Missing Fractions and Mixed Number Sums | Grade 6 Math | LevelKite

You already know how to add and subtract fractions with unlike denominators. In this lesson you use that skill in two new ways.

First, you will find a missing number in a fraction sentence. A fraction sentence is an equation with fractions, such as $\frac{3}{8} + \frac{1}{2} = \frac{7}{8}$ . It says the amounts on the two sides of the equals sign are the same.

Second, you will add and subtract mixed numbers. A mixed number is a whole number and a fraction written together, such as $2\frac{3}{4}$ .

One condition holds through this entire lesson: every number in a sentence measures the same whole. Picture one number line where the distance from $0$ to $1$ never changes. That distance is your one whole. This matters because you can only count parts together when they are the same size. A matching denominator is not enough by itself: twelfths of one whole are smaller than twelfths of a bigger whole.

Lay a full paper strip along your number line from $0$ to $1$ . Its length is one whole. Now take a piece that is $\frac{7}{8}$ of that whole and cut it into two smaller pieces. One is $\frac{1}{2}$ of the whole strip. The other is $\frac{3}{8}$ of the same whole strip.

The two smaller pieces are the parts. The piece before you cut it is the total. Joining the parts rebuilds the total. Taking one part away from the total leaves the other part. In this way addition and subtraction undo each other, so this one cut gives you several true sentences:

$\frac{3}{8} + \frac{1}{2} = \frac{7}{8}$ (part plus part makes the total)
$\frac{7}{8} - \frac{1}{2} = \frac{3}{8}$ (total minus one part leaves the other part)
$\frac{7}{8} - \frac{3}{8} = \frac{1}{2}$ (total minus the other part leaves the first part)

Now leave one number out of a sentence. You may see the gap shown as a blank. Here, let $n$ stand for the missing number. The letter $n$ is a variable: a symbol for a number you do not know yet.

To find $n$ , first decide whether it is the total or a part. When every number measures the same whole, the shape of the sentence tells you:

In an addition sentence, the two numbers being added are the parts. The number on the other side of the equals sign is the total.
In a subtraction sentence, the number you subtract from is the total. The number you take away and the number left over are the parts.

The table shows four sentences with a missing number, all made from the strip. In each row, the table renames $\frac{1}{2}$ as $\frac{4}{8}$ , so both fractions count eighths of the same whole. Notice that the operation you use depends on what is missing, not on the sign in the sentence.

Four sentences from the strip. Every number measures the same whole.

Sentence	The missing number is	How to find it	Result
$n + \frac{1}{2} = \frac{7}{8}$	a part	total minus known part: $\frac{7}{8} - \frac{1}{2} = \frac{7}{8} - \frac{4}{8}$	$n = \frac{3}{8}$
$\frac{3}{8} + \frac{1}{2} = n$	the total	part plus part: $\frac{3}{8} + \frac{1}{2} = \frac{3}{8} + \frac{4}{8}$	$n = \frac{7}{8}$
$\frac{7}{8} - n = \frac{1}{2}$	a part	total minus known part: $\frac{7}{8} - \frac{1}{2} = \frac{7}{8} - \frac{4}{8}$	$n = \frac{3}{8}$
$n - \frac{1}{2} = \frac{3}{8}$	the total	part plus part: $\frac{1}{2} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8}$	$n = \frac{7}{8}$

Key concept

Two parts and one total of the same whole

An addition or subtraction sentence links two parts and one total. When every number in the sentence measures the same whole:

a missing total equals the sum of the two parts
a missing part equals the total minus the known part

In that same whole, you can also split a total into parts, or join parts into a total, without changing its value. That is how you will work with mixed numbers: split them into wholes and fraction parts, then add or subtract parts that are the same size.

\text{part} + \text{part} = \text{total}

Worked example

Find the missing part in a subtraction sentence

Complete the sentence $\frac{5}{6} - n = \frac{1}{4}$ . The variable $n$ stands for the missing number, and all three numbers measure the same whole.

Decide what n is. All three numbers measure the same whole, so the sentence shows one total and its two parts. The number you subtract from, five sixths, is the total. The number taken away, n, and the number left over, one fourth, are the parts.
$\underbrace{\frac{5}{6}}_{\text{total}} - \underbrace{n}_{\text{part}} = \underbrace{\frac{1}{4}}_{\text{part}}$
A missing part equals the total minus the known part. Both sentences describe the same total split into the same two parts of one whole, so the new sentence is true for the same value of n.
$n = \frac{5}{6} - \frac{1}{4}$
Sixths and fourths are different-sized parts, so you cannot count them together yet. Rename both fractions as twelfths of the same whole. Multiply the numerator and denominator of five sixths by 2, and of one fourth by 3.
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \qquad \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Both fractions measure the same whole and now count twelfths, so the parts are equal in size. Subtract the counts: taking three twelfths from ten twelfths leaves seven twelfths.
$n = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}$
Check by putting seven twelfths in place of n in the original sentence, still counting twelfths of the same whole. Taking that part away from the total should leave the other part, one fourth, and it does.
$\frac{5}{6} - \frac{7}{12} = \frac{10}{12} - \frac{7}{12} = \frac{3}{12} = \frac{1}{4}$

Answer

The missing number is $n = \frac{7}{12}$ , so the completed sentence is $\frac{5}{6} - \frac{7}{12} = \frac{1}{4}$ .

Now use parts and totals with mixed numbers. A mixed number is already a sum. For example, $2\frac{3}{4}$ means $2 + \frac{3}{4}$ : two wholes and three fourths of one more whole of the same size. On the number line it sits three fourths of the way from $2$ to $3$ .

To add two mixed numbers that measure the same whole, split each one into its whole-number part and its fraction part. Adding in a different order does not change a sum. So you can add the wholes together and the fraction parts together, once the fraction parts count equal-sized parts of that whole.

Sometimes the fraction parts add up to $1$ whole or more. Then regroup: trade a full set of equal parts for $1$ whole. For example, $\frac{4}{4}$ of a whole is exactly $1$ whole, so the trade does not change the total.

Worked example

Add mixed numbers and regroup

Add $2\frac{3}{4} + 1\frac{2}{3}$ . Both mixed numbers measure the same whole.

Split each mixed number into its whole-number part and its fraction part. A mixed number means its wholes plus its fraction of one more whole of the same size, so the value does not change.
$2\frac{3}{4} + 1\frac{2}{3} = 2 + \frac{3}{4} + 1 + \frac{2}{3}$
Fourths and thirds are different-sized parts. Rename both fraction parts as twelfths. Both measure the same whole, so these twelfths are equal in size and you can count them together.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \qquad \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
Add the wholes, then add the twelfths. You may add in this order because changing the order of addition does not change the sum, and every part measures the same whole.
$2 + 1 = 3 \qquad \frac{9}{12} + \frac{8}{12} = \frac{17}{12}$
Seventeen twelfths is more than 1 whole, so regroup. Twelve twelfths of a whole make exactly 1 whole, and five twelfths are left over. Join that extra whole with the 3 wholes.
$3 + \frac{17}{12} = 3 + \frac{12}{12} + \frac{5}{12} = 4\frac{5}{12}$
Check the size on the same number line. The whole-number parts alone add to 3, so the sum must be more than 3. The first number is less than 3 and the second is less than 2, so the sum must be less than 3 plus 2, which is 5. The answer fits.
$3 < 4\frac{5}{12} < 5$

Answer

The sum of $2\frac{3}{4}$ and $1\frac{2}{3}$ is $4\frac{5}{12}$ .

Subtracting mixed numbers uses the same split into wholes and fraction parts. The number you subtract from is the total. You take away the wholes and the fraction part of the second number.

Sometimes the total does not show enough fraction parts to take away. Then regroup in the other direction: trade $1$ whole from the total for a full set of equal parts of that same whole. The total keeps its value, because the full set is still exactly $1$ whole.

The trade below rewrites $3\frac{1}{4}$ as $2\frac{5}{4}$ , which means $2 + \frac{5}{4}$ . It is the same amount, written with more fourths showing.

3\frac{1}{4} = 2 + \frac{4}{4} + \frac{1}{4} = 2\frac{5}{4}

Worked example

Subtract mixed numbers and regroup

Subtract $8\frac{2}{5} - 3\frac{2}{3}$ . Both mixed numbers measure the same whole.

Rename the fraction parts as fifteenths. Both measure the same whole, so these fifteenths are equal-sized parts. The whole-number parts stay as they are. The problem becomes 8 and six fifteenths minus 3 and ten fifteenths.
$\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \qquad \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
Compare the fraction parts. Both count fifteenths of the same whole, and 6 is less than 10. So the total does not show enough fifteenths to take ten fifteenths away yet.
$8\frac{6}{15} - 3\frac{10}{15} \qquad \frac{6}{15} < \frac{10}{15}$
Regroup the total. Trade 1 of its 8 wholes for fifteen fifteenths. Fifteen fifteenths of a whole are exactly 1 whole, so the total keeps its value.
$8\frac{6}{15} = 7 + \frac{15}{15} + \frac{6}{15} = 7\frac{21}{15}$
Taking away 3 and ten fifteenths means taking away 3 wholes and also ten fifteenths of the same whole. Now the total shows enough fifteenths, so subtract wholes from wholes and fifteenths from fifteenths. That leaves 4 and eleven fifteenths.
$7 - 3 = 4 \qquad \frac{21}{15} - \frac{10}{15} = \frac{11}{15}$
Check with addition. The answer and the amount you took away are the two parts of the total, all measured in the same whole. Adding them should rebuild 8 and two fifths, and it does.
$4\frac{11}{15} + 3\frac{10}{15} = 7\frac{21}{15} = 8\frac{6}{15} = 8\frac{2}{5}$

Answer

The difference $8\frac{2}{5} - 3\frac{2}{3}$ is $4\frac{11}{15}$ .

A common mistake. In $8\frac{6}{15} - 3\frac{10}{15}$ , it is tempting to subtract the smaller fraction from the larger one: $8 - 3 = 5$ and $\frac{10}{15} - \frac{6}{15} = \frac{4}{15}$ , giving $5\frac{4}{15}$ .

This fails because order matters in subtraction. You must take $\frac{10}{15}$ away from the total. The mistake takes $\frac{6}{15}$ away from $\frac{10}{15}$ instead, which answers a different question. Adding back shows the error. With every number in fifteenths of the same whole, $5\frac{4}{15} + 3\frac{10}{15} = 8\frac{14}{15}$ . That is more than the total, $8\frac{6}{15}$ . Regroup $1$ whole as $\frac{15}{15}$ first, then subtract.

Here is the idea to keep. Every addition or subtraction sentence links two parts and one total, and every number in it must measure the same whole. Add the parts to find a missing total. Subtract the known part from the total to find a missing part.

Mixed numbers follow the same idea. Split them into wholes and fraction parts, rename the fraction parts so they count equal-sized parts of that whole, and regroup $1$ whole as a full set of parts when you need to. Then check your answer: put it back into the sentence, or test whether its size makes sense.

Next, you will use these skills to solve word problems and to find mistakes in fraction addition and subtraction.