Add and Subtract Fractions with Common Denominators

Model Fraction Addition

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

quarters 1/4+2/4=3/4

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. 1/4 \(\dfrac{1}{4}\)
Add two more \(\dfrac{1}{4}\) pieces. 2/4 \(
\begin{align}
+\dfrac{2}{4} \\
\text{___}
\end{align}
\)
The result is \(\dfrac{3}{4}\). 3/4 \(\dfrac{3}{4}\)


So again, we see that

\(\dfrac{1}{4}+\dfrac{2}{4}=\dfrac{3}{4}\)


Example 4.52

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

Solution
Start with one three \(\dfrac{1}{8}\) pieces. 3/8 \(\dfrac{3}{8}\)
Add two \(\dfrac{1}{8}\) pieces. 2/8 \(\dfrac{2}{8}\)
How many \(\dfrac{1}{8}\) pieces are there? 5/8 \(\dfrac{5}{8}\)


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



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

If \(a\), \(b\), 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?

7/12 - 2/12 = 5/12

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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