1. Operations with Real Numbers

1.3 Fractions

Learning Objectives

By the end of this section it is expected that you will be able to:

  • Multiply and divide fractions
  • Simplifying expressions with fraction bar
  • Add or subtract fractions with a common denominator
  • Add or subtract fractions with different denominators
  • Use the order of operations to simplify complex fractions

Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with \frac{3}{4}.

A rectangle made up of four squares in a row. The first three squares are shaded.

Now we’ll take \frac{1}{2} of \frac{3}{4}.

A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.

Notice that now, the whole is divided into 8 equal parts. So \frac{1}{2}\cdot\frac{3}{4}=\frac{3}{8}.

To multiply fractions, we multiply the numerators and multiply the denominators.

Fraction Multiplication

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

\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 1, we will multiply negative and a positive, so the product will be negative.

EXAMPLE 1

Multiply: -\phantom{\rule{0.2em}{0ex}}\frac{11}{12}\cdot \frac{5}{7}.

Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

-\phantom{\rule{0.2em}{0ex}}\frac{11}{12}\cdot\frac{5}{7}
Determine the sign of the product; multiply. -\phantom{\rule{0.2em}{0ex}}\frac{11\cdot5}{12\cdot7}
Are there any common factors in the numerator and the demoninator? No. -\phantom{\rule{0.2em}{0ex}}\frac{55}{84}

TRY IT 1

Multiply: -\phantom{\rule{0.2em}{0ex}}\frac{10}{28}\cdot\frac{8}{15}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{4}{21}

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as \frac{a}{1}. So, for example, 3=\frac{3}{1}.

EXAMPLE 2

Multiply: -\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\cdot \left(-20x\right).

Solution

Determine the sign of the product. The signs are the same, so the product is positive.

-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\cdot \left(-20x\right)
Write 20x as a fraction. \frac{12}{5}\cdot \left(\frac{20x}{1}\right)
Multiply.
Rewrite 20 to show the common factor 5 and divide it out. .
Simplify. 48x

TRY IT 2

Multiply: \frac{11}{3}\cdot \left(-9a\right).

Show answer

-33a

Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of \frac{2}{3} is \frac{3}{2}.

Notice that \frac{2}{3}\cdot \frac{3}{2}=1. A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of -\phantom{\rule{0.2em}{0ex}}\frac{10}{7} is -\phantom{\rule{0.2em}{0ex}}\frac{7}{10}, since -\phantom{\rule{0.2em}{0ex}}\frac{10}{7}\cdot \left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{10}\right)=1.

Reciprocal

The reciprocal of \frac{a}{b} is \frac{b}{a}.

A number and its reciprocal multiply to one \frac{a}{b}\cdot \frac{b}{a}=1.

To divide fractions, we multiply the first fraction by the reciprocal of the second.

Fraction Division

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

\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}

We need to say b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0 to be sure we don’t divide by zero!

EXAMPLE 3

Find the quotient: -\phantom{\rule{0.2em}{0ex}}\frac{7}{8}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right).

Solution
-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right)
To divide, multiply the first fraction by the reciprocal of the second. -\phantom{\rule{0.2em}{0ex}}\frac{7}{18}\cdot -\phantom{\rule{0.2em}{0ex}}\frac{27}{14}
Determine the sign of the product, and then multiply.. \frac{7\cdot 27}{18\cdot 14}
Rewrite showing common factors. .
Remove common factors. \frac{3}{2\cdot 2}
Simplify. \frac{3}{4}

TRY IT 3

Find the quotient: -\phantom{\rule{0.2em}{0ex}}\frac{7}{27}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{35}{36}\right).

Show answer

\frac{4}{15}

There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

  • “To multiply fractions, multiply the numerators and multiply the denominators.”
  • “To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

This is an image with two columns. The first column reads “One fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the centre of each one representing one half. The halves are labeled “one half”. Under this is the equation “2 times 1 fourth”. Under this is another equation “two over 1 times 1 fourth.” Under this is the fraction two fourths and under this is the fraction one half. The next column reads “there are eight quarters in two dollars.” Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation “two over one divided by one fourth.” Under this is two over one times four over one. Under this is the answer “8”.

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

Complex Fraction

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

\frac{\frac{6}{7}}{3}\phantom{\rule{1em}{0ex}}\frac{\frac{3}{4}}{\frac{5}{8}}\phantom{\rule{1em}{0ex}}\frac{\frac{x}{2}}{\frac{5}{6}}

To simplify a complex fraction, we remember that the fraction bar means division. For example, the complex fraction \frac{\frac{3}{4}}{\frac{5}{8}} means \frac{3}{4}\div \frac{5}{8}.

EXAMPLE 4

Simplify: \frac{\frac{3}{4}}{\frac{5}{8}}.

Solution
\frac{\frac{3}{4}}{\frac{5}{8}}
Rewrite as division. \frac{3}{4}\div \frac{5}{8}
Multiply the first fraction by the reciprocal of the second. \frac{3}{4}\cdot \frac{8}{5}
Multiply. \frac{3\cdot 8}{4\cdot 5}
Look for common factors. .
Divide out common factors and simplify. \frac{6}{5}

TRY IT 4

Simplify: \frac{\frac{2}{3}}{\frac{5}{6}}.

Show answer

\frac{4}{5}

Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression \frac{5-3}{7+1}, we first simplify the numerator and the denominator separately. Then we divide.

\frac{5-3}{7+1}
\frac{2}{8}
\frac{1}{4}

HOW TO: Simplify an Expression with a Fraction Bar

  1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Simplify the fraction.

EXAMPLE 5

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

Solution
\frac{4-2\left(3\right)}{{2}^{2}+2}
Use the order of operations to simpliy the numerator and the denominator. \frac{4-6}{4+2}
Simplify the numerator and the denominator. \frac{-2}{6}
Simplify. A negative divided by a positive is negative. -\phantom{\rule{0.2em}{0ex}}\frac{1}{3}

TRY IT 5

Simplify: \frac{6-3\left(5\right)}{{3}^{2}+3}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.
\begin{array}{cccccc}\frac{-1}{3}=-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\hfill & & & & & \frac{\text{negative}}{\text{positive}}=\text{negative}\hfill \\ \frac{1}{-3}=-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}\hfill & & & & & \frac{\text{positive}}{\text{negative}}=\text{negative}\hfill \end{array}

Placement of Negative Sign in a Fraction

For any positive numbers a and b,

\frac{-a}{b}=\frac{a}{-b}=\phantom{\rule{0.2em}{0ex}}-\frac{a}{b}

EXAMPLE 6

Simplify: \frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}.

Solution
\begin{array}{cccc}& \frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}\end{array}
Multiply. \frac{-12+\left(-12\right)}{-6-2}
Simplify. \frac{-24}{-8}
Divide. 3

TRY IT 6

Simplify: \frac{8\left(-2\right)+4\left(-3\right)}{-5\left(2\right)+3}.

Show answer

4

Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Fraction Addition and Subtraction

If a,b,\phantom{\rule{0.2em}{0ex}}\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}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{a}{c}-\phantom{\rule{0.2em}{0ex}}\frac{b}{c}=\frac{a-b}{c}

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

EXAMPLE 7

Find the sum: \frac{3}{7}+\frac{2}{7}.

Solution
\frac{3}{7}+\frac{2}{7}
Add the numerators and place the sum over the common denominator.

Simplify.

 \frac{3+2}{7}

\frac{5}{7}

TRY IT 7

Find the sum: \frac{5}{9}+\frac{2}{9}.

Show answer

\frac{7}{9}

EXAMPLE 8

Find the difference: -\phantom{\rule{0.2em}{0ex}}\frac{23}{24}-\phantom{\rule{0.2em}{0ex}}\frac{13}{24}.

Solution
-\phantom{\rule{0.2em}{0ex}}\frac{23}{24}-\phantom{\rule{0.2em}{0ex}}\frac{13}{24}
Subtract the numerators and place the difference over the common denominator. \frac{-23-13}{24}
Simplify. \frac{-36}{24}
Simplify. Remember, -\phantom{\rule{0.2em}{0ex}}\frac{a}{b}=\frac{-a}{b}. -\phantom{\rule{0.2em}{0ex}}\frac{3}{2}

TRY IT 8

Find the difference: -\phantom{\rule{0.2em}{0ex}}\frac{19}{28}-\phantom{\rule{0.2em}{0ex}}\frac{7}{28}.

Show answer

-\phantom{\rule{0.2em}{0ex}}\frac{26}{28}

Now we will do an example that has both addition and subtraction.

EXAMPLE 9

Simplify: \frac{3}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}.

Solution
Add and subtract fractions—do they have a common denominator? Yes. \frac{3}{8}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\right)-\phantom{\rule{0.2em}{0ex}}\frac{1}{8}
Add and subtract the numerators and place the difference over the common denominator. \frac{3+\left(-5\right)-1}{8}
Simplify left to right. \frac{-2-1}{8}
Simplify. -\phantom{\rule{0.2em}{0ex}}\frac{3}{8}

TRY IT 9

Simplify: \frac{2}{5}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{9}\right)-\phantom{\rule{0.2em}{0ex}}\frac{7}{9}.

Show answer

-1

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Least Common Denominator

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

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

EXAMPLE 10

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

Solution

In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have “Step 1. Do they have a common denominator? No – rewrite each fraction with the LCD (least common denominator).” To the right of this, we have the statement “No. Find the LCD 12, 18.” To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have “Change into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!” To the right of this, we have 7/12 plus 5/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21/36 plus 10/36.The next step reads “Step 2. Add or subtract the fractions.” The hint reads “Add.” And we have 31/36.The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”

TRY IT 10

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

Show answer

\frac{79}{60}

HOW TO: Add or Subtract Fractions

  1. Do they have a common denominator?
    • Yes—go to step 2.
    • No—rewrite each fraction with the LCD (least common denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
  2. Add or subtract the fractions.
  3. Simplify, if possible.

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked “missing factors.” The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).

In (Example 10), the LCD, 36, has two factors of 2 and two factors of 3.

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2

We will apply this method as we subtract the fractions in the next example.

EXAMPLE 11

Subtract: \frac{7}{15}-\phantom{\rule{0.2em}{0ex}}\frac{19}{24}.

Solution

Do the fractions have a common denominator? No, so we need to find the LCD.

Find the LCD.\phantom{\rule{5em}{0ex}}.
Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
Rewrite as equivalent fractions with the LCD. .
Simplify. .
Subtract. \phantom{\rule{2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{39}{120}
Check to see if the answer can be simplified. \phantom{\rule{2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{13\cdot 3}{40\cdot 3}
Both 39 and 120 have a factor of 3.
Simplify. \phantom{\rule{2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{13}{40}

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

TRY IT 11

Subtract: \frac{13}{24}-\phantom{\rule{0.2em}{0ex}}\frac{17}{32}.

Show answer

\frac{1}{96}

We now have all four operations for fractions. The table below summarizes fraction operations.

Summary of Fraction Operations
Fraction Operation Sample Equation What to Do
Fraction multiplication \frac{a}{b} \cdot \frac{c}{d}=\frac{ac}{bd} Multiply the numerators and multiply the denominators
Fraction division \frac{a}{b} \div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c} Multiply the first fraction by the reciprocal of the second.
Fraction addition \frac{a}{c}+\frac{b}{c}=\frac{a+b}{c} Add the numerators and place the sum over the common denominator.
Fraction subtraction \frac{a}{c}-\phantom{\rule{0.2em}{0ex}}\frac{b}{c}=\frac{a-b}{c} Subtract the numerators and place the difference over the common denominator.

To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction \frac{\frac{3}{4}}{\frac{5}{8}} by dividing \frac{3}{4} by \frac{5}{8}.

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.

EXAMPLE 12

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

Solution

In this figure, we have a table with directions on the left and mathematical statements on the right. On the first line, we have “Step 1. Simplify the numerator. Remember one half squared means one half times one half.” To the right of this, we have the quantity (1/2) squared all over the quantity (4 plus 3 squared). Then, we have 1/4 over the quantity (4 plus 3 squared).The next line’s direction reads “Step 2. Simplify the denominator.” To the right of this, we have 1/4 over the quantity (4 plus 9), under which we have 1/4 over 13.The final step is “Step 3. Divide the numerator by the denominator. Simplify if possible. Remember, thirteen equals thirteen over 1.” To the right we have 1/4 divided by 13. Then we have 1/4 times 1/13, which equals 1/52.

TRY IT 12

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

Show answer

\frac{1}{90}

HOW TO: Simplify Complex Fractions

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator. Simplify if possible.

EXAMPLE 13

Simplify: \frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}}.

Solution

It may help to put parentheses around the numerator and the denominator.

\frac{\left(\frac{1}{2}+\frac{2}{3}\right)}{\left(\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}\right)}
Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12). \frac{\left(\frac{3}{6}+\frac{4}{6}\right)}{\left(\frac{9}{12}-\phantom{\rule{0.2em}{0ex}}\frac{2}{12}\right)}
Simplify. \frac{\left(\frac{7}{6}\right)}{\left(\frac{7}{12}\right)}
Divide the numerator by the denominator. \frac{7}{6}\div \frac{7}{12}
Simplify. \frac{7}{6}\cdot\frac{12}{7}
Divide out common factors. \frac{7\cdot6\cdot2}{6\cdot7}
Simplify. 2

TRY IT 13

Simplify: \frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}}.

Show answer

2

Key Concepts

  • Fraction Division: If a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d are numbers where b\ne 0,c\ne 0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0, then \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}. To divide fractions, multiply the first fraction by the reciprocal of the second.
  • Fraction Multiplication: If a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d are numbers where b\ne 0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0, then \frac{a}{b} \cdot \frac{c}{d}=\frac{ac}{bd}. To multiply fractions, multiply the numerators and multiply the denominators.
  • Placement of Negative Sign in a Fraction: For any positive numbers a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b, \frac{-a}{b}=\frac{a}{-b}=-\phantom{\rule{0.2em}{0ex}}\frac{a}{b}.
  • Fraction Addition and Subtraction: If a,b,\phantom{\rule{0.2em}{0ex}}\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} and \frac{a}{c}-\phantom{\rule{0.2em}{0ex}}\frac{b}{c}=\frac{a-b}{c}.
    To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
  • Strategy for Adding or Subtracting Fractions
    1. Do they have a common denominator?
      Yes—go to step 2.
      No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator.
    2. Add or subtract the fractions.
    3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
  • Simplify Complex Fractions
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Divide the numerator by the denominator. Simplify if possible.

Glossary

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

1.3 Exercise Set

In the following exercises, multiply.

  1. \frac{3}{4}\cdot \frac{9}{10}
  2. -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\cdot\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\right)
  3. -\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\cdot \frac{3}{10}
  4. \left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{15}\right) \cdot \left(\frac{9}{20}\right)
  5. \left(-\phantom{\rule{0.2em}{0ex}}\frac{63}{84}\right)\cdot \left(-\phantom{\rule{0.2em}{0ex}}\frac{44}{90}\right)
  6. 4\cdot \frac{5}{11}
  7. -8\cdot \left(\frac{17}{4}\right)

In the following exercises, divide.

  1. \frac{3}{4}\div \frac{2}{3}
  2. -\phantom{\rule{0.2em}{0ex}}\frac{7}{9}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{4}\right)
  3. \frac{5}{18}\div \left(-\phantom{\rule{0.2em}{0ex}}\frac{15}{24}\right)
  4. -5\div \frac{1}{2}
  5. \frac{3}{4}\div \left(-12\right)

In the following exercises, simplify.

  1. \frac{-\phantom{\rule{0.2em}{0ex}}\frac{8}{21}}{\frac{12}{35}}
  2. \frac{-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}}{2}
  3. \frac{22+3}{10}
  4. \frac{48}{24-15}
  5. \frac{-6+6}{8+4}
  6. \frac{4\cdot3}{6\cdot6}
  7. \frac{{4}^{2}-1}{25}
  8. \frac{8\cdot3+2\cdot9}{14+3}
  9. \frac{5\cdot6-3\cdot4}{4\cdot5-2\cdot3}
  10. \frac{{5}^{2}-{3}^{2}}{3-5}
  11. \frac{7\cdot4-2\left(8-5\right)}{9\cdot3-3\cdot5}
  12. \frac{9\left(8-2\right)-3\left(15-7\right)}{6\left(7-1\right)-3\left(17-9\right)}

In the following exercises, add.

  1. \frac{6}{13}+\frac{5}{13}
  2. -\phantom{\rule{0.2em}{0ex}}\frac{3}{16}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{16}\right)
  3. -\phantom{\rule{0.2em}{0ex}}\frac{8}{17}+\frac{15}{17}
  4. \frac{6}{13}+\left(-\phantom{\rule{0.2em}{0ex}}\frac{10}{13}\right)+\left(-\phantom{\rule{0.2em}{0ex}}\frac{12}{13}\right)
In the following exercises, subtract.
  1. \frac{11}{15}-\phantom{\rule{0.2em}{0ex}}\frac{7}{15}
  2. \frac{11}{12}-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}
  3. \frac{19}{21}-\phantom{\rule{0.2em}{0ex}}\frac{4}{21}
  4. -\phantom{\rule{0.2em}{0ex}}\frac{3}{5}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\right)
  5. -\phantom{\rule{0.2em}{0ex}}\frac{7}{9}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\right)

In the following exercises, add or subtract.

  1. \frac{1}{2}+\frac{1}{7}
  2. \frac{1}{3}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{1}{9}\right)
  3. \frac{7}{12}+\frac{5}{8}
  4. \frac{7}{12}-\phantom{\rule{0.2em}{0ex}}\frac{9}{16}
  5. \frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{3}{8}
  6. -\phantom{\rule{0.2em}{0ex}}\frac{11}{30}+\frac{27}{40}
  7. -\phantom{\rule{0.2em}{0ex}}\frac{13}{30}+\frac{25}{42}
  8. -\phantom{\rule{0.2em}{0ex}}\frac{39}{56}-\phantom{\rule{0.2em}{0ex}}\frac{22}{35}
  9. -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}-\left(-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\right)
  10. 1+\frac{7}{8}

In the following exercises, simplify.

  1. \frac{{2}^{3}+{4}^{2}}{{\left(\frac{2}{3}\right)}^{2}}
  2. \frac{{\left(\frac{3}{5}\right)}^{2}}{{\left(\frac{3}{7}\right)}^{2}}
  3. \frac{2}{\frac{1}{3}+\frac{1}{5}}
  4. \frac{\frac{7}{8}-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}}
  5. \frac{1}{2}+\frac{2}{3}\cdot \frac{5}{12}1-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}\div \frac{1}{10}
  6. 1-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}\div \frac{1}{10}
  7. \frac{2}{3}+\frac{1}{6}+\frac{3}{4}
  8. \frac{3}{8}-\phantom{\rule{0.2em}{0ex}}\frac{1}{6}+\frac{3}{4}
  9. 12\left(\frac{9}{20}-\phantom{\rule{0.2em}{0ex}}\frac{4}{15}\right)
  10. \frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}
  11. \left(\frac{5}{9}+\frac{1}{6}\right)\div \left(\frac{2}{3}-\phantom{\rule{0.2em}{0ex}}\frac{1}{2}\right)
  1. Decorating. Kayla is making covers for the throw pillows on her sofa. For each pillow cover, she needs \frac{1}{2} yard of print fabric and \frac{3}{8} yard of solid fabric. What is the total amount of fabric Kayla needs for each pillow cover?
  2. Baking. A recipe for chocolate chip cookies calls for \frac{3}{4} cup brown sugar. Leona wants to double the recipe. a) How much brown sugar will Leona need? Show your calculation. b) Measuring cups usually come in sets of \frac{1}{4},\frac{1}{3},\frac{1}{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}1 cup. Draw a diagram to show two different ways that Leona could measure the brown sugar needed to double the cookie recipe.
  3. Portions. Regin purchased a bulk package of candy that weighs 5 pounds. He wants to sell the candy in little bags that hold \frac{1}{4} pound. How many little bags of candy can he fill from the bulk package?

Answers:

  1. \frac{27}{40}
  2. \frac{1}{4}
  3. -\phantom{\rule{0.2em}{0ex}}\frac{1}{6}
  4. -\phantom{\rule{0.2em}{0ex}}\frac{21}{50}
  5. \frac{11}{30}
  6. \frac{20}{11}
  7. -34
  8. \frac{9}{8}
  9. \frac{4}{9}.
  10. -\phantom{\rule{0.2em}{0ex}}\frac{4}{9}
  11. -10
  12. -\phantom{\rule{0.2em}{0ex}}\frac{1}{16}
  13. -\phantom{\rule{0.2em}{0ex}}\frac{10}{9}
  14. -\phantom{\rule{0.2em}{0ex}}\frac{2}{5}
  15. \frac{5}{2}
  16. \frac{16}{3}
  17. 0
  18. \frac{1}{3}
  19. \frac{3}{5}
  20. 2\frac{8}{17}
  21. 1\frac{2}{7}
  22. -8
  23. \frac{11}{6}
  24. \frac{5}{2}
  25. \frac{11}{13}
  26. -\phantom{\rule{0.2em}{0ex}}\frac{5}{8}
  27. \frac{7}{17}
  28. -\phantom{\rule{0.2em}{0ex}}\frac{16}{13}
  29. \frac{4}{15}
  30. \frac{1}{2}
  31. \frac{5}{7}
  32. \frac{1}{5}
  33. -\phantom{\rule{0.2em}{0ex}}\frac{2}{9}
  34. \frac{9}{14}
  35. \frac{4}{9}
  36. \frac{29}{24}
  37. \frac{1}{48}
  38. \frac{7}{24}
  39. \frac{37}{120}
  40. \frac{17}{105}
  41. -\phantom{\rule{0.2em}{0ex}}\frac{53}{40}
  42. \frac{1}{12}
  43. \frac{15}{8}
  44. \frac{4x+3}{12}
  45. \frac{49}{25}
  46. \frac{15}{4}
  47. \frac{5}{21}
  48. \frac{7}{9}
  49. -5
  50. \frac{19}{12}
  51. \frac{23}{24}
  52. \frac{11}{5}
  53. 1
  54. \frac{13}{3}
  55. \frac{7}{8} yard
  56. a) 1\frac{1}{2} cups b) answers will vary
  57. 20 bags

Attributions

This chapter has been adapted from “Add and Subtract Fractions” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

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Business/Technical Mathematics Copyright © 2021 by Izabela Mazur and Kim Moshenko is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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