• 1 hour = 60 minutes

Solution

For this rate, we are only expected to change hours to minutes – the denominator unit.

Rewrite the conversion factor into a fraction such that the hours are in the numerator position and the minutes represent the denominator \(\frac{1\;\mathrm{hr}}{60\;\mathrm{min}}\). Only by orienting it this way will the hour units cancel out.

\[\require{cancel} \frac{45\;\mathrm{km}}{\cancel{\mathrm{hr}}} \times \frac{1\;\cancel{\mathrm{hr}}}{60\;\mathrm{min}}\]

\[\frac{45\;\mathrm{km}}{60\;\mathrm{min}} = \boxed{0.75 \frac{\;\mathrm{km}}{\;\mathrm{min}} \quad\text{or}\quad \frac{0.75\;\mathrm{km}}{\;\mathrm{min}}}\]

Therefore, 45 kilometers per hour is 0.75 kilometers per minute.

Example 2: Both units

Convert 2.3 gallons per minute to milliliters per second.

Note:

• 1 minute = 60 seconds
• 1 liter = 1000 milliliters
• 1 gallon = 3.78541 liters

Solution

We need to change gallons → milliliters and minutes → seconds. There is no direct pathway from gallons to mL, but there is a direct pathway from minutes to seconds.

Based on the conversion factors given, to convert gallons to mL we must convert gallons → liters (L) → mL — two conversion factors are required. Coupled with this, we will also include the third conversion factor to change minutes into seconds.

\[\frac{2.3\;\cancel{\color{blue}{\mathrm{gal}}}}{\cancel{\color{red}{\mathrm{min}}}} \times \frac{3.78541\;\cancel{\color{orange}{\mathrm{L}}}}{1\;\cancel{\color{blue}{\mathrm{gal}}}} \times \frac{1000\;\mathrm{mL}}{1\;\cancel{\color{orange}{\mathrm{L}}}} \times \frac{1\;\cancel{\color{red}{\mathrm{min}}}}{60\;\mathrm{sec}} \]

\[\frac{2.3 \times 3.78541 \times 1000\;\mathrm{mL}}{60\;\mathrm{sec}} = \frac{8706.443\;\mathrm{mL}}{60\;\mathrm{sec}}\]

\[\approx 145.1 \frac{\mathrm{mL}}{\mathrm{sec}}\]

Therefore, 2.3 gallons per minute is approximately 145.1 milliliters per second.

Example 3: Unusual case

The density of the rare earth metal platinum is 21.4 grams per cubic centimeter. Convert this quantity to ounces per cubic inches.

Note:

• 1 ounce = 28.3495 grams
• 1 inch = 2.54 centimeters

Solution

Density is a measure of mass per volume. Here, the density of platinum is in \(\frac{\mathrm{g}}{\mathrm{cm}^3}\) (metric units) and it needs to be made into \(\frac{\mathrm{oz}}{\mathrm{in}^3}\) (imperial units).

We are given a direct conversion factor that changes grams → ounces. However, the conversion factor that relates inches and centimeters cannot be used directly; 1 inch = 2.54 centimeters must be modified to reflect cubic units. Thus, just as we did here, we will apply the same strategy to this conversion factor:

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

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

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

Now we can apply both conversion ratios to the conversion of this quantity:

\[\frac{21.4\;\cancel{\color{red}{\mathrm{g}}}}{\cancel{\color{blue}{\mathrm{cm}^3}}} \times \frac{16.387064\;\cancel{\color{blue}{\mathrm{cm}^3}}}{1\;\mathrm{in}^3} \times \frac{1\;\mathrm{oz}}{28.3495\;\cancel{\color{red}{\mathrm{g}}}}\]

\[\frac{350.683...\;\mathrm{oz}}{28.3495\;\mathrm{in}^3}\]

\[\approx 12.7\;\frac{\mathrm{oz}}{\mathrm{in}^3}\]

Therefore, 21.4 grams per cubic centimeter is approximately 12.7 ounces per cubic inches.

Supplementary Video