4.4 Add and Subtract Fractions with Common Denominators

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

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

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

Start with one $\frac{1}{4}$ piece.		$The mathematical fraction 1/4, representing one-fourth, is clearly displayed on a white background.$
Add two more $\frac{1}{4}$ pieces.		$A mathematical expression featuring a plus sign followed by the fraction two over four, with a blank line or underscore beneath it, indicating an operation or a problem to be solved.$
The result is $\frac{3}{4}$ .		$A white background with the fraction 3/4 written in black text on the right side.$

So again, we see that

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

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.

Example 3

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

	$\frac{x}{3} + \frac{2}{3}$
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: $- \frac{9}{d} + \frac{3}{d} .$

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.	$- \frac{6}{d}$

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).

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

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, $\frac{5}{12} .$

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.

Example 8

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

	$\frac{23}{24} - \frac{14}{24}$
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: $\frac{y}{6} - \frac{1}{6} .$

	$\frac{y}{6} - \frac{1}{6}$
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: $- \frac{10}{x} - \frac{4}{x} .$

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

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

Key Concepts

Fraction Addition
- If $a, b,$ and $c$ are numbers where $c not equal to 0$ , then $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$ .
- To add fractions, add the numerators and place the sum over the common denominator.
Fraction Subtraction
- If $a, b,$ and $c$ are numbers where $c not equal to 0$ , then $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$ .
- To subtract fractions, subtract the numerators and place the difference over the common denominator.

Practice Makes Perfect

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

$\frac{2}{5} + \frac{1}{5}$

$\frac{3}{10} + \frac{4}{10}$

A circle is divided into 10 equal pieces. 7 of the pieces are shaded.

\frac{7}{10}

$\frac{1}{6} + \frac{3}{6}$

$\frac{3}{8} + \frac{3}{8}$

A circle is divided into 8 equal pieces. 6 of the pieces are shaded.

\frac{3}{4}

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{4}{9} + \frac{1}{9}$

$\frac{2}{9} + \frac{5}{9}$

$\frac{7}{9}$

$\frac{6}{13} + \frac{7}{13}$

$\frac{9}{15} + \frac{7}{15}$

$\frac{16}{15}$

$\frac{x}{4} + \frac{3}{4}$

$\frac{y}{3} + \frac{2}{3}$

$\frac{y + 2}{3}$

$\frac{7}{p} + \frac{9}{p}$

$\frac{8}{q} + \frac{6}{q}$

$\frac{14}{q}$

$\frac{8 b}{9} + \frac{3 b}{9}$

$\frac{5 a}{7} + \frac{4 a}{7}$

$\frac{9 a}{7}$

$\frac{−12 y}{8} + \frac{3 y}{8}$

$\frac{−11 x}{5} + \frac{7 x}{5}$

$\frac{−4 x}{5}$

$- \frac{1}{8} + (- \frac{3}{8})$

$- \frac{1}{8} + (- \frac{5}{8})$

$- \frac{3}{4}$

$- \frac{3}{16} + (- \frac{7}{16})$

$- \frac{5}{16} + (- \frac{9}{16})$

$- \frac{7}{8}$

$- \frac{8}{17} + \frac{15}{17}$

$- \frac{9}{19} + \frac{17}{19}$

$\frac{8}{19}$

$\frac{6}{13} + (- \frac{10}{13}) + (- \frac{12}{13})$

$\frac{5}{12} + (- \frac{7}{12}) + (- \frac{11}{12})$

$- \frac{13}{12}$

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

$\frac{5}{8} - \frac{2}{8}$

$\frac{5}{6} - \frac{2}{6}$

A circle is divided into eight sections, three of which are shaded.

\frac{1}{2}

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{4}{5} - \frac{1}{5}$

$\frac{4}{5} - \frac{3}{5}$

$\frac{1}{5}$

$\frac{11}{15} - \frac{7}{15}$

$\frac{9}{13} - \frac{4}{13}$

$\frac{5}{13}$

$\frac{11}{12} - \frac{5}{12}$

$\frac{7}{12} - \frac{5}{12}$

$\frac{1}{6}$

$\frac{4}{21} - \frac{19}{21}$

$- \frac{8}{9} - \frac{16}{9}$

$- \frac{8}{3}$

$\frac{y}{17} - \frac{9}{17}$

$\frac{x}{19} - \frac{8}{19}$

$\frac{x - 8}{19}$

$\frac{5 y}{8} - \frac{7}{8}$

$\frac{11 z}{13} - \frac{8}{13}$

$\frac{11 z - 8}{13}$

$- \frac{8}{d} - \frac{3}{d}$

$- \frac{7}{c} - \frac{7}{c}$

$- \frac{14}{c}$

$- \frac{23}{u} - \frac{15}{u}$

$- \frac{29}{v} - \frac{26}{v}$

$- \frac{55}{v}$

$\frac{6 c}{7} - \frac{5 c}{7}$

$\frac{12 d}{11} - \frac{9 d}{11}$

$\frac{3 d}{11}$

$\frac{−4 r}{13} - \frac{5 r}{13}$

$\frac{−7 s}{3} - \frac{7 s}{3}$

$- \frac{14 s}{3}$

$- \frac{3}{5} - (- \frac{4}{5})$

$- \frac{3}{7} - (- \frac{5}{7})$

$\frac{2}{7}$

$- \frac{7}{9} - (- \frac{5}{9})$

$- \frac{8}{11} - (- \frac{5}{11})$

$- \frac{3}{11}$

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

$- \frac{5}{18} \cdot \frac{9}{10}$

$- \frac{3}{14} \cdot \frac{7}{12}$

$- \frac{1}{8}$

$\frac{n}{5} - \frac{4}{5}$

$\frac{6}{11} - \frac{s}{11}$

$\frac{6 - s}{11}$

$- \frac{7}{24} + \frac{2}{24}$

$- \frac{5}{18} + \frac{1}{18}$

$- \frac{2}{9}$

$\frac{8}{15} \div \frac{12}{5}$

$\frac{7}{12} \div \frac{9}{28}$

$\frac{49}{27}$

Everyday Math

Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has $\frac{6}{10}$ of a pound of nuts and $\frac{3}{10}$ of a pound of raisins. How much trail mix can he make?

Baking Janet needs $\frac{5}{8}$ of a cup of flour for a recipe she is making. She only has $\frac{3}{8}$ of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?

$\frac{1}{4}$ cup

Writing Exercises

Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know.
Bits in case: $\frac{1}{16}$ , $\frac{1}{8}$ , ___, ___, $\frac{5}{16}$ , $\frac{3}{8}$ , ___, $\frac{1}{2}$ , $\frac{9}{16}$ , $\frac{5}{8}$ .
Bits that fell out: $\frac{7}{16}$ , $\frac{3}{16}$ , $\frac{1}{4}$ .

After a party, Lupe has $\frac{5}{12}$ of a cheese pizza, $\frac{4}{12}$ of a pepperoni pizza, and $\frac{4}{12}$ of a veggie pizza left. Will all the slices fit into $1$ pizza box? Explain your reasoning.

No, adding up the number of pieces gives $\frac{13}{12}$ , which is greater than 1. (Answers may vary.)

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Fraction skills self-assessment table. Users rate their ability to model and add/subtract fractions with common denominators: 'Confidently', 'With some help', or 'No - I don't get it!' — **Figure 4.10**

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Start with three $\frac{1}{8}$ pieces.		$The image displays the fraction 3/8, written vertically with the numeral 3 above a horizontal line and the numeral 8 below it, centered on a white background.$
Add two $\frac{1}{8}$ pieces.		$A mathematical expression shows a plus sign followed by the fraction two-eighths (+ 2/8) with a horizontal line underneath, likely indicating part of a larger sum or equation.$
How many $\frac{1}{8}$ pieces are there?		$The fraction 5/8 is displayed against a plain white background.$

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

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

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