Units of Length
LEARNING outcomes
- Define units of length and convert from one to another
- Perform arithmetic calculations on units of length
- Solve application problems involving units of length
Suppose you want to purchase tubing for a project, and you see two signs in a hardware store: $1.88 for 2 feet of tubing and $5.49 for 3 yards of tubing. If both types of tubing will work equally well for your project, which is the better price? You need to know about two units of measurement, yards and feet, in order to determine the answer.
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. Below are examples to show measurement in each of these units.
| Unit | Description | Image |
| Inch/Inches | Some people donate their hair to be made into wigs for cancer patients who have lost hair as a result of treatment. One company requires hair donations to be at least 8 inches long. |  |
| Frame size of a bike: the distance from the center of the crank to the top of the seat tube. Frame size is usually measured in inches. This frame is 16 inches. |  |
|
| Foot/Feet | Rugs are typically sold in standard lengths. One typical size is a rug that is 8 feet wide and 11 feet long. This is often described as an 8 by 11 rug. |  |
| Yard/Yards | Soccer fields vary some in their size. An official field can be any length between 100 and 130 yards. |  |
| Mile/Miles | A marathon is 26.2 miles long. One marathon route is shown in the map to the right. |  |
| Unit Equivalents | Conversion Factors (longer to shorter units of measurement) | Conversion Factors(shorter to longer units of measurement) |
| 1 foot = 12 inches | [latex] \displaystyle \frac{12\ \text{inches}}{1\ \text{foot}}[/latex] | [latex] \displaystyle \frac{1\text{ foot}}{12\text{ inches}}[/latex] |
| 1 yard = 3 feet | [latex] \displaystyle \frac{3\text{ feet}}{1\text{ yard}}[/latex] | [latex] \displaystyle \frac{\text{1 yard}}{\text{3 feet}}[/latex] |
| 1 mile = 5,280 feet | [latex] \displaystyle \frac{5,280\text{ feet}}{1\text{ mile}}[/latex] | [latex] \displaystyle \frac{\text{1 mile}}{\text{5,280 feet}}[/latex] |
recall simplifying fractions
Conversion factors must contain the same number in the top and in the bottom in order to equal 1. Since these are ratios of equal values, they are also fractions. And since a fraction is a division problem, we can apply the same principle we use to simplify fractions,[latex]\dfrac{a}{a}=1[/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 youd 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]. 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. 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.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].
Answer: The shortcut for converting mixed numbers to improper fractions is based on the way we add fractions. That is, a mixed number such as [latex]3\dfrac{1}{2}[/latex] is actually an addition problem, [latex]3 + \dfrac{1}{2}[/latex]. Let's add those numbers together. [latex] \begin{array}{rcl} &&3 + \dfrac{1}{2} & \\ &=&\dfrac{3}{1}+\dfrac{1}{2} & \text{write the whole number as a fraction} \\ &=&\dfrac{6}{2}+\dfrac{1}{2} & \text{rewrite as equivalent fractions with the same denominators} \\ &=&\dfrac{7}{2} & \text{add fractions with common denominators} \end{array}[/latex]
Example
How many inches are in [latex] \displaystyle 3\frac{1}{2}[/latex] feet?Answer: Begin by reasoning about your answer. Since a foot is longer than an inch, this means the answer would be greater than [latex] \displaystyle 3\frac{1}{2}[/latex]. Find the conversion factor that compares inches and feet, with inches in the numerator, and multiply.
[latex]3\frac{1}{2}\text{feet}\cdot\frac{12\text{ inches}}{1\text{foot}}=\text{? inches}[/latex]
Rewrite the mixed number as an improper fraction before multiplying.[latex]\frac{7}{2}\text{feet}\cdot\frac{12\text{ inches}}{1\text{foot}}=\text{? inches}[/latex]
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}{2}\cancel{\text{feet}}\cdot\frac{12\text{ inches}}{\cancel{1\text{foot}}}=\text{? inches}[/latex]
Rewrite as multiplication of numerators and denominators.[latex]\frac{7\cdot12\text{ inches}}{2}=\frac{84\text{ inches}}{2}=42\text{ inches}[/latex]
There are 42 inches in [latex] \displaystyle 3\frac{1}{2}[/latex] feet.[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{foor}}{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. Here is another example of a length conversion using the factor label method.Convert an improper fraction to a mixed number
To convert a mixed number to an improper fraction, you'll need to consider how many times the denominator divides into the numerator and if there is a remainder. Ex. Convert the improper fraction [latex]\dfrac{13}{5}[/latex] to a mixed number. The fraction bar is a division symbol.[latex]\dfrac{13}{5} = 13 \div 5[/latex]. [latex]13 \div 5 = 2[/latex] with [latex]3[/latex] left over, which we place over the divisor, [latex]5[/latex]. That is, the result of the division is [latex]2[/latex] and [latex]\dfrac{3}{5}[/latex], which we write as the mixed number [latex]2\dfrac{3}{5}[/latex].Example
How many yards is [latex]7[/latex] feet?Answer: Start by reasoning about the size of your answer. Since a yard is longer than a foot, there will be fewer yards. So your answer will be less than [latex]7[/latex]. Find the conversion factor that compares feet and yards, with yards in the numerator.
[latex]7\text{ feet}\cdot\dfrac{1\text{ yard}}{3\text{ feet}}=\text{? yards}[/latex]
Cancel the similar units, feet and feet, leaving only yards.[latex]7\cancel{\text{ feet}}\cdot\dfrac{1\text{ yard}}{3\cancel{\text{ feet}}}=\text{? yards}[/latex]
[latex]7\cdot\dfrac{1\text{ yard}}{3}=\text{? yards}[/latex]
7 feet equals [latex] 2\dfrac{1}{3}[/latex] yards.how was the final answer obtained?
Let's look at the last two lines more closely. [latex]7\cdot\dfrac{1\text{ yard}}{3}=\text{? yards} \longrightarrow \text{ 7 feet equals } 2\dfrac{1}{3}[/latex] yards. Here's how it was done. [latex] \begin{array}{rcl} &&7\cdot\dfrac{1\text{ yard}}{3} & \\ &=& \dfrac{7}{1}\cdot\dfrac{1\text{ yard}}{3} & \text{write the whole number as fraction} \\ &=& \dfrac{7\cdot1\text{ yard}}{3} & \text{multiply the fractions} \\ &=& \dfrac{7}{3}\cdot\dfrac{1\text{ yard}}{1}& \text{separate the factors into two fractions} \\ &=& \dfrac{7}{3} \text{ yards} & \text{simplify}\\ &=& 2\dfrac{1}{3} \text{ yards} & \text{convert the improper fraction to a mixed number}. \end{array}[/latex]Try It
[ohm_question]117507-986-992-993[/ohm_question]Apply Unit Conversions With Length
There are times when you will need to perform computations on measurements that are given in different units. For example, consider the tubing problem given earlier. You must decide which of the two options is a better price, and you have to compare prices given in different unit measurements. In order to compare, you need to convert the measurements into one single, common unit of measurement. To be sure you have made the computation accurately, think about whether the unit you are converting to is smaller or larger than the number you have. Its relative size will tell you whether the number you are trying to find is greater or lesser than the given number.Example
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?Answer: 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.
[latex]15\text{ feet}+30\text{ feet}+26\text{ feet}=71\text{ feet}[/latex]
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. Use the conversion factor [latex]\frac{1\text{ yard}}{3\text{ feet}}[/latex][latex]\frac{71\text{ feet}}{1}\cdot\frac{1\text{ yard}}{3\text{ feet}}=\text{? yards}[/latex]
[latex]\frac{71\cancel{\text{ feet}}}{1}\cdot\frac{1\text{ yard}}{3\cancel{\text{ feet}}}={23}\frac{2}{3}\text{ yards}[/latex]
Try It
[ohm_question]126605[/ohm_question]Example
Two runners were comparing how much they had trained earlier that day. Jo said, "According to my pedometer, I ran 8.3 miles." Alex said, "That's a little more than what I ran. I ran 8.1 miles." How many more feet did Jo run than Alex?Answer: You need to find the difference between the distance Jo ran and the distance Alex ran. Since both distances are given in the same unit, you can subtract and keep the unit the same.
[latex]8.3\text{ miles}-8.1\text{ miles}=0.2\text{ mile}[/latex]
[latex]0.2\text{ mile}=\frac{2}{10}\text{ mile}[/latex]
Since the problem asks for the difference in feet, you must convert from miles to feet. How many feet is 0.2 mile? Reason about the size of your answer. Since a mile is longer than a foot, the distance when expressed as feet will be a number greater than 0.2.[latex]\frac{2}{10}\text{ mile}=[/latex] ___ feet
Use the conversion factor [latex] \displaystyle \frac{5,280\text{ feet}}{1\text{ mile}}[/latex].[latex]\frac{2\text{mile}}{10}\cdot\frac{5,280\text{ feet}}{1\text{ mile}}[/latex] = ___ feet
[latex]\frac{2\cancel{\text{mile}}}{10}\cdot\frac{5,280\text{ feet}}{1\cancel{\text{ mile}}}[/latex] = ___ feet
[latex]\frac{2}{10}\cdot\frac{5,280\text{ feet}}{1}[/latex] = ___ feet
Multiply. Divide.[latex] \displaystyle \frac{2\bullet \text{5,280 feet}}{10\bullet 1}[/latex]= ___ feet
[latex] \displaystyle \frac{10,560\text{ feet}}{10}[/latex]= ___ feet
[latex] \displaystyle \frac{\text{10,560 feet}}{\text{10}}[/latex]= 1,056 feet
Jo ran 1,056 feet further than Alex.Example
You are walking through a hardware store and notice two sales on tubing:- 3 yards of Tubing A costs $5.49.
- Tubing B sells for $1.88 for 2 feet.
Answer: Find the unit price for each tubing. This will make it easier to compare.
Tubing A
Find the cost per yard of Tubing A by dividing the cost of 3 yards of the tubing by 3.3 yards = $5.49
[latex]\frac{5.49\div3}{3\text{ yards}\div3}=\frac{\$1.83}{1\text{ yard}}[/latex]
Tubing B is sold by the foot. Find the cost per foot by dividing $1.88 by 2 feet.Tubing B
2 feet = $1.88
[latex]\frac{1.88\div2}{2\text{ feet}\div2}=\frac{\$0.94}{1\text{ foot}}[/latex]
To compare the prices, you need to have the same unit of measure. Use the conversion factor [latex] \displaystyle \frac{3\text{ feet}}{1\text{ yard}}[/latex], cancel and multiply.[latex]\frac{\$0.94}{1\text{ foot}}\cdot\frac{3\text{ feet}}{1\text{ yard}}=\frac{\$\text{____}}{\text{____ yard}}[/latex]
[latex]\frac{\$0.94}{1\cancel{\text{ foot}}}\cdot\frac{3\cancel{\text{ feet}}}{1\text{ yard}}=\frac{\$2.82}{1\text{ yard}}[/latex]
[latex]\$2.82\text{ per yard}[/latex]
Compare prices for 1 yard of each tubing. Tubing A: $1.83 per yard Tubing B: $2.82 per yard Tubing A is less expensive than Tubing B.Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Units of Length. Provided by: Monterey Institute for Technology and Education (MITE) Authored by: Developmental Math 2014An Open Program. License: CC BY: Attribution.
- meter-tape-measure-measure-gage. Authored by: EME. License: CC0: No Rights Reserved.
- Question ID 117507. Authored by: Volpe, Amy. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 986. Authored by: Lippman, David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 126605. Authored by: Day, Alyson. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
