# 4.4 Add and subtract fractions with common denominators  (Page 2/2)

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

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

Doing the Manipulative Mathematics activity "Model Fraction Subtraction" will help you develop a better understanding of subtracting fractions.

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. Use a model to find each difference. Show a diagram to illustrate your model.

$\frac{7}{8}-\frac{4}{8}$

$\frac{3}{8}$ , models may differ.

Use a model to find each difference. Show a diagram to illustrate your model.

$\frac{5}{6}-\frac{4}{6}$

$\frac{1}{6}$ , models may differ

## 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 $a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c$ are numbers where $c\ne 0,$ then

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

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

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. $\frac{3}{8}$

Find the difference: $\frac{19}{28}-\frac{7}{28}.$

$\frac{3}{7}$

Find the difference: $\frac{27}{32}-\frac{11}{32}.$

$\frac{1}{2}$

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.

Find the difference: $\frac{x}{7}-\frac{2}{7}.$

$\frac{x-2}{7}$

Find the difference: $\frac{y}{14}-\frac{13}{14}.$

$\frac{y-13}{14}$

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. $-\frac{14}{x}$

Find the difference: $-\frac{9}{x}-\frac{7}{x}.$

$-\frac{16}{x}$

Find the difference: $-\frac{17}{a}-\frac{5}{a}.$

$-\frac{22}{a}$

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

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

## Solution

 $\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}$ Combine the numerators over the common denominator. $\frac{3+\left(-5\right)-1}{8}$ 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. $-\frac{3}{8}$

Simplify: $\frac{2}{5}+\left(-\frac{4}{5}\right)-\frac{3}{5}.$

−1

Simplify: $\frac{5}{9}+\left(-\frac{4}{9}\right)-\frac{7}{9}.$

$-\frac{2}{3}$

## Key concepts

• If $a,b,$ and $c$ are numbers where $c\ne 0$ , then $\frac{a}{c}+\frac{b}{c}=\frac{a+c}{c}$ .
• To add fractions, add the numerators and place the sum over the common denominator.
• Fraction Subtraction
• If $a,b,$ and $c$ are numbers where $c\ne 0$ , then $\frac{a}{c}-\frac{b}{c}=\frac{a-c}{c}$ .
• To subtract fractions, subtract the numerators and place the difference over the common denominator.

## Practice makes perfect

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

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

$\frac{3}{10}+\frac{4}{10}$ $\frac{7}{10}$

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

$\frac{3}{8}+\frac{3}{8}$ $\frac{3}{4}$

Add Fractions with a Common Denominator

In the following exercises, find each sum.

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

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

$\frac{7}{9}$

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

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

$\frac{16}{15}$

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

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

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

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

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

$\frac{14}{q}$

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

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

$\frac{9a}{7}$

$\frac{-12y}{8}+\frac{3y}{8}$

$\frac{-11x}{5}+\frac{7x}{5}$

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

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

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

$-\frac{3}{4}$

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

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

$-\frac{7}{8}$

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

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

$\frac{8}{19}$

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

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

$-\frac{13}{12}$

Model Fraction Subtraction

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

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

$\frac{5}{6}-\frac{2}{6}$ $\frac{1}{2}$

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

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

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

$\frac{1}{5}$

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

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

$\frac{5}{13}$

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

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

$\frac{1}{6}$

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

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

$-\frac{8}{3}$

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

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

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

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

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

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

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

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

$-\frac{14}{c}$

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

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

$-\frac{55}{v}$

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

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

$\frac{3d}{11}$

$\frac{-4r}{13}-\frac{5r}{13}$

$\frac{-7s}{3}-\frac{7s}{3}$

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

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

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

$\frac{2}{7}$

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

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

$-\frac{3}{11}$

Mixed Practice

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

$-\frac{5}{18}·\frac{9}{10}$

$-\frac{3}{14}·\frac{7}{12}$

$-\frac{1}{8}$

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

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

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

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

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

$-\frac{2}{9}$

$\frac{8}{15}÷\frac{12}{5}$

$\frac{7}{12}÷\frac{9}{28}$

$\frac{49}{27}$

## 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?

$\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{cup}$

## 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.

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.

## Self check

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

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?

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