- Choose the appropriate units of measurement for a given problem or situation
- Convert between units of measurement using conversion factors
- Perform basic arithmetic operations on units of length, weight, and capacity
- Apply knowledge of units of length, weight, and capacity to solve real-world problems.
Length
units of length
Length is the distance from one end of an object to the other end, or from one object to another. The system for measuring length in the United States is based on the four customary units of length: inch, foot, yard, and mile.
| Unit Equivalents | Conversion Factors (longer to shorter units of measurement) | Conversion Factors (shorter to longer units of measurement) |
| [latex]1[/latex] foot [latex]= 12[/latex] inches | [latex]\displaystyle \frac{12\ \text{inches}}{1\ \text{foot}}[/latex] | [latex]\displaystyle \frac{1\text{ foot}}{12\text{ inches}}[/latex] |
| [latex]1[/latex] yard [latex]= 3[/latex] feet | [latex]\displaystyle \frac{3\ \text{feet}}{1\ \text{yard}}[/latex] | [latex]\displaystyle \frac{1\text{ yard}}{3\text{ feet}}[/latex] |
| [latex]1[/latex] mile [latex]= 5,280[/latex] feet | [latex]\displaystyle \frac{5,280\text{ feet}}{1\text{ mile}}[/latex] | [latex]\displaystyle \frac{\text{1 mile}}{\text{5,280 feet}}[/latex] |
Convert Between Different Units of Length
You can use the conversion factors to convert a measurement, such as feet, to another type of measurement, such as inches.
Note that there are many more inches for a measurement than there are feet for the same measurement, as feet is a longer unit of measurement. You could use the conversion factor [latex]\displaystyle \frac{\text{12 inches}}{\text{1 foot}}[/latex].
If a length is measured in feet, and you’d like to convert the length to yards, you can think, “I am converting from a shorter unit to a longer one, so the length in yards will be less than the length in feet.” You could use the conversion factor [latex]\displaystyle \frac{\text{1 yard}}{\text{3 feet}}[/latex].
If a distance is measured in miles, and you want to know how many feet it is, you can think, “I am converting from a longer unit of measurement to a shorter one, so the number of feet would be greater than the number of miles.” You could use the conversion factor [latex]\displaystyle \frac{5,280\text{ feet}}{1\text{ mile}}[/latex].
factor label method
You can use the factor label method (also known as dimensional analysis) to convert a length from one unit of measure to another using the conversion factors. In the factor label method, you multiply by unit fractions to convert a measurement from one unit to another.
Convert a mixed number to an improper fraction
You can use a handy shortcut to convert a mixed number to its equivalent fractional form. First multiply the whole number part by the denominator of the fraction then add the numerator to the result. Finally place that number over the denominator.
[latex]a\dfrac{b}{c}=\dfrac{ac+b}{c}[/latex].
Ex. Convert [latex]3\dfrac{1}{2}[/latex] to an improper fraction.
[latex]3\dfrac{1}{2}=\dfrac{3\cdot 2 + 1}{2}=\dfrac{7}{2}[/latex].
Study the example below to see how the factor label method can be used to convert [latex]\displaystyle 3\frac{1}{2}[/latex] feet into an equivalent number of inches.
Notice that by using the factor label method you can cancel the units out of the problem, just as if they were numbers. You can only cancel if the unit being canceled is in both the numerator and denominator of the fractions you are multiplying.
[latex]\frac{7}{2}\cancel{\text{feet}}\cdot\frac{12\text{ inches}}{\cancel{1\text{foot}}}=\text{? inches}[/latex]
What if you had used the wrong conversion factor?
[latex]\frac{7}{2}\text{feet}\cdot\frac{1\text{foot}}{12\text{ inches}}=\text{? inches}[/latex]?
You could not cancel the feet because the unit is not the same in both the numerator and the denominator. So if you complete the computation, you would still have both feet and inches in the answer and no conversion would take place.