2.03 Unit conversions
Ideas
Unit conversions
Proportions allow us to convert measurements with different units. We can use a conversion factor, which is written as a ratio in fraction form with equivalent measurements in the numerator and denominator. The first conversion values we may see are the metric system relationships:
1 \text{ km} = 1000 \text{ m}
1 \text{ m} = 100 \text{ cm}
1 \text{ cm} = 10 \text{ mm}
The conversion factors for the metric system are summarized in the conversion chart below:
In the United States, we will more regularly see relationships from the U.S. customary units:
1 \text{ mi} = 1760 \text{ yards}
1 \text{ yard} = 3 \text{ ft}
1 \text{ ft} = 12 \text{ in}
If we want to convert 8 feet into inches, the conversion value is 1 \text{ ft} = 12 \text{ in}.
| \displaystyle 8 \text{ } \cancel{\text{ft}} \cdot \dfrac{12 \text{ in} }{1 \text{ } \cancel{\text{ft}}} | \displaystyle = | \displaystyle 96 \text{ in} | Multiply the given by the conversion factor |
Notice that the unit of the given number is \text {ft} and the conversion factor has the unit \text {ft} in the denominator. This causes the \text {ft} to divide out leaving just the unit, \text {in}.
However, if we want to convert 96 inches into feet, the conversion value is the same, but we put the feet in the numerator.
| \displaystyle 96\text{ in}\cdot \dfrac{1 \text{ ft}}{12\text{ in}} | \displaystyle = | \displaystyle \dfrac {96 \text{ } \cancel{\text{in}}} {12\text{ } \cancel{\text{in}}} \cdot 1\text{ ft} | Multiply the given by the conversion factor |
| \displaystyle = | \displaystyle 8\text{ ft} | Simplify the fraction by dividing and cancelling the units |
We can also solve by setting up a proportion:
| \displaystyle \dfrac{96 \text{ in}}{x \text{ ft}} | \displaystyle = | \displaystyle \dfrac{12 \text{ in}}{1 \text{ ft}} | Setting up our proportion |
| \displaystyle 96 \cdot 1 | \displaystyle = | \displaystyle x \cdot 12 | Means Extremes Property |
| \displaystyle 96 | \displaystyle = | \displaystyle 12x | Simplify the multiplication |
| \displaystyle 8 | \displaystyle = | \displaystyle x | Divide by 12 |
Fortunately, we don't have to memorize these conversions. As long as we are given the conversion factor, we can use what we know about proportions to convert between units of measurement.
Examples
Example 1
To convert meters to inches, we can use the following table:
| Meters | Inches |
|---|---|
| 1 | 39.4 |
| 78.8 | |
| 3 | |
| 4 | |
| 197 |
Complete the table.
Using the table, convert 13 meters to inches.
Using the table, convert 709.2 inches to meters.
Example 2
To ride the scariest rollercoasters in an amusement park, Wynsleth needs to be over 160\text{ cm} tall.
If Wynsleth is 4\text{ ft } 11\text{ in} and 1 \text{ in} = 2.54 \text{ cm}, is she tall enough?
Conversion of units can be done by applying ratios and proportional relationships. If we know a relationship between the units, we can create a conversion factor equal to 1 where the units of the denominator match the given units in the problem.
\cancel{\text{Given units}} \cdot \dfrac{\text{Desired units}}{ \cancel{\text{Given units}}} = \text{Desired units}
We can also set up a proportion to convert between units.