Section 1.1: Length
Length is the distance from one end of an object to the other end, or from one object to another. For example, the length of a letter-sized piece of paper is 11 inches. The system for measuring length in the United States is based on the four customary units of length: inch, foot, yard, and mile. You can use any of these four U.S. customary measurement units to describe the length of something, but it makes more sense to use certain units for certain purposes. For example, it makes more sense to describe the length of a rug in feet rather than miles, and to describe a marathon in miles rather than inches.
The table below shows equivalents and conversion factors for the four customary units of measurement of length.
| Unit Equivalent | Conversion Factor (longer to shorter) |
Conversion Factor (shorter to longer) |
| 1 foot = 12 inches | [latex]\frac{12~inches}{1~foot}[/latex] | [latex]\frac{1~foot}{12~inches}[/latex] |
| 1 yard = 3 feet | [latex]\frac{3~feet}{1~yard}[/latex] | [latex]\frac{1~yard}{3~feet}[/latex] |
| 1 mile = 5280 feet | [latex]\frac{5280~feet}{1~mile}[/latex] | [latex]\frac{1~mile}{5280~feet}[/latex] |
You can use the conversion factors to convert a measurement, such as feet, to another type of measurement, such as inches. Note that each of these conversion factors is a ratio of equal values, so each conversion factor equals 1. Multiplying a measurement by a conversion factor does not change the size of the measurement at all since it is the same as multiplying by 1; it just changes the units that you are using to measure.
There are many more inches for a measurement than there are feet for the same measurement, as feet is a longer unit of measurement. So, you could use the conversion factor [latex]\frac{12~inches}{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]\frac{1~yard}{3~feet}[/latex].
Dimensional Analysis: The Factor-Label Method
You can use the factor label method 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. Study the example below to see how the factor label method can be used to convert a measurement given in feet into an equivalent number of inches.
Example 1
How many inches are in [latex]3\frac{1}{2}[/latex] feet?
Solution: Begin by reasoning about your answer. Since a foot is longer than an inch, this means the answer would be greater than [latex]3\frac{1}{2}[/latex] feet. Now state your givens and your goal.
Given: [latex]3\frac{1}{2}[/latex] feet
Goal: Convert to inches, or equivalently, solve [latex]3\frac{1}{2}[/latex] feet = ________inches
1) Find the conversion factor that compares inches and feet, with “inches” in the
numerator, and multiply.
[latex]3\frac{1}{2}[/latex] feet x [latex]\frac{12~inches}{1 foot}[/latex] = ________inches
2) Rewrite the mixed number as an improper fraction before multiplying.
[latex]\frac{7}{2}[/latex] feet x [latex]\frac{12 inches}{1 foot}[/latex] = ________inches (Since [latex]3\frac{1}{2}[/latex] = [latex]\frac{7}{2}[/latex])
3) You can cancel similar units when they appear in the numerator and the denominator. So here, cancel the similar units “feet” and “foot.” This eliminates this unit from the problem.
[latex]\frac{7~feet}{2}[/latex] x [latex]\frac{12~inches}{1~foot}[/latex] = [latex]\frac{7~\bcancel{feet}}{2}[/latex] x [latex]\frac{12~inches}{1~\bcancel{foot}}[/latex] = [latex]\frac{7}{2}[/latex] x [latex]\frac{12~inches}{1}[/latex]
4) Rewrite as multiplication of numerators and denominators. Then divide.
[latex]\frac{7}{2}[/latex] x [latex]\frac{12~inches}{1}[/latex] = [latex]\frac{7~x~12~inches}{2~x~1}[/latex] = [latex]\frac{84}{2}[/latex] inches = 42 inches
Answer: [latex]3\frac{1}{2}[/latex] feet equals 42 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 cancelled is in both the numerator and denominator of the fractions you are multiplying. In the problem above, you cancelled feet
and foot leaving you with inches, which is what you were trying to find. What if you had used the wrong conversion factor as shown below?
[latex]\frac{7~feet}{2}[/latex] x [latex]\frac{1~foot}{12~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.
Example 2
An interior decorator needs border trim for a home she is wallpapering. She needs 15 feet of border trim for the living room, 30 feet of border trim for the bedroom, and 26 feet of border trim for the dining room. How many yards of border trim does she need?
Solution: State your givens and your goal.
Given: 15 feet of border trim for the living room, 30 feet of border trim for the bedroom, and 26 feet of border trim for the dining room
Goal: How many yards of border trim does she need?
1) You need to find the total length of border trim that is needed for all three rooms in the house. Since the measurements for each room are given in feet, you can add the numbers.
15 feet + 30 feet + 26 feet = 71 feet
2) Observe that our goal is the number of yards, not the number of feet. So we need to convert. How many yards is 71 feet? (Reason about the size of your answer. Since a yard is longer than a foot, there will be fewer yards. Expect your answer to be less than 71.)
71 feet = _____ yards
3) Use the conversion factor [latex]\frac{1~yard}{3~feet}[/latex]
[latex]\frac{71~feet}{1}[/latex] x [latex]\frac{1~yard}{3~feet}[/latex] = _____ yard
4) Since “feet” is in the numerator and denominator, you can cancel this unit. Then rewrite as multiplication of numerators and denominators.
[latex]\frac{71~feet}{1}[/latex] x [latex]\frac{1~yard}{3~feet}[/latex] = [latex]\frac{71~\bcancel{feet}}{1}[/latex] x [latex]\frac{1~yard}{3~\bcancel{feet}}[/latex] = [latex]\frac{71~x~1~yard}{1~x~3}[/latex]
5) Multiply and then rewrite the improper fraction as a mixed number.
[latex]\frac{71~x~1~yard}{1~x~3}[/latex] = [latex]\frac{71}{3}[/latex] yards = [latex]23\frac{2}{3}[/latex]
Answer: The interior designer needs [latex]23\frac{2}{3}[/latex] yards of border trim.
At times, you will need to make unit conversions for measurements in two or three dimensions (think finding an area of a field, or the volume of a container). The following is an example for converting units in two dimensions using an area problem.
Example 3
A landscaper is installing artificial turf for a football field. The dimensions of the football field are 120 yards long by 50 yards wide. His supplier sells artificial turf by the square foot. How many square feet of turf is needed for the field?
Solution: State your givens and your goal.
Given: The dimensions of the football field are 120 yards long by 50 yards wide.
Goal: How many square feet of turf is needed for the field?
1) You need to find the total area of the field that needs to be covered. Since both length and width are given in yards, you can multiply. (Note: The resulting units are in square yards.)
120 yards × 50 yards = 6,000 square yards
2) We are asked to find the number of square feet needed. So convert square yards to square feet. (Reason about the size of your answer. Since a square foot is smaller than a square yard, there will be more square feet. Expect your answer to be larger than 6000.)
[latex]\frac{6,000~yd^2}{1}[/latex] x [latex]\frac{3~ft}{1~yd}[/latex] x [latex]\frac{3~ft}{1~yd}[/latex] = [latex]\frac{6,000~\bcancel{yd}~\bcancel{yd}}{1}[/latex] x [latex]\frac{3~ft}{1~\bcancel{yd}}[/latex] x [latex]\frac{3~ft}{1~\bcancel{yd}}[/latex] = [latex]\frac{6,000~x~3~ft~x~3~ft}{1~x~1~x~1}[/latex] = 54,000 ft2
Answer: The landscaper needs 54,000 square feet of artificial turf for the football field.
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 cancelled is in both the numerator and denominator of the fractions you are multiplying. In the problem above, you cancelled feet and foot leaving you with inches, which is what you were trying to find.
What if you had used the wrong conversion factor as shown below?
[latex]\frac{7~feet}{2}[/latex]x[latex]\frac{1~foot}{12~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.
Section 1.1 You Try Problems:
A) Use the Factor-Label Method to determine the number of feet in [latex]2\frac{1}{2}[/latex] miles.
B) A fence company is measuring a rectangular area in order to install a fence around its perimeter. If the length of the rectangular area is 130 yards and the width is 75 feet, what is the total length of the distance to be fenced?
1.1 – Answers to You Try Problems
a) 13,200 feet
b) 930 feet or 310 yards