Add and Subtract Mixed Numbers

Model addition of mixed numbers with a common denominator

We’ve added and subtracted proper and improper fractions, but not mixed numbers yet. Think about addition using money: if Ron has $1$ dollar and $1$ quarter, he has $1\tfrac{1}{4}$ dollars; if Don has $2$ dollars and $1$ quarter, he has $2\tfrac{1}{4}$ dollars. Together they’d have $3$ dollars and $2$ quarters — that is, $3\tfrac{2}{4}$ dollars, which is the same as $3\tfrac{1}{2}$ dollars (since two quarters is half a dollar).

When you added the dollars, then added the quarters, you were adding the whole numbers and adding the fractions separately:

Example. Model $2\tfrac{1}{3} + 1\tfrac{2}{3}$ and give the sum.

Using fraction circles: two wholes and one $\tfrac{1}{3}$ piece, plus one whole and two $\tfrac{1}{3}$ pieces, gives three wholes and three $\tfrac{1}{3}$ pieces — that’s $3\tfrac{3}{3}$, which equals $4$ wholes. So $2\tfrac{1}{3} + 1\tfrac{2}{3} = 4$.

Example. Model $1\tfrac{3}{5} + 2\tfrac{3}{5}$ and give the sum as a mixed number.

One whole and three fifths, plus two wholes and three fifths, gives three wholes and six fifths: $3\tfrac{6}{5}$. Since $\tfrac{6}{5}$ is equivalent to $1\tfrac{1}{5}$, we add that whole to the $3$ to get $4\tfrac{1}{5}$.

Add mixed numbers with a common denominator

Modeling with fraction circles illustrates the process for adding mixed numbers: add the whole numbers and add the fractions, then simplify the result if possible.

Example. Add: $3\tfrac{4}{9} + 2\tfrac{2}{9}$.

Add the whole numbers: $3 + 2 = 5$. Add the fractions: $\tfrac{4}{9} + \tfrac{2}{9} = \tfrac{6}{9}$. Simplify the fraction by removing the common factor of $3$:

Sometimes the sum of the fractions is an improper fraction, and we need to rewrite it as a mixed number and combine it with the whole-number total.

Example. Find the sum $9\tfrac{5}{9} + 5\tfrac{7}{9}$.

Add the whole numbers and add the fractions: $14\tfrac{12}{9}$. Rewrite $\tfrac{12}{9}$ as the mixed number $1\tfrac{3}{9}$, and add the extra whole: $14 + 1\tfrac{3}{9} = 15\tfrac{3}{9}$. Simplify the fraction:

An alternate method is to convert the mixed numbers to improper fractions first, then add — this is usually written horizontally.

Example. Add by converting to improper fractions: $3\tfrac{7}{8} + 4\tfrac{3}{8}$.

Convert: $\tfrac{31}{8} + \tfrac{35}{8}$. Add the fractions: $\tfrac{66}{8}$. Rewrite as a mixed number and simplify:

Since the problem was given in mixed-number form, we write the sum as a mixed number too.

Model subtraction of mixed numbers

Think of pizzas to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half. You have to cut it into at least two pieces first, then give him half:

Example. Use a model to subtract: $1 - \tfrac{1}{3}$.

Rewrite the whole as $\tfrac{3}{3}$ (cutting it into three pieces since the fraction has denominator $3$). Take away $\tfrac{1}{3}$: there are $\tfrac{2}{3}$ left.

What if we start with more than one whole?

Example. Use a model to subtract: $2 - 1\tfrac{2}{5}$.

Rewrite $2$ as $1\tfrac{5}{5}$ (cutting one of the wholes into fifths, so you have $1$ whole and $\tfrac{5}{5}$). Take away $1\tfrac{2}{5}$: there is $\tfrac{3}{5}$ left.

What if you start with a mixed number and need to subtract a fraction larger than its fraction part? Think of needing three quarters for a parking meter, but having only a $1 bill and one quarter. You could change the$1 bill into $4$ quarters — now you have $5$ quarters and can put in $3$. This models “borrowing”: taking one whole from the whole-number part and adding it to the fraction part as an improper fraction.

Example. Use a model to subtract: $1\tfrac{1}{4} - \tfrac{3}{4}$.

Rewrite: one whole and one fourth is $\tfrac{4}{4} + \tfrac{1}{4} = \tfrac{5}{4}$. Take away $\tfrac{3}{4}$: there’s $\tfrac{2}{4}$ left, which simplifies to $\tfrac{1}{2}$.

Subtract mixed numbers with a common denominator

Now we subtract mixed numbers without a model — though it may help to picture the model in your mind as you follow the steps.

Example. Find the difference: $5\tfrac{3}{5} - 2\tfrac{4}{5}$.

Since $\tfrac{3}{5}$ is less than $\tfrac{4}{5}$, we take $1$ from the $5$ and add it to the $\tfrac{3}{5}$: $5 + \tfrac{3}{5} \to 4\tfrac{8}{5}$ (since $\tfrac{5}{5} + \tfrac{3}{5} = \tfrac{8}{5}$). Subtract the fractions: $\tfrac{8}{5} - \tfrac{4}{5} = \tfrac{4}{5}$. Subtract the whole numbers: $4 - 2 = 2$.

Just as with addition, we can also subtract mixed numbers by converting them first to improper fractions.

Example. Find the difference by converting to improper fractions: $9\tfrac{6}{11} - 7\tfrac{10}{11}$.

Convert: $\tfrac{105}{11} - \tfrac{87}{11}$. Subtract the numerators: $\tfrac{18}{11}$. Rewrite as a mixed number:

When the answer will clearly be negative — because the second mixed number is bigger than the first — it’s easier to subtract using improper fractions rather than borrowing with mixed numbers.

Example. Subtract: $3\tfrac{5}{11} - 4\tfrac{3}{4}$.

We can see the answer will be negative, since we’re subtracting a bigger number from a smaller one. Convert to equivalent fractions with the LCD of $44$, then to improper fractions: $\tfrac{152}{44} - \tfrac{209}{44}$. Subtract:

Add and subtract mixed numbers with different denominators

To add or subtract mixed numbers with different denominators, we first convert the fractions to equivalent fractions with the LCD, then follow the same steps as above.

Example. Add: $2\tfrac{1}{2} + 5\tfrac{2}{3}$.

The LCD of $2$ and $3$ is $6$. Rewrite as equivalent fractions with denominator $6$: $2\tfrac{3}{6} + 5\tfrac{4}{6}$. Add: $7\tfrac{7}{6}$. Since $\tfrac{7}{6}$ is improper, rewrite in simplest form:

Example. Subtract: $4\tfrac{3}{4} - 2\tfrac{7}{8}$.

The LCD of $4$ and $8$ is $8$. Rewrite the first mixed number with denominator $8$: $4\tfrac{6}{8}$. Since $\tfrac{6}{8}$ is less than $\tfrac{7}{8}$, borrow one whole from the $4$: $3\tfrac{14}{8} - 2\tfrac{7}{8}$. Subtract:

Key terms

mixed number addition — add the whole-number parts and the fraction parts separately (after matching denominators), then simplify, combining any improper fraction result into the whole-number total. borrowing (in subtraction of mixed numbers) — when the top fraction is smaller than the bottom fraction, take one whole from the top whole-number part and add it to the top fraction, forming an improper fraction, before subtracting.

This section is adapted from Prealgebra 2e, Section 4.6: Add and Subtract Mixed Numbers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: described the fraction-circle, dollar/quarter, and parking-meter models in prose instead of reproducing the diagrams and three-column model/notation tables; omitted the Be Prepared quiz, Manipulative Mathematics callout, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.