Unit Conversion with Ratios

Have you ever wondered how to figure out your height in centimeters if you only know it in inches? Or how to convert a speed limit from miles to kilometers? We can solve these problems easily using ratios and proportional reasoning.

What is a Conversion Ratio?

A conversion ratio (or conversion factor) is a fraction where the numerator and the denominator represent equal amounts, just in different units. Because the top and bottom are equal in value, the fraction represents exactly $1$. Multiplying a measurement by $1$ changes the units without changing the actual amount.

For example, since $1 \text{ inch} = 2.54 \text{ centimeters}$, our conversion ratio is: $\frac{2.54 \text{ cm}}{1 \text{ in}}$

To convert units, multiply your starting measurement by a conversion ratio that puts the unit you want on top and the unit you want to cancel out on the bottom.

Step-by-Step Examples

Example 1: Inches to Centimeters

Question: How many centimeters are in $3.5$ inches?

1. Find the conversion fact: $1 \text{ inch} = 2.54 \text{ cm}$
2. Set up the ratio: We want centimeters, so we put $\text{cm}$ on the top of the fraction and inches on the bottom. $3.5 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}}$
3. Multiply: The "inches" units cancel out. $3.5 \times 2.54 = 8.89 \text{ cm}$

Example 2: Miles to Kilometers

Question: Convert $5$ miles to kilometers.

1. Find the conversion fact: $1 \text{ mile} \approx 1.61 \text{ km}$
2. Set up the ratio: $5 \text{ mi} \times \frac{1.61 \text{ km}}{1 \text{ mi}}$
3. Multiply: $5 \times 1.61 = 8.05 \text{ km}$

Example 3: Kilograms to Pounds

Question: Convert $10 \text{ kg}$ to pounds ($\text{lbs}$).

1. Find the conversion fact: $1 \text{ kg} \approx 2.2 \text{ lbs}$
2. Set up the ratio: We want pounds, so $\text{lbs}$ goes on top. $10 \text{ kg} \times \frac{2.2 \text{ lbs}}{1 \text{ kg}}$
3. Multiply: $10 \times 2.2 = 22 \text{ lbs}$

The Golden Rule of Unit Conversion

Always write out your units! When you set up your multiplication, the unit you are converting from should be in the denominator of your ratio so that it cancels out beautifully.

$\text{Starting Unit} \times \frac{\text{Desired Unit}}{\text{Starting Unit}} = \text{Desired Unit}$