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

A municipal trash facility allows a person to throw away a maximum of 30 pounds of trash per week. Last week, 140 people threw away the maximum allowable trash. How many tons of trash did this equal?

Answer:   Determine the total trash for the week expressed in pounds. If 140 people each throw away 30 pounds, you can find the total by multiplying.

[latex]140\cdot30\text{ pounds}=4,200\text{ pounds}[/latex]

Then convert 4,200 pounds to tons. Reason about your answer. Since a ton is heavier than a pound, expect your answer to be a number less than 4,200.

[latex]4,200\text{ pounds}=\text{___ tons}[/latex]

Find the conversion factor appropriate for the situation:

[latex] \displaystyle \frac{1\text{ ton}}{2,000\text{ pounds}}[/latex]

[latex]\frac{4,200\text{ pounds}}{1}\cdot\frac{1\text{ ton}}{2,000\text{ pounds}}=\text{___ tons}[/latex]

[latex]\frac{4,200\cancel{\text{ pounds}}}{1}\cdot\frac{1\text{ ton}}{2,000\cancel{\text{ pounds}}}=\text{___ tons}[/latex]

[latex]\frac{4,200}{1}\cdot\frac{1\text{ ton}}{2,000}=\text{___ tons}[/latex]

Multiply and simplify.

[latex]\frac{4,200\cdot1\text{ ton}}{1\cdot2,000}=\text{___ tons}[/latex]

[latex]\frac{4,200\text{ ton}}{2,000}=\text{____ tons}[/latex]

[latex]\frac{4,200\text{ ton}}{2,000}=2\frac{1}{10}\text{ tons}[/latex]

The total amount of trash generated is [latex] \displaystyle 2\frac{1}{10}[/latex] tons.

Example

How many pints is [latex] \displaystyle 2\frac{3}{4}[/latex] gallons?

Answer: Begin by reasoning about your answer. Since a gallon is larger than a pint, expect the answer in pints to be a number greater than [latex] \displaystyle 2\frac{3}{4}[/latex].

[latex]2\frac{3}{4}\text{ gallons}=\text{___ pints}[/latex]

The table above does not contain a conversion factor for gallons and pints, so you cannot convert it in one step. However, you can use quarts as an intermediate unit, as shown here. Set up the equation so that two sets of labels cancel— gallons and quarts.

[latex]\frac{11\text{ gallons}}{4}\cdot\frac{4\text{ quarts}}{1\text{ gallon}}\cdot\frac{2\text{ pints}}{1\text{ quart}}=\text{___ pints}[/latex]

[latex]\frac{11\cancel{\text{ gallons}}}{4}\cdot\frac{4\cancel{\text{ quarts}}}{1\cancel{\text{ gallon}}}\cdot\frac{2\text{ pints}}{1\cancel{\text{ quart}}}=\text{___ pints}[/latex]

 [latex]\frac{11}{4}\cdot{4}{1}\cdot\frac{2\text{ pints}}{1}=\text{___ pints}[/latex]

Multiply and simplify.

[latex]\frac{11\cdot4\cdot2\text{ pints}}{4\cdot1\cdot1}=\text{___ pints}[/latex]

[latex]\frac{88\text{ pints}}{4}=22\text{ pints}[/latex]

[latex] \displaystyle 2\frac{3}{4}[/latex] gallons is 22 pints.

Example

How many gallons is 32 fluid ounces?

Answer: Begin by reasoning about your answer. Since gallons is a larger unit than fluid ounces, expect the answer to be less than 32.

[latex]32\text{ fluid ounces}=\text{___ gallons}[/latex]

The table above does not contain a conversion factor for gallons and fluid ounces, so you cannot convert it in one step. Use a series of intermediate units, as shown here.

[latex]\frac{32\text{ fl oz}}{1}\cdot\frac{1\text{ cup}}{8\text{ fl oz}}\cdot\frac{1\text{ qt}}{2\text{ pt}}\cdot\frac{1\text{ gal}}{4\text{ qt}}=\text{___ gal}[/latex]

Cancel units that appear in both the numerator and denominator.

[latex]\frac{32\cancel{\text{ fl oz}}}{1}\cdot\frac{1\cancel{\text{ cup}}}{8\cancel{\text{ fl oz}}}\cdot\frac{1\cancel{\text{ qt}}}{2\cancel{\text{ pt}}}\cdot\frac{1\text{ gal}}{4\cancel{\text{ qt}}}=\text{___ gal}[/latex]

[latex]\frac{32}{1}\cdot\frac{1}{8}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1\text{ gal}}{4}=\text{____ gal}[/latex]

Multiply and simplify.

[latex]\frac{32\cdot1\cdot1\cdot1\cdot1\text{ gal}}{1\cdot8\cdot2\cdot2\cdot4}=\text{___ gal}[/latex]

[latex] \displaystyle \frac{32\text{ gal}}{\text{128}}=\frac{1}{4}\text{ gal}[/latex]

32 fluid ounces is the same as [latex]\frac{1}{4}[/latex] gallon.

Exercises

Sven and Johanna were hosting a potluck dinner. They did not ask their guests to tell them what they would be bringing, and three people ended up bringing soup. Erin brought 1 quart, Richard brought 3 pints, and LeVar brought 9 cups. How much soup did they have total?

Answer: Since the problem asks for the total amount of soup, you must add the three quantities. Before adding, you must convert the quantities to the same unit. The problem does not require a particular unit, so you can choose. Cups might be the easiest computation.

[latex]1\text{ quart}+3\text{ pints}+9\text{ cups}[/latex]

This is given in the table of equivalents.

[latex]1\text{ quart}=4\text{ cups}[/latex]

Use the factor label method to convert pints to cups.

[latex]\frac{3\text{ pints}}{1}\cdot\frac{2\text{ cups}}{1\text{ pint}}=\text{___ cups}[/latex]

[latex]\frac{3\cancel{\text{ pints}}}{1}\cdot\frac{2\text{ cups}}{1\cancel{\text{ pint}}}=\text{6 cups}[/latex]

  Add the 3 quantities.

[latex]4\text{ cups}+6\text{ cups}+9\text{ cups}=19\text{ cups}[/latex]

There are 19 cups of soup for the dinner.

Exercises

Natasha is making lemonade to bring to the beach. She has two containers. One holds one gallon and the other holds 2 quarts. If she fills both containers, how many cups of lemonade will she have?

Answer: This problem requires you to find the sum of the capacity of each container and then convert that sum to cups.

[latex]1\text{ gallon}+2\text{ quarts}=\text{___ cups}[/latex]

First, find the sum in quarts. 1 gallon is equal to 4 quarts.

[latex]4\text{ quarts}+2\text{ quarts}=6\text{ quarts}[/latex]

Since the problem asks for the capacity in cups, convert 6 quarts to cups. Cancel units that appear in both the numerator and denominator. Multiply.

[latex]\frac{6\text{ quarts}}{1}\cdot\frac{2\text{ pints}}{1\text{ quart}}\cdot\frac{2\text{ cups}}{1\text{ pint}}=\text{____ cups}[/latex]

[latex]\frac{6\cancel{\text{ quarts}}}{1}\cdot\frac{2\cancel{\text{ pints}}}{1\cancel{\text{ quart}}}\cdot\frac{2\text{ cups}}{1\cancel{\text{ pint}}}=\text{____ cups}[/latex]

[latex]6\times2\times2=24\text{ cups}[/latex]

Natasha will have 24 cups of lemonade.