Add and Subtract Fractions with Common Denominators

Model Fraction Addition

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

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

$\begin{array}{ccc} \dfrac{1}{4} & \dfrac{2}{4} & \dfrac{3}{4} \\ \text { one quarter +} & \text { two quarters =} & \text { three quarters } \end{array}$

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

Start with one $\dfrac{1}{4}$ piece.		$\dfrac{1}{4}$
Add two more $\dfrac{1}{4}$ pieces.		$\begin{align}+\dfrac{2}{4} \\\text{___}\end{align}$
The result is $\dfrac{3}{4}$ .		$\dfrac{3}{4}$

So again, we see that

$\dfrac{1}{4}+\dfrac{2}{4}=\dfrac{3}{4}$

Add Fractions with a Common Denominator

Example 4.52 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

$a$ ,

$b$ , and

$c$ are numbers where

$c≠0$ , then

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

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

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

$\dfrac{7}{12}-\dfrac{2}{12}=\dfrac{5}{12}$

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

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

Again, we have five twelfths, $\dfrac{5}{12}$ .

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

$\dfrac{a}{c}-\dfrac{b}{c}=\dfrac{a-b}{c}$

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

Source: Rice University, https://openstax.org/books/prealgebra/pages/4-4-add-and-subtract-fractions-with-common-denominators
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