Customary Units: Fractions, Mixed Units and Problems | Grade 5 Math | LevelKite

You already know how to convert, which means to change a measurement from one unit to another. In this lesson, some amounts are fractions or mixed numbers. A mixed number is a whole number and a fraction together, such as $1\frac{1}{2}$ .

Start with what the amount means. $1\frac{1}{2}$ feet is 1 whole foot and half of another foot.

1 whole foot is $12$ inches.
Half of a foot is half of $12$ inches. That is $6$ inches.
Together you have $12 + 6 = 18$ inches.

So $1\frac{1}{2}$ feet and $18$ inches are the same length. The length did not change. Only the unit changed.

Finding half of $12$ is the same as multiplying $\frac{1}{2} \times 12$ . The table shows more lengths this way.

The same length in words, in feet and in inches

In words	In feet	Number sentence	In inches
one fourth of a foot	$\frac{1}{4}$	$\frac{1}{4} \times 12 = 3$	$3$
one half of a foot	$\frac{1}{2}$	$\frac{1}{2} \times 12 = 6$	$6$
three fourths of a foot	$\frac{3}{4}$	$\frac{3}{4} \times 12 = 9$	$9$
one and one half feet	$1\frac{1}{2}$	$1\frac{1}{2} \times 12 = 12 + 6 = 18$	$18$
two and one fourth feet	$2\frac{1}{4}$	$2\frac{1}{4} \times 12 = 24 + 3 = 27$	$27$

Key concept

One large unit is a fixed group of small units

A conversion fact tells how many small units make one large unit, such as $1\text{ foot} = 12\text{ inches}$ or $1\text{ pound} = 16\text{ ounces}$ . Converting changes the unit. It never changes the amount.

To change to a smaller unit, multiply by the number of small units in one large unit. Each large unit becomes that many small units.
To change to a larger unit, divide by that same number. Each full group of small units makes one large unit.

This works for whole numbers, fractions and mixed numbers. The same number also tells you how to trade between units when you add or subtract.

Worked example

A mixed number of pounds in ounces

A bag of apples weighs $2\frac{3}{4}$ pounds. How many ounces does it weigh?

Decide whether to multiply or divide. Ounces are smaller than pounds, and 1 pound is 16 ounces. Each pound becomes 16 ounces, so multiply the number of pounds by 16.
$2\tfrac{3}{4} \times 16$
Split the mixed number into its whole part and its fraction part. You can do this because 2 and three fourths pounds means 2 whole pounds plus three fourths of another pound.
$2\tfrac{3}{4} = 2 + \tfrac{3}{4}$
Convert the whole part. 2 pounds is 2 groups of 16 ounces.
$2 \times 16 = 32$
Convert the fraction part. One fourth of 16 is 4, so three fourths of 16 is 3 groups of 4.
$\tfrac{3}{4} \times 16 = 3 \times 4 = 12$
Add the ounces from the two parts. The total, 44, is a bigger number than the number of pounds. That makes sense here, because an ounce is smaller than a pound.
$32 + 12 = 44$

Answer

The bag of apples weighs $44$ ounces. That is the same weight as $2\frac{3}{4}$ pounds.

A fraction can also show up in the answer. This can happen when you change to a larger unit and the small units do not fill a whole number of large units.

How many feet is $30$ inches? Feet are larger than inches, so divide by $12$ . Two groups of $12$ make $24$ inches, and $6$ inches are left over. Those $6$ inches are $6$ of the $12$ inches in a foot, so they are $\frac{6}{12}$ of a foot. That is the same as $\frac{1}{2}$ of a foot.

So $30$ inches is $2\frac{1}{2}$ feet. Here is the same work as a number sentence.

30 \div 12 = 2\tfrac{6}{12} = 2\tfrac{1}{2}

You can also write $30$ inches as $2$ feet $6$ inches. This measurement is in mixed units, which means it uses two units together. It says $2$ whole feet and $6$ more inches.

To add or subtract mixed units, work with one unit at a time. Put inches with inches and feet with feet. Then check the smaller unit.

Adding: if you have $12$ inches or more, trade $12$ inches for $1$ foot. Keep trading until fewer than $12$ inches are left.
Subtracting: if you do not have enough inches to take away, first trade $1$ foot for $12$ inches.

This trading is called regrouping. It never changes the amount, because $12$ inches and $1$ foot are the same length. Other pairs of units regroup with their own conversion fact, such as $16$ ounces for $1$ pound.

The table adds $3$ feet $8$ inches and $2$ feet $7$ inches.

3 feet 8 inches + 2 feet 7 inches = 6 feet 3 inches

What you do	Feet	Inches
Write the first length	$3$	$8$
Write the second length	$2$	$7$
Add feet to feet and inches to inches	$3 + 2 = 5$	$8 + 7 = 15$
Regroup: trade $12$ of the $15$ inches for $1$ foot	$5 + 1 = 6$	$15 - 12 = 3$

Worked example

Subtracting feet and inches

A board is $7$ feet $4$ inches long. You cut off a piece that is $2$ feet $9$ inches long. How long is the board that is left?

Start with the inches, the smaller unit. You must take away 9 inches, but the board's length shows only 4 inches. That is not enough, so you need to regroup.
$4 < 9$
Trade 1 of the 7 feet for 12 inches. That leaves 6 feet. The 12 new inches join the 4 inches to make 16 inches. The length is unchanged, because 1 foot and 12 inches are the same length.
$7\text{ feet } 4\text{ inches} = 6\text{ feet } 16\text{ inches}$
Subtract inches from inches. Now 16 inches is enough to take 9 inches away.
$16 - 9 = 7$
Subtract feet from feet. Use 6 feet, not 7, because you traded 1 foot away.
$6 - 2 = 4$
Check by adding the two pieces back together. 4 feet 7 inches plus 2 feet 9 inches is 6 feet 16 inches. Trade 12 of the 16 inches for 1 foot, and you get the 7 feet 4 inches you started with.
$6\text{ feet } 16\text{ inches} = 7\text{ feet } 4\text{ inches}$

Answer

The board that is left is $4$ feet $7$ inches long.

Some problems need a conversion and another operation, such as adding, subtracting, multiplying or dividing. Three questions help you plan.

What does the problem ask for, and in which unit?
Do the measurements use the same unit? You can add or subtract the numbers in two measurements only when they count the same unit. If the units are different, convert first.
Is your result in the unit the problem asks for? If it is not, convert at the end.

Worked example

Punch: convert, add, then convert again

You mix $1\frac{1}{2}$ gallons of juice with $3$ quarts of soda water to make punch. How many gallons of punch do you have?

Plan. The juice is in gallons and the soda water is in quarts, so you cannot add the two numbers yet. Put both amounts in quarts. That works well here, because one and one half gallons is a whole number of quarts. Then add, and change the total to gallons, because the problem asks for gallons.
Change the juice to quarts. 1 gallon is 4 quarts, so each gallon becomes 4 quarts: multiply by 4. The 1 whole gallon is 4 quarts. The half gallon is half of 4 quarts, which is 2 quarts.
$1\tfrac{1}{2} \times 4 = 4 + 2 = 6$
Both amounts now count quarts, so add them to find the total amount of punch in quarts.
$6 + 3 = 9$
Change 9 quarts to gallons. Gallons are larger than quarts, so divide by 4. Two groups of 4 quarts make 8 quarts, and 1 quart is left over. That 1 quart is one fourth of a gallon.
$9 \div 4 = 2\tfrac{1}{4}$

Answer

You have $2\frac{1}{4}$ gallons of punch. That is the same amount as $2$ gallons $1$ quart.

Takeaway. One large unit is always the same group of small units. That one fact does three jobs.

It converts any amount, even a fraction or a mixed number. Multiply by the number in the fact to change to a smaller unit. Divide by it to change to a larger unit.
It tells you how to regroup mixed units, such as $12$ inches for $1$ foot.
It lets you put measurements in the same unit before you add or subtract them in a longer problem.

Each time, the amount stays the same. Only the unit changes.