Add and Subtract Fractions with Common Denominators

Model fraction addition

How many quarters are pictured below: one quarter plus two more quarters?

Quarters are really fractions of a dollar — another way to say fourths. One quarter plus two quarters equals three quarters. We can model the same fact with fraction circles: start with one $\tfrac{1}{4}$ piece, add two more $\tfrac{1}{4}$ pieces, and the result is $\tfrac{3}{4}$ of a whole circle. So again, $\tfrac{1}{4} + \tfrac{2}{4} = \tfrac{3}{4}$.

Example. Use a model to find the sum $\tfrac{3}{8} + \tfrac{2}{8}$.

Start with three $\tfrac{1}{8}$ pieces. Add two more $\tfrac{1}{8}$ pieces. Count how many $\tfrac{1}{8}$ pieces there are in total: five. The model shows that $\tfrac{3}{8} + \tfrac{2}{8} = \tfrac{5}{8}$.

Add fractions with a common denominator

The example above shows that to add same-size pieces — meaning fractions that already share a denominator — we simply add the number of pieces.

Example. Find the sum $\tfrac{3}{5} + \tfrac{1}{5}$.

Add the numerators and place the sum over the common denominator: $\tfrac{3+1}{5} = \tfrac{4}{5}$.

Example. Find the sum $\tfrac{x}{3} + \tfrac{2}{3}$.

Add the numerators and place the sum over the common denominator: $\tfrac{x+2}{3}$. Note this can’t be simplified any further — since $x$ and $2$ aren’t like terms, they can’t be combined.

Example. Find the sum $-\tfrac{9}{d} + \tfrac{3}{d}$.

Rewrite the first fraction with the negative sign in the numerator, since $-\tfrac{a}{b} = \tfrac{-a}{b}$: $\tfrac{-9}{d} + \tfrac{3}{d}$. Add the numerators: $\tfrac{-9+3}{d} = \tfrac{-6}{d}$. Rewrite with the negative sign in front of the fraction:

Example. Find the sum $\tfrac{2n}{11} + \tfrac{5n}{11}$.

Add the numerators and place the sum over the common denominator: $\tfrac{2n+5n}{11}$. Combine like terms in the numerator:

Example. Find the sum $-\tfrac{3}{12} + \left(-\tfrac{5}{12}\right)$.

Add the numerators and place over the common denominator: $\tfrac{-3+(-5)}{12} = \tfrac{-8}{12}$. Simplify by removing the common factor of $4$:

Model fraction subtraction

Subtracting fractions with common denominators works much like adding them. Think of a pizza cut into $12$ slices. Suppose five pieces are eaten for dinner, leaving seven ($\tfrac{7}{12}$ of the pizza). If Leonardo then eats two of the remaining pieces ($\tfrac{2}{12}$), five pieces are left:

Example. Use fraction circles to find the difference $\tfrac{4}{5} - \tfrac{1}{5}$.

Start with four $\tfrac{1}{5}$ pieces. Take away one $\tfrac{1}{5}$ piece. Count how many fifths are left: three. So $\tfrac{4}{5} - \tfrac{1}{5} = \tfrac{3}{5}$.

Subtract fractions with a common denominator

We subtract fractions with a common denominator in much the same way as we add them.

Example. Find the difference $\tfrac{23}{24} - \tfrac{14}{24}$.

Subtract the numerators and place the difference over the common denominator: $\tfrac{23-14}{24} = \tfrac{9}{24}$. Simplify by removing the common factor of $3$:

Example. Find the difference $\tfrac{y}{6} - \tfrac{1}{6}$.

Subtract the numerators and place the difference over the common denominator: $\tfrac{y-1}{6}$. The fraction is already simplified, since $y$ and $1$ are not like terms and cannot be combined.

Example. Find the difference $-\tfrac{10}{x} - \tfrac{4}{x}$.

Rewrite the first fraction as $\tfrac{-10}{x}$. Subtract the numerators: $\tfrac{-10-4}{x} = \tfrac{-14}{x}$. Rewrite with the negative sign in front of the fraction:

Now let’s combine addition and subtraction in one expression.

Example. Simplify: $\tfrac{3}{8} + \left(-\tfrac{5}{8}\right) - \tfrac{1}{8}$.

Combine all the numerators over the common denominator: $\tfrac{3+(-5)-1}{8}$. Simplify the numerator working left to right: $3 + (-5) = -2$, then $-2 - 1 = -3$. Rewrite with the negative sign in front:

Key terms

fraction addition — for a common denominator $c \neq 0$, $\tfrac{a}{c} + \tfrac{b}{c} = \tfrac{a+b}{c}$: add the numerators and keep the denominator. fraction subtraction — for a common denominator $c \neq 0$, $\tfrac{a}{c} - \tfrac{b}{c} = \tfrac{a-b}{c}$: subtract the numerators and keep the denominator.

This section is adapted from Prealgebra 2e, Section 4.4: Add and Subtract Fractions with Common Denominators 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 and quarter-coin addition and subtraction models in prose instead of reproducing the diagrams; omitted the Be Prepared quiz, Manipulative Mathematics callouts, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.