Add and Subtract Fractions with Common Denominators

4.4 Add and Subtract Fractions with Common Denominators

The topics covered in this section are:

  1. Model fraction addition
  2. Add fractions with a common denominator
  3. Model fraction subtraction
  4. Subtract fractions with a common denominator

4.4.1 Model Fraction Addition

How many quarters are pictured? One quarter plus $2$ quarters equals $3$ quarters.

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

$\frac{1}{4}$$\frac{2}{4}$$\frac{3}{4}$
one quarter$+$two quarters$=$three quarters

Let’s use fraction circles to model the same example, $\frac{1}{4} + \frac{1}{2}$.

Start with one $\frac{1}{4}$ piece..$\LARGE \frac{1}{4}$
Add two more $\frac{1}{4}$ pieces..$\LARGE + \frac{2}{4}$
________
The result is $\frac{3}{4}$..$\LARGE \frac{3}{4}$

So again, we see that

$\LARGE \frac{1}{4} + \frac{2}{4} = \frac{3}{4}$

Example 1

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

Solution

Start with three $\frac{1}{8}$ pieces..$\LARGE \frac{3}{8}$
Add two $\frac{1}{8}$ pieces..$\LARGE + \frac{2}{8}$
________
How many $\frac{1}{8}$ pieces are tehre?.$\LARGE \frac{5}{8}$

There are five $\frac{1}{8}$ pieces, or five-eighths. The model shows that $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.

4.4.2 Add Fractions with a Common Denominator

Example 1 shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

FRACTION ADDITION

If $a,b$, and $c$ are numbers where $c \neq 0$, then

$\LARGE \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$

To add fractions with a common denominator, add the numerators and place the sum over the common denominator.

Example 2

Find the sum: $\frac{3}{5} + \frac{1}{5}$.

Solution

$\frac{3}{5} + \frac{1}{5}$
Add the numerators and place the sum over the common denominator.$\frac{3+1}{5}$
Simplify.$\frac{4}{5}$

Example 3

Find the sum: $\frac{x}{3} + \frac{2}{3}$.

Solution

$\frac{x}{3} + \frac{2}{3}$
Add the numerators and place the sum over the common denominator.$\frac{x+2}{3}$

Note that we cannot simplify this fraction any more. Since $x$ and $2$ are not like terms, we cannot combine them.

Example 4

Find the sum: $- \frac{9}{d} + \frac{3}{d}$.

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

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

$- \frac{9}{d} + \frac{3}{d}$
Rewrite the first fraction with the negative in the numerator.$\frac{-9}{d} + \frac{3}{d}$
Add the numerators and place the sum over the common denominator.$\frac{-9+3}{d}$
Simplify the numerator.$\frac{-6}{d}$
Rewrite with negative sign in front of the fraction.$- \frac{6}{d}$

Example 5

Find the sum: $\frac{2n}{11} + \frac{5n}{11}$.

Solution

$\frac{2n}{11} + \frac{5n}{11}$
Add the numerators and place the sum over the common denominator.$\frac{2n+5n}{11}$
Combine like terms.$\frac{7n}{11}$

Example 6

Find the sum: $- \frac{3}{12} + (- \frac{5}{12} )$.

Solution

$- \frac{3}{12} + (- \frac{5}{12} )$
Add the numerators and place the sum over the common denominator.$\frac{-3+(-5)}{12}$
Add.$\frac{-8}{12}$
Simplify the fraction.$- \frac{2}{3}$

4.4.3 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 $\frac{7}{12}$ of the pizza) left in the box. If Leonardo eats $2$ of these remaining pieces (or $\frac{2}{12}$ of the pizza), how much is left? There would be $5$ pieces left (or $\frac{5}{12}$ of the pizza).

$\LARGE \frac{7}{12} – \frac{2}{12} = \frac{5}{12}$

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

Start with seven $\frac{1}{12}$ pieces. Take away two $\frac{1}{12}$ 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, $\frac{5}{12}$.

Example 7

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

Solution

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

4.4.4 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 $a,b$, and $c$ are numbers where $c \neq 0$, then

$\LARGE \frac{a}{c} – \frac{b}{c} = \frac{a-b}{c}$

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

Example 8

Find the difference: $\frac{23}{24} – \frac{14}{24}$.

Solution

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

Example 9

Find the difference: $\frac{y}{6} – \frac{1}{6}$.

Solution

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

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

Example 10

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

Solution

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

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

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

Example 11

Simplify: $\frac{3}{8} + (- \frac{5}{8}) – \frac{1}{8}$.

Solution

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