Subtract from a Whole and Add Three or More Fractions | Grade 5 Math | LevelKite

You already know how to add or subtract two fractions with unlike denominators. The denominator is the bottom number of a fraction, and it tells the size of the parts. Unlike denominators are different from each other, so the parts are different sizes. The numerator is the top number, and it counts the parts. To add or subtract, you first rename the fractions. To rename is to write the same amount with a new denominator, such as $\frac{1}{2} = \frac{2}{4}$ .

In this lesson you will use that same move in three new places:

to subtract a fraction from a whole number (a number with no fraction part, such as 3), as in $3 - \frac{2}{5}$
to add three or more fractions, as in $\frac{1}{2} + \frac{1}{6} + \frac{1}{4}$
to find a missing fraction, as in $\square + \frac{1}{4} = \frac{5}{6}$

Key concept

Same-size parts first

You can add or subtract fractions only when they count parts of the same size. So first rename the numbers until the parts you add or subtract share one denominator. This works for a whole number, for three or more fractions, and for a problem with a missing fraction.

A whole number does not show a denominator, but you can give it one. Choose the denominator of the fraction you want to subtract.

Say you want to take two fifths away from 1. Think of 1 whole cut into 5 equal parts. Each part is one fifth, and all 5 parts make the whole. So one whole is five fifths: $1 = \frac{5}{5}$ . Now the parts match. Five fifths take away two fifths is three fifths: $1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$ .

A bigger whole number works the same way, and you have two choices. Take 3:

Rename every whole. Three wholes make $5 + 5 + 5 = 15$ fifths, so three is fifteen fifths: $3 = \frac{15}{5}$ .
Rename one whole. Keep 2 wholes as they are and cut only the third one. Then three is two wholes and five fifths: $3 = 2 + \frac{5}{5}$ .

Both names stand for the same amount as 3.

Worked example

Subtract a fraction from a whole number

What is $3 - \frac{2}{5}$ ?

The fraction counts fifths, so you need fifths to subtract from. Take one whole out of the 3 and rename it as five fifths. Keep the other 2 wholes as they are.
$3 = 2 + \frac{5}{5}$
Subtract two fifths from the five fifths. The denominators match, so the parts are the same size. Subtract the numerators and keep the denominator: 5 take away 2 is 3.
$\frac{5}{5} - \frac{2}{5} = \frac{3}{5}$
Join the 2 wholes you kept with the three fifths that are left. A whole number and a fraction written side by side make a mixed number. It means the two amounts together.
$2 + \frac{3}{5} = 2\frac{3}{5}$
Check with the other choice: rename all 3 wholes as 15 fifths. Taking away 2 fifths leaves 13 fifths. Ten of those fifths make 2 wholes, and 3 fifths are left over, so the two ways agree.
$\frac{15}{5} - \frac{2}{5} = \frac{13}{5} = \frac{10}{5} + \frac{3}{5} = 2\frac{3}{5}$

Answer

$3 - \frac{2}{5}$ is $2\frac{3}{5}$ , which is two wholes and three fifths. Renaming one whole and renaming all three wholes give the same amount.

An addend is a number that you add, and the sum is the total you get. You have added two addends before. With three or more, the rule about same-size parts still holds. You need one common denominator for all of them. A common denominator is a number that every denominator divides into evenly, with nothing left over.

Look at $\frac{1}{2} + \frac{1}{6} + \frac{1}{4}$ . Halves, sixths and fourths are three different sizes of part, so you cannot add the numerators yet. A denominator that fits only two of the fractions is not enough. It must fit all three.

Worked example

Add three fractions

What is $\frac{1}{2} + \frac{1}{6} + \frac{1}{4}$ ?

Find one denominator that fits all three addends. Count by the largest denominator, 6, and test each number you say. The first number, 6, fails because 4 does not divide into it evenly. The next number, 12, works because 2, 6 and 4 all divide into it evenly.
$12 \div 2 = 6, \quad 12 \div 6 = 2, \quad 12 \div 4 = 3$
Rename each addend in twelfths. Multiply the numerator and the denominator by the same number, so the amount does not change. The divisions in the first step tell you which number to use: 6, 2 and 3. One half is six twelfths, one sixth is two twelfths, and one fourth is three twelfths.
$\frac{1 \times 6}{2 \times 6} = \frac{6}{12}, \quad \frac{1 \times 2}{6 \times 2} = \frac{2}{12}, \quad \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Every addend now counts twelfths, so add the three numerators and keep the denominator 12. Adding 6, 2 and 3 gives 11.
$\frac{6}{12} + \frac{2}{12} + \frac{3}{12} = \frac{11}{12}$

Answer

The sum of $\frac{1}{2}$ , $\frac{1}{6}$ and $\frac{1}{4}$ is $\frac{11}{12}$ , or eleven twelfths.

The same steps work for four addends, or for more. Take $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}$ . The largest denominator is 16, and 2, 4 and 8 all divide into 16 evenly. So rename every addend in sixteenths. One half is eight sixteenths, one fourth is four sixteenths, and one eighth is two sixteenths. The last addend is already one sixteenth. Now add the numerators: $8 + 4 + 2 + 1 = 15$ . The sum is fifteen sixteenths.

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = \frac{8}{16} + \frac{4}{16} + \frac{2}{16} + \frac{1}{16} = \frac{15}{16}

Sometimes a number sentence, a line of math with an equals sign, has a box where a fraction is missing. One example is $\square + \frac{1}{4} = \frac{5}{6}$ . Your job is to find the fraction that makes the sentence true.

Addition and subtraction are inverse operations. That means each one undoes the other. You have seen this with whole numbers. Since $5 + 4 = 9$ , you also know that $9 - 4 = 5$ and $9 - 5 = 4$ . All three sentences use the same three numbers.

That is how you find a missing number:

The sentence $\square + 4 = 9$ adds 4 to the box. Undo that by subtracting 4: $\square = 9 - 4 = 5$ . The box can also come second, as in $4 + \square = 9$ . You still subtract the 4.
The sentence $\square - 4 = 5$ takes 4 away from the box. Undo that by adding 4 back: $\square = 5 + 4 = 9$ .
The sentence $9 - \square = 5$ takes the box away from 9 and leaves 5. So the box and the 5 make 9 together. That is $\square + 5 = 9$ , so $\square = 9 - 5 = 4$ .

Fractions follow the same rules.

Worked example

Find a missing fraction

Find the missing fraction: $\square + \frac{1}{4} = \frac{5}{6}$ .

Read the sentence: some fraction plus one fourth makes five sixths. The sentence adds one fourth to the box, so undo that with the inverse operation. Subtract one fourth from five sixths.
$\square = \frac{5}{6} - \frac{1}{4}$
Sixths and fourths are different sizes of part. Both 6 and 4 divide into 12 evenly, so rename both fractions in twelfths. Five sixths is ten twelfths, and one fourth is three twelfths.
$\frac{5 \times 2}{6 \times 2} = \frac{10}{12}, \quad \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
The parts match now, so subtract the numerators and keep the denominator 12. Ten take away 3 is 7.
$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$
Check by putting seven twelfths in the box. One fourth is three twelfths, so seven twelfths plus three twelfths is ten twelfths. Ten twelfths is the same amount as five sixths, so the sentence is true.
$\frac{7}{12} + \frac{3}{12} = \frac{10}{12} = \frac{5}{6}$

Answer

The missing fraction is $\frac{7}{12}$ , because $\frac{7}{12} + \frac{1}{4} = \frac{5}{6}$ .

Missing fractions in subtraction sentences

Number sentence	Same shape with whole numbers	How to find the box	Missing fraction
$\square - \frac{1}{3} = \frac{1}{2}$	$\square - 4 = 5$	Add $\frac{1}{3}$ back: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$	$\frac{5}{6}$
$\frac{4}{5} - \square = \frac{1}{2}$	$9 - \square = 5$	Start minus what is left: $\frac{4}{5} - \frac{1}{2} = \frac{8}{10} - \frac{5}{10} = \frac{3}{10}$	$\frac{3}{10}$

The big idea: add or subtract only when the parts are the same size. To subtract from a whole number, rename it, or one whole taken from it, as a fraction. To add three or more fractions, give them all one common denominator. To find a missing fraction, use the inverse operation, then add or subtract as usual.

Next, you will compare the answers to two fraction problems, sometimes without working them out. You will also find and fix mistakes, and use easy numbers such as $\frac{1}{2}$ and 1 to show that a wrong answer cannot be right.