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Unit Conversion with Ratios

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 11. Multiplying a measurement by 11 changes the units without changing the actual amount.

For example, since 1ย inch=2.54ย centimeters1 \text{ inch} = 2.54 \text{ centimeters}, our conversion ratio is: 2.54ย cm1ย in\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 3.53.5 inches?

  1. Find the conversion fact: 1ย inch=2.54ย cm1 \text{ inch} = 2.54 \text{ cm}
  2. Set up the ratio: We want centimeters, so we put cm\text{cm} on the top of the fraction and inches on the bottom. 3.5ย inร—2.54ย cm1ย in3.5 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}}
  3. Multiply: The "inches" units cancel out. 3.5ร—2.54=8.89ย cm3.5 \times 2.54 = 8.89 \text{ cm}

Example 2: Miles to Kilometers

Question: Convert 55 miles to kilometers.

  1. Find the conversion fact: 1ย mileโ‰ˆ1.61ย km1 \text{ mile} \approx 1.61 \text{ km}
  2. Set up the ratio: 5ย miร—1.61ย km1ย mi5 \text{ mi} \times \frac{1.61 \text{ km}}{1 \text{ mi}}
  3. Multiply: 5ร—1.61=8.05ย km5 \times 1.61 = 8.05 \text{ km}

Example 3: Kilograms to Pounds

Question: Convert 10ย kg10 \text{ kg} to pounds (lbs\text{lbs}).

  1. Find the conversion fact: 1ย kgโ‰ˆ2.2ย lbs1 \text{ kg} \approx 2.2 \text{ lbs}
  2. Set up the ratio: We want pounds, so lbs\text{lbs} goes on top. 10ย kgร—2.2ย lbs1ย kg10 \text{ kg} \times \frac{2.2 \text{ lbs}}{1 \text{ kg}}
  3. Multiply: 10ร—2.2=22ย lbs10 \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.

Startingย Unitร—Desiredย UnitStartingย Unit=Desiredย Unit\text{Starting Unit} \times \frac{\text{Desired Unit}}{\text{Starting Unit}} = \text{Desired Unit}