Adding and Subtracting Mixed Numbers | Grade 7 Math | LevelKite

A mixed number, such as $3\frac{1}{4}$ , has two parts: a whole number and a proper fraction written right after it. A proper fraction is a positive fraction less than $1$ : its numerator (top number) is smaller than its denominator (bottom number). The mixed number $3\frac{1}{4}$ means $3 + \frac{1}{4}$ , which is three wholes and one more fourth. The $3$ is its whole part and the $\frac{1}{4}$ is its fraction part.

You already know how to add and subtract fractions. This lesson builds on that skill so you can add and subtract mixed numbers. Every mixed number here is positive, and each subtraction takes a smaller number away from a larger one, so no answer is negative.

Because a mixed number is a whole part plus a fraction part, adding two mixed numbers means adding four pieces: two whole parts and two fraction parts. Addition lets you change the order and the grouping of its pieces without changing the total. So you can gather the whole parts together and the fraction parts together, then add each group. Watch this happen with $2\frac{1}{5} + 1\frac{2}{5}$ .

2\frac{1}{5} + 1\frac{2}{5} = \left(2 + \frac{1}{5}\right) + \left(1 + \frac{2}{5}\right) = (2 + 1) + \left(\frac{1}{5} + \frac{2}{5}\right) = 3 + \frac{3}{5} = 3\frac{3}{5}

Key concept

Wholes with wholes, fractions with fractions

A positive mixed number is its whole part plus its fraction part. So to add or subtract two mixed numbers, give the fraction parts a common denominator (one denominator shared by both), then combine wholes with wholes and fractions with fractions.

Sometimes you must regroup: trade $1$ whole for an equal fraction such as $\frac{4}{4}$ or $\frac{12}{12}$ , or trade such a fraction back for $1$ whole. The number keeps its value.

Adding: if the fraction parts total $1$ or more, move $1$ whole from that total into the whole part.
Subtracting: if the first fraction part is smaller than the second, move $1$ whole from the first whole part into its fraction part.

Worked example

Adding with different denominators

Find $2\frac{3}{4} + 1\frac{5}{6}$ .

Write each mixed number as its whole part plus its fraction part. Then gather the whole parts together and the fraction parts together. Changing the order and the grouping in addition keeps the total the same.
$2\frac{3}{4} + 1\frac{5}{6} = (2 + 1) + \left(\frac{3}{4} + \frac{5}{6}\right)$
Give the fraction parts a common denominator. The multiples of 4 are 4, 8, 12, and so on. The multiples of 6 are 6, 12, and so on. The first one they share is 12, so use twelfths. Multiplying a numerator and its denominator by the same nonzero number is the same as multiplying the fraction by 1, so each fraction keeps its value.
$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}, \quad \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
Add the fraction parts. Both now count twelfths, so add the numerators and keep the denominator 12.
$\frac{9}{12} + \frac{10}{12} = \frac{19}{12}$
Regroup. Nineteen twelfths is more than 1 whole, because 12 twelfths make 1 whole. Split off those 12 twelfths as 1 whole, which leaves 7 twelfths. Two fraction parts are each less than 1, so their total is less than 2: adding two mixed numbers never moves more than 1 whole.
$\frac{19}{12} = \frac{12}{12} + \frac{7}{12} = 1 + \frac{7}{12}$
Add the whole parts, including the 1 whole from regrouping, then attach the fraction that is left. Give a final answer in simplest form, which means the numerator and denominator share no factor greater than 1. Here 7 and 12 share none, so you are done.
$(2 + 1) + 1 + \frac{7}{12} = 4 + \frac{7}{12} = 4\frac{7}{12}$

Answer

Adding $2\frac{3}{4}$ and $1\frac{5}{6}$ gives $4\frac{7}{12}$ . That is reasonable: rounding both numbers up gives $3 + 2 = 5$ , and the exact answer is a little less than $5$ .

Subtraction works part by part too. Taking away $1\frac{1}{5}$ means taking away $1$ whole and $\frac{1}{5}$ more, so you subtract wholes from wholes and fractions from fractions. For example, $4\frac{3}{5} - 1\frac{1}{5} = 3\frac{2}{5}$ , because $4 - 1 = 3$ and $\frac{3}{5} - \frac{1}{5} = \frac{2}{5}$ .

A snag appears in $5\frac{1}{4} - 2\frac{3}{4}$ . The first fraction part, $\frac{1}{4}$ , is smaller than the $\frac{3}{4}$ you must take away. The first number is large enough, but most of it sits in its whole part. So you regroup: move $1$ of the $5$ wholes into the fraction part as $\frac{4}{4}$ .

The table renames three mixed numbers this way. In each row the whole part drops by $1$ and the numerator grows by the denominator, because the denominator tells how many equal pieces make $1$ whole. A renamed form has a fraction part greater than $1$ , so it is a working form and not a final answer, but it equals the original number exactly.

Moving 1 whole into the fraction part

Mixed number	Take out 1 whole	Write that 1 as a fraction	Renamed form
$5\frac{1}{4}$	$4 + 1 + \frac{1}{4}$	$4 + \frac{4}{4} + \frac{1}{4}$	$4\frac{5}{4}$
$3\frac{2}{9}$	$2 + 1 + \frac{2}{9}$	$2 + \frac{9}{9} + \frac{2}{9}$	$2\frac{11}{9}$
$8\frac{3}{10}$	$7 + 1 + \frac{3}{10}$	$7 + \frac{10}{10} + \frac{3}{10}$	$7\frac{13}{10}$

Worked example

Subtracting with regrouping

Find $5\frac{1}{4} - 2\frac{3}{4}$ .

Compare the fraction parts. They already share the denominator 4, but the first number has only 1 fourth and you must take away 3 fourths. One fourth is too small, so regroup before you subtract.
$\frac{1}{4} < \frac{3}{4}$
Rename the first number as in the first row of the table: move 1 of its 5 wholes into the fraction part as 4 fourths. Because 4 fourths equal 1 whole, the number keeps its value.
$5\frac{1}{4} = 4 + \frac{4}{4} + \frac{1}{4} = 4\frac{5}{4}$
Subtract wholes from wholes and fourths from fourths. Taking away 2 wholes and then 3 fourths takes away all of the second number.
$4\frac{5}{4} - 2\frac{3}{4} = (4 - 2) + \left(\frac{5}{4} - \frac{3}{4}\right) = 2\frac{2}{4}$
Write the fraction part in simplest form. Dividing the numerator and the denominator by their common factor 2 removes a factor of 2 over 2, which equals 1, so the value stays the same.
$2\frac{2}{4} = 2\frac{2 \div 2}{4 \div 2} = 2\frac{1}{2}$
Check by adding back the number you took away. If your answer is right, the total returns the starting number. Use the form in fourths so the fraction parts match. Five fourths is 1 whole and 1 fourth, so the total is 5 and 1 fourth.
$2\frac{2}{4} + 2\frac{3}{4} = 4\frac{5}{4} = 5\frac{1}{4}$

Answer

Subtracting $2\frac{3}{4}$ from $5\frac{1}{4}$ leaves $2\frac{1}{2}$ .

A tempting shortcut that fails. In $5\frac{1}{4} - 2\frac{3}{4}$ , you might subtract the wholes, $5 - 2 = 3$ , then subtract the smaller fraction from the larger one, $\frac{3}{4} - \frac{1}{4} = \frac{2}{4}$ , and write $3\frac{2}{4}$ .

That swap changes the problem. It treats $\frac{3}{4}$ as part of what you start with and $\frac{1}{4}$ as what you take away, so it really answers $5\frac{3}{4} - 2\frac{1}{4}$ . You start with more and take away less, so the result is too large. The check shows it: $3\frac{2}{4} + 2\frac{3}{4} = 5\frac{5}{4} = 6\frac{1}{4}$ , not $5\frac{1}{4}$ .

Subtraction depends on order, so never swap the fraction parts. Regroup $1$ whole instead.

Worked example

Miles left on a trail

A hiking trail is $7\frac{1}{6}$ miles long. You have hiked $3\frac{3}{4}$ miles of it. How many miles of the trail are left?

The distance left is the trail length minus the distance you have hiked. Both distances are in miles, so you can subtract them.
$7\frac{1}{6} - 3\frac{3}{4}$
Give the fraction parts a common denominator. The first multiple that 6 and 4 share is 12, so use twelfths again. Multiply each numerator and its denominator by the same nonzero number, which keeps each value.
$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}, \quad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
The problem is now 7 and 2 twelfths minus 3 and 9 twelfths. Two twelfths is smaller than 9 twelfths, so regroup: move 1 whole of the 7 into the fraction part as 12 twelfths, which equal 1 whole.
$7\frac{2}{12} = 6 + \frac{12}{12} + \frac{2}{12} = 6\frac{14}{12}$
Subtract wholes from wholes and twelfths from twelfths. The numbers 5 and 12 share no factor greater than 1, so the fraction part is in simplest form.
$6\frac{14}{12} - 3\frac{9}{12} = (6 - 3) + \left(\frac{14}{12} - \frac{9}{12}\right) = 3\frac{5}{12}$
Check by adding back the miles you have hiked. Fourteen twelfths is 1 whole and 2 twelfths, and 2 twelfths simplifies to 1 sixth, so the total returns the trail length.
$3\frac{5}{12} + 3\frac{9}{12} = 6\frac{14}{12} = 7\frac{2}{12} = 7\frac{1}{6}$

Answer

There are $3\frac{5}{12}$ miles of the trail left.

Takeaway. A positive mixed number is a whole part plus a fraction part. Give the fraction parts a common denominator, then add or subtract wholes with wholes and fractions with fractions. If the fraction parts total $1$ or more when you add, or the first fraction part is smaller than the second when you subtract, regroup $1$ whole. Regrouping changes how a number is written, not its value. Next, you will divide fractions and mixed numbers.