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.

Example

How many yards is 7 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 7. Find the conversion factor that compares feet and yards, with yards in the numerator.

[latex]7\text{ feet}\cdot\frac{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\frac{1\text{ yard}}{3\cancel{\text{ feet}}}=\text{? yards}[/latex]

[latex]7\cdot\frac{1\text{ yard}}{3}=\text{? yards}[/latex]

7 feet equals [latex] \displaystyle 2\frac{1}{3}[/latex] yards.

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]

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. Either tubing is acceptable for your project. Which tubing is less expensive?

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.