How to Add Improper Fractions With Different Denominators

Fractions

24 Add and Subtract Fractions with Different Denominators

Learning Objectives

By the end of this section, you will be able to:

Find the least common denominator (LCD)
Convert fractions to equivalent fractions with the LCD
Add and subtract fractions with different denominators
Identify and use fraction operations
Use the order of operations to simplify complex fractions
Evaluate variable expressions with fractions

Before you get started, take this readiness quiz.

Find two fractions equivalent to $\frac{5}{6}.$
If you missed this problem, review (Figure).
Simplify: $\frac{1+5·3}{{2}^{2}+4}.$
If you missed this problem, review (Figure).

Find the Least Common Denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?

Let's think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that's not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals cents and one dime equals cents, so the sum is cents. See (Figure).

Together, a quarter and a dime are worth cents, or $\frac{35}{100}$ of a dollar.

A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.

Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is Since there are cents in one dollar, cents is $\frac{25}{100}$ and cents is $\frac{10}{100}.$ So we add $\frac{25}{100}+\frac{10}{100}$ to get $\frac{35}{100},$ which is cents.

You have practiced adding and subtracting fractions with common denominators. Now let's see what you need to do with fractions that have different denominators.

First, we will use fraction tiles to model finding the common denominator of $\frac{1}{2}$ and $\frac{1}{3}.$

We'll start with one $\frac{1}{2}$ tile and $\frac{1}{3}$ tile. We want to find a common fraction tile that we can use to match both $\frac{1}{2}$ and $\frac{1}{3}$ exactly.

If we try the $\frac{1}{4}$ pieces, of them exactly match the $\frac{1}{2}$ piece, but they do not exactly match the $\frac{1}{3}$ piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into two pieces, each labeled 1 fourth. Underneath the second rectangle are two pieces, each labeled 1 fourth. These rectangles together are longer than the rectangle labeled as 1 third.

If we try the $\frac{1}{5}$ pieces, they do not exactly cover the $\frac{1}{2}$ piece or the $\frac{1}{3}$ piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into three pieces, each labeled 1 sixth. Underneath the second rectangle is an equally sized rectangle split vertically into 2 pieces, each labeled 1 sixth.

If we try the $\frac{1}{6}$ pieces, we see that exactly of them cover the $\frac{1}{2}$ piece, and exactly of them cover the $\frac{1}{3}$ piece.

If we were to try the $\frac{1}{12}$ pieces, they would also work.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into 6 pieces, each labeled 1 twelfth. Underneath the second rectangle is an equally sized rectangle split vertically into 4 pieces, each labeled 1 twelfth.

Even smaller tiles, such as $\frac{1}{24}$ and $\frac{1}{48},$ would also exactly cover the $\frac{1}{2}$ piece and the $\frac{1}{3}$ piece.

The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of $\frac{1}{2}$ and $\frac{1}{3}$ is

Notice that all of the tiles that cover $\frac{1}{2}$ and $\frac{1}{3}$ have something in common: Their denominators are common multiples of and the denominators of $\frac{1}{2}$ and $\frac{1}{3}.$ The least common multiple (LCM) of the denominators is and so we say that is the least common denominator (LCD) of the fractions $\frac{1}{2}$ and $\frac{1}{3}.$

Doing the Manipulative Mathematics activity "Finding the Least Common Denominator" will help you develop a better understanding of the LCD.

Least Common Denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.

Find the LCD for the fractions $\frac{7}{12}$ and $\frac{5}{18}.$

Find the least common denominator for the fractions: $\frac{7}{12}$ and $\frac{11}{15}.$

Find the least common denominator for the fractions: $\frac{13}{15}$ and $\frac{17}{5}.$

To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM.

Find the least common denominator (LCD) of two fractions.

Factor each denominator into its primes.
List the primes, matching primes in columns when possible.
Bring down the columns.
Multiply the factors. The product is the LCM of the denominators.
The LCM of the denominators is the LCD of the fractions.

Find the least common denominator for the fractions $\frac{8}{15}$ and $\frac{11}{24}.$

Solution

To find the LCD, we find the LCM of the denominators.

Find the LCM of and

The top line shows 15 equals 3 times 5. The next line shows 24 equals 2 times 2 times 2 times 3. The 3s are lined up vertically. The next line shows LCM equals 2 times 2 times 2 times 3 times 5. The last line shows LCM equals 120.

The LCM of and is So, the LCD of $\frac{8}{15}$ and $\frac{11}{24}$ is

Find the least common denominator for the fractions: $\frac{13}{24}$ and $\frac{17}{32}.$

Find the least common denominator for the fractions: $\frac{9}{28}$ and $\frac{21}{32}.$

224

Convert Fractions to Equivalent Fractions with the LCD

Earlier, we used fraction tiles to see that the LCD of $\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$ is We saw that three $\frac{1}{12}$ pieces exactly covered $\frac{1}{4}$ and two $\frac{1}{12}$ pieces exactly covered $\frac{1}{6},$ so

$\frac{1}{4}=\frac{3}{12}\text{and}\frac{1}{6}=\frac{2}{12}.$

On the left is a rectangle labeled 1 fourth. Below it is an identical rectangle split vertically into 3 equal pieces, each labeled 1 twelfth. On the right is a rectangle labeled 1 sixth. Below it is an identical rectangle split vertically into 2 equal pieces, each labeled 1 twelfth.

We say that $\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{3}{12}$ are equivalent fractions and also that $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{2}{12}$ are equivalent fractions.

We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.

Equivalent Fractions Property

If are whole numbers where $b\ne 0,c\ne 0,\phantom{\rule{0.2em}{0ex}}\text{then}$

$\frac{a}{b}=\frac{a·c}{b·c}\phantom{\rule{1.0em}{0ex}}\text{and}\phantom{\rule{1.0em}{0ex}}\frac{a·c}{b·c}=\frac{a}{b}$

To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let's see how to change $\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$ to equivalent fractions with denominator without using models.

Convert $\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{6}$ to equivalent fractions with denominator their LCD.

Change to equivalent fractions with the LCD:

$\frac{3}{4}$ and $\frac{5}{6},\phantom{\rule{0.2em}{0ex}}\text{LCD}=12$

$\frac{9}{12},\frac{10}{12}$

Change to equivalent fractions with the LCD:

$-\frac{7}{12}$ and $\frac{11}{15},\phantom{\rule{0.2em}{0ex}}\text{LCD}=60$

$-\frac{35}{60},\frac{44}{60}$

Convert two fractions to equivalent fractions with their LCD as the common denominator.

Find the LCD.
For each fraction, determine the number needed to multiply the denominator to get the LCD.
Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.
Simplify the numerator and denominator.

Change to equivalent fractions with the LCD:

$\frac{13}{24}$ and $\frac{17}{32},$ LCD

$\frac{52}{96},\frac{51}{96}$

Change to equivalent fractions with the LCD:

$\frac{9}{28}$ and $\frac{27}{32},$ LCD

$\frac{72}{224},\frac{189}{224}$

Add and Subtract Fractions with Different Denominators

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

Add or subtract fractions with different denominators.

Find the LCD.
Convert each fraction to an equivalent form with the LCD as the denominator.
Add or subtract the fractions.
Write the result in simplified form.

Add: $\frac{1}{2}+\frac{1}{3}.$

Add: $\frac{1}{4}+\frac{1}{3}.$

$\frac{7}{12}$

Add: $\frac{1}{2}+\frac{1}{5}.$

$\frac{7}{10}$

Subtract: $\frac{1}{2}-\left(-\frac{1}{4}\right).$

Simplify: $\frac{1}{2}-\left(-\frac{1}{8}\right).$

$\frac{5}{8}$

Simplify: $\frac{1}{3}-\left(-\frac{1}{6}\right).$

$\frac{1}{2}$

Add: $\frac{7}{12}+\frac{5}{18}.$

Add: $\frac{7}{12}+\frac{11}{15}.$

$\frac{79}{60}$

Add: $\frac{13}{15}+\frac{17}{20}.$

$\frac{103}{60}$

When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The "missing" factors of each denominator are the numbers you need.

The first line says 12 equals 2 times 2 times 3. There is a blank space next to the 3. The next line says 18 equals 2 times 3 times 3. There is a blank space between the 2 and the first 3. There are red lines drawn from the blank spaces. This is labeled as missing factors. There is a horizontal line. Below the line, it says LCD equals 2 times 2 times 3 times 3. Below this, it says LCD equals 36.

The LCD, has factors of and factors of

Twelve has two factors of but only one of —so it is 'missing' one We multiplied the numerator and denominator of $\frac{7}{12}$ by to get an equivalent fraction with denominator

Eighteen is missing one factor of —so you multiply the numerator and denominator $\frac{5}{18}$ by to get an equivalent fraction with denominator We will apply this method as we subtract the fractions in the next example.

Subtract: $\frac{7}{15}-\frac{19}{24}.$

Subtract: $\frac{13}{24}-\frac{17}{32}.$

$\frac{1}{96}$

Subtract: $\frac{21}{32}-\frac{9}{28}.$

$\frac{75}{224}$

Add: $-\phantom{\rule{0.2em}{0ex}}\frac{11}{30}+\frac{23}{42}.$

Add: $-\frac{13}{42}+\frac{17}{35}.$

$\frac{37}{210}$

Add: $-\frac{19}{24}+\frac{17}{32}.$

$-\frac{25}{96}$

In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

Add: $\frac{3}{5}+\frac{x}{8}.$

Add: $\frac{y}{6}+\frac{7}{9}.$

$\frac{3y+14}{18}$

Add: $\frac{x}{6}+\frac{7}{15}.$

$\frac{5x+14}{30}$

Identify and Use Fraction Operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

Summary of Fraction Operations

Fraction multiplication: Multiply the numerators and multiply the denominators.

$\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}$

Fraction division: Multiply the first fraction by the reciprocal of the second.

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$

Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

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

Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

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

Simplify:

ⓐ $-\frac{1}{4}+\frac{1}{6}$
ⓑ $-\frac{1}{4}÷\frac{1}{6}$

Solution

First we ask ourselves, "What is the operation?"

ⓐThe operation is addition.

Do the fractions have a common denominator? No.

	$-\frac{1}{4}+\frac{1}{6}$
Find the LCD.
Rewrite each fraction as an equivalent fraction with the LCD.
Simplify the numerators and denominators.	$-\frac{3}{12}+\frac{2}{12}$
Add the numerators and place the sum over the common denominator.	$-\frac{1}{12}$
Check to see if the answer can be simplified. It cannot.

ⓑThe operation is division. We do not need a common denominator.

	$-\frac{1}{4}÷\frac{1}{6}$
To divide fractions, multiply the first fraction by the reciprocal of the second.	$-\frac{1}{4}·\frac{6}{1}$
Multiply.	$-\frac{6}{4}$
Simplify.	$-\frac{3}{2}$

Simplify each expression:

ⓐ $-\frac{3}{4}-\frac{1}{6}$
ⓑ $-\frac{3}{4}·\frac{1}{6}$

Simplify each expression:

ⓐ $\frac{5}{6}÷\left(-\frac{1}{4}\right)$
ⓑ $\frac{5}{6}-\left(-\frac{1}{4}\right)$

Simplify:

ⓐ $\frac{5}{x}-\frac{3}{10}$
ⓑ $\frac{5}{x}·\frac{3}{10}$

Simplify:

ⓐ $\frac{3a}{4}-\frac{8}{9}$
ⓑ $\frac{3a}{4}·\frac{8}{9}$

Simplify:

ⓐ $\frac{4k}{5}+\frac{5}{6}$
ⓑ $\frac{4k}{5}÷\frac{5}{6}$

Use the Order of Operations to Simplify Complex Fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}=\frac{3}{4}÷\frac{5}{8}$

Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

Simplify complex fractions.

Simplify the numerator.
Simplify the denominator.
Divide the numerator by the denominator.
Simplify if possible.

Simplify: $\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}.$

Simplify: $\frac{{\left(\frac{1}{3}\right)}^{2}}{{2}^{3}+2}$ .

$\frac{1}{90}$

Simplify: $\frac{1+{4}^{2}}{{\left(\frac{1}{4}\right)}^{2}}$ .

272

Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}.$

Simplify: $\frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\frac{1}{3}}$ .

Simplify: $\frac{\frac{2}{3}-\frac{1}{2}}{\frac{1}{4}+\frac{1}{3}}$ .

$\frac{2}{7}$

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Evaluate $y-\frac{5}{6}$ when $y=-\frac{2}{3}.$

Key Concepts

Find the least common denominator (LCD) of two fractions.
1. Factor each denominator into its primes.
2. List the primes, matching primes in columns when possible.
3. Bring down the columns.
4. Multiply the factors. The product is the LCM of the denominators.
5. The LCM of the denominators is the LCD of the fractions.
Equivalent Fractions Property
Convert two fractions to equivalent fractions with their LCD as the common denominator.
1. Find the LCD.
2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
4. Simplify the numerator and denominator.
Add or subtract fractions with different denominators.
1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.
Summary of Fraction Operations
- Fraction multiplication: Multiply the numerators and multiply the denominators.
  $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$
- Fraction division: Multiply the first fraction by the reciprocal of the second.
  $\frac{a}{b}+\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$
- Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
  $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$
- Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
  $\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}$
Simplify complex fractions.
1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator.
4. Simplify if possible.

Practice Makes Perfect

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

$\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$

$\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{2}{5}$

$\frac{7}{12}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{8}$

$\frac{9}{16}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{7}{12}$

$\frac{13}{30}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{25}{42}$

$\frac{23}{30}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{48}$

240

$\frac{21}{35}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{39}{56}$

$\frac{18}{35}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{33}{49}$

245

$\frac{2}{3}\text{,}\phantom{\rule{0.2em}{0ex}}\frac{1}{6},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$

$\frac{2}{3}\text{,}\phantom{\rule{0.2em}{0ex}}\frac{1}{4},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{3}{5}$

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{4},\phantom{\rule{0.2em}{0ex}}\text{LCD}=12$

$\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{5},\phantom{\rule{0.2em}{0ex}}\text{LCD}=20$

$\frac{5}{20},\frac{4}{20}$

$\frac{5}{12}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{7}{8},\phantom{\rule{0.2em}{0ex}}\text{LCD}=24$

$\frac{7}{12}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{8},\phantom{\rule{0.2em}{0ex}}\text{LCD}=24$

$\frac{14}{24},\frac{15}{24}$

$\frac{13}{16}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{−}\phantom{\rule{0.1em}{0ex}}\frac{11}{12},\phantom{\rule{0.2em}{0ex}}\text{LCD}=48$

$\frac{11}{16}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{−}\phantom{\rule{0.1em}{0ex}}\frac{5}{12},\phantom{\rule{0.2em}{0ex}}\text{LCD}=48$

$\frac{33}{48},-\frac{20}{48}$

$\frac{1}{3},\frac{5}{6},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{3}{4},\phantom{\rule{0.2em}{0ex}}\text{LCD}=12$

$\frac{1}{3},\frac{3}{4},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{3}{5},\phantom{\rule{0.2em}{0ex}}\text{LCD}=60$

$\frac{20}{60},\frac{45}{60},\frac{36}{60}$

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{1}{3}+\frac{1}{5}$

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

$\frac{9}{20}$

$\frac{1}{2}+\frac{1}{7}$

$\frac{1}{3}+\frac{1}{8}$

$\frac{11}{24}$

$\frac{1}{3}-\left(-\frac{1}{9}\right)$

$\frac{1}{4}-\left(-\frac{1}{8}\right)$

$\frac{3}{8}$

$\frac{1}{5}-\left(-\frac{1}{10}\right)$

$\frac{1}{2}-\left(-\frac{1}{6}\right)$

$\frac{2}{3}$

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

$\frac{3}{4}+\frac{2}{5}$

$\frac{23}{20}$

$\frac{7}{12}+\frac{5}{8}$

$\frac{5}{12}+\frac{3}{8}$

$\frac{19}{24}$

$\frac{7}{12}-\frac{9}{16}$

$\frac{7}{16}-\frac{5}{12}$

$\frac{1}{48}$

$\frac{11}{12}-\frac{3}{8}$

$\frac{5}{8}-\frac{7}{12}$

$\frac{1}{24}$

$\frac{2}{3}-\frac{3}{8}$

$\frac{5}{6}-\frac{3}{4}$

$\frac{1}{12}$

$-\frac{11}{30}+\frac{27}{40}$

$-\frac{9}{20}+\frac{17}{30}$

$\frac{7}{60}$

$-\frac{13}{30}+\frac{25}{42}$

$-\frac{23}{30}+\frac{5}{48}$

$-\frac{53}{80}$

$-\frac{39}{56}-\frac{22}{35}$

$-\frac{33}{49}-\frac{18}{35}$

$-\frac{291}{245}$

$-\frac{2}{3}-\left(-\frac{3}{4}\right)$

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

$\frac{1}{20}$

$-\frac{9}{16}-\left(-\frac{4}{5}\right)$

$-\frac{7}{20}-\left(-\frac{5}{8}\right)$

$\frac{11}{40}$

$1+\frac{7}{8}$

$1+\frac{5}{6}$

$\frac{11}{6}$

$1-\frac{5}{9}$

$1-\frac{3}{10}$

$\frac{7}{10}$

$\frac{x}{3}+\frac{1}{4}$

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

$\frac{3y+4}{6}$

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

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

$\frac{4x-5}{20}$

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$-\frac{3}{8}÷\left(-\frac{3}{10}\right)$

$-\frac{5}{12}÷\left(-\frac{5}{9}\right)$

$-\frac{3}{8}+\frac{5}{12}$

$-\frac{1}{8}+\frac{7}{12}$

$\frac{11}{24}$

$\frac{5}{6}-\frac{1}{9}$

$\frac{5}{9}-\frac{1}{6}$

$\frac{7}{18}$

$\frac{3}{8}·\left(-\frac{10}{21}\right)$

$\frac{7}{12}·\left(-\frac{8}{35}\right)$

$-\frac{2}{15}$

$-\frac{7}{15}-\frac{y}{4}$

$-\frac{3}{8}-\frac{x}{11}$

$\frac{-33-8x}{88}$

$\frac{11}{12a}·\frac{9a}{16}$

$\frac{10y}{13}·\frac{8}{15y}$

$\frac{16}{39}$

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{{\left(\frac{1}{5}\right)}^{2}}{2+{3}^{2}}$

$\frac{{\left(\frac{1}{3}\right)}^{2}}{5+{2}^{2}}$

$\frac{1}{81}$

$\frac{{2}^{3}+{4}^{2}}{{\left(\frac{2}{3}\right)}^{2}}$

$\frac{{3}^{3}-{3}^{2}}{{\left(\frac{3}{4}\right)}^{2}}$

$\frac{{\left(\frac{3}{5}\right)}^{2}}{{\left(\frac{3}{7}\right)}^{2}}$

$\frac{{\left(\frac{3}{4}\right)}^{2}}{{\left(\frac{5}{8}\right)}^{2}}$

$\frac{36}{25}$

$\frac{2}{\frac{1}{3}+\frac{1}{5}}$

$\frac{5}{\frac{1}{4}+\frac{1}{3}}$

$\frac{60}{7}$

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

$\frac{\frac{3}{4}+\frac{1}{2}}{\frac{5}{6}-\frac{2}{3}}$

$\frac{15}{2}$

$\frac{\frac{7}{8}-\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}}$

$\frac{\frac{3}{4}-\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}$

$\frac{3}{13}$

Mixed Practice

In the following exercises, simplify.

$\frac{1}{2}+\frac{2}{3}·\frac{5}{12}$

$\frac{1}{3}+\frac{2}{5}·\frac{3}{4}$

$\frac{19}{30}$

$1-\frac{3}{5}÷\frac{1}{10}$

$1-\frac{5}{6}÷\frac{1}{12}$

−9

$\frac{2}{3}+\frac{1}{6}+\frac{3}{4}$

$\frac{2}{3}+\frac{1}{4}+\frac{3}{5}$

$\frac{91}{60}$

$\frac{3}{8}-\frac{1}{6}+\frac{3}{4}$

$\frac{2}{5}+\frac{5}{8}-\frac{3}{4}$

$\frac{11}{40}$

$12\left(\frac{9}{20}-\frac{4}{15}\right)$

$8\left(\frac{15}{16}-\frac{5}{6}\right)$

$\frac{5}{6}$

$\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}$

$\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$

$\left(\frac{5}{9}+\frac{1}{6}\right)÷\left(\frac{2}{3}-\frac{1}{2}\right)$

$\left(\frac{3}{4}+\frac{1}{6}\right)÷\left(\frac{5}{8}-\frac{1}{3}\right)$

$\frac{22}{7}$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$4{p}^{2}q$ when $p=-\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q=\frac{5}{9}$

$5{m}^{2}n\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=-\frac{2}{5}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=\frac{1}{3}$

$\frac{4}{15}$

$2{x}^{2}{y}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}x=-\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\text{and}y=-\frac{1}{2}$

$8{u}^{2}{v}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=-\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}v=-\frac{1}{2}$

$-\frac{9}{16}$

$\frac{u+v}{w}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=-4,v=-8,w=2$

$\frac{m+n}{p}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=-6,n=-2,p=4$

−2

$\frac{a+b}{a-b}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=-3,b=8$

$\frac{r-s}{r+s}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}r=10,s=-5$

Everyday Math

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs $\frac{3}{16}$ yard of print fabric and $\frac{3}{8}$ yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $1\frac{1}{4}$ cups of sugar for the chocolate chip cookies, and $1\frac{1}{8}$ cups for the oatmeal cookies How much sugar does she need altogether?

$\text{She needs}\phantom{\rule{0.2em}{0ex}}2\frac{3}{8}\phantom{\rule{0.2em}{0ex}}\text{cups}$

Writing Exercises

Explain why it is necessary to have a common denominator to add or subtract fractions.

Explain how to find the LCD of two fractions.

Answers will vary.

Self Check

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

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

Glossary

least common denominator (LCD): The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

How to Add Improper Fractions With Different Denominators

Source: https://opentextbc.ca/prealgebraopenstax/chapter/add-and-subtract-fractions-with-different-denominators/

How to Add Improper Fractions With Different Denominators

24 Add and Subtract Fractions with Different Denominators

Learning Objectives

Find the Least Common Denominator

Convert Fractions to Equivalent Fractions with the LCD

Add and Subtract Fractions with Different Denominators

Identify and Use Fraction Operations

Use the Order of Operations to Simplify Complex Fractions

Evaluate Variable Expressions with Fractions

Key Concepts

Practice Makes Perfect

Everyday Math

Writing Exercises

Self Check

Glossary

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