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.