Subtracting Fractions With Common Denominators

Learning Outcomes

Use fraction circles to find the difference between two fractions with like denominators
Subtract fractions with a like denominator without fraction circles
Subtract fractions with like denominators that contain variables

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into $12$ slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or ${\Large\frac{7}{12}}$ of the pizza) left in the box. If Leonardo eats $2$ of these remaining pieces (or ${\Large\frac{2}{12}}$ of the pizza), how much is left? There would be $5$ pieces left (or ${\Large\frac{5}{12}}$ of the pizza).

${\Large\frac{7}{12}}-{\Large\frac{2}{12}}={\Large\frac{5}{12}}$

Let’s use fraction circles to model the same example, ${\Large\frac{7}{12}}-{\Large\frac{2}{12}}$ .

Start with seven ${\Large\frac{1}{12}}$ pieces. Take away two ${\Large\frac{1}{12}}$ pieces. How many twelfths are left?

The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange.
Again, we have five twelfths, ${\Large\frac{5}{12}}$ .

Example

Use fraction circles to find the difference: ${\Large\frac{4}{5}}-{\Large\frac{1}{5}}$

Solution:
Start with four ${\Large\frac{1}{5}}$ pieces. Take away one ${\Large\frac{1}{5}}$ piece. Count how many fifths are left. There are three ${\Large\frac{1}{5}}$ pieces left.

The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.

Try It

Subtract Fractions with a Common Denominator

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

Fraction Subtraction

If $a,b,\text{ and }c$ are numbers where $c\ne 0$ , then

${\Large\frac{a}{c}}-{\Large\frac{b}{c}}={\Large\frac{a-b}{c}}$

To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Example

Find the difference: ${\Large\frac{23}{24}}-{\Large\frac{14}{24}}$

Show Solution

Solution:

${\Large\frac{23}{24}}-{\Large\frac{14}{24}}$
Subtract the numerators and place the difference over the common denominator.	${\Large\frac{23 - 14}{24}}$
Simplify the numerator.	${\Large\frac{9}{24}}$
Simplify the fraction by removing common factors.	${\Large\frac{3}{8}}$

Try It

Watch the following video for more examples of subtracting fractions with like denominators.

Example

Find the difference: ${\Large\frac{y}{6}}-{\Large\frac{1}{6}}$

Show Solution

Solution:

	${\Large\frac{y}{6}}-{\Large\frac{1}{6}}$
Subtract the numerators and place the difference over the common denominator.	${\Large\frac{y - 1}{6}}$

The fraction is simplified because we cannot combine the terms in the numerator.

Try it

Example

Find the difference: ${\Large-\frac{10}{x}-\frac{4}{x}}$

Show Solution

Solution:
Remember, the fraction ${\Large-\frac{10}{x}}$ can be written as ${\Large\frac{-10}{x}}$

$-\frac{10}{x}-\frac{4}{x}$
Subtract the numerators.	${\Large\frac{-10 - 4}{x}}$
Simplify.	${\Large\frac{-14}{x}}$
Rewrite with the negative sign in front of the fraction.	${\Large-\frac{14}{x}}$

Try It

Now lets do an example that involves both addition and subtraction.

Example

Simplify: ${\Large\frac{3}{8}}+\left(-{\Large\frac{5}{8}}\right)-{\Large\frac{1}{8}}$

Show Solution

Solution:

	$\Large\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}$
Combine the numerators over the common denominator.	${\Large\frac{3+\left(-5\right)-1}{8}}$
Simplify the numerator, working left to right.	${\Large\frac{-2 - 1}{8}}$
Subtract the terms in the numerator.	${\Large\frac{-3}{8}}$
Rewrite with the negative sign in front of the fraction.	${\Large-\frac{3}{8}}$

Try It

In the next video we show more examples of subtracting fractions with a common denominator. Make note of the second example, it addresses a common mistake made by students when simplifying fractions with variables.

Module 4: Fractions