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Guias de estudo > Prealgebra

Dividing Fractions

Learning Outcomes

  • Use a model to describe the result of dividing a fraction by a fraction
  • Use an algorithm to divide fractions

Why is 12÷3=4?12\div 3=4? We previously modeled this with counters. How many groups of 33 counters can be made from a group of 1212 counters?

Four red ovals are shown. Inside each oval are three grey circles. There are 44 groups of 33 counters. In other words, there are four 3s3\text{s} in 1212. So, 12÷3=412\div 3=4. What about dividing fractions? Suppose we want to find the quotient: 12÷16\frac{1}{2}\div \frac{1}{6}. We need to figure out how many 16s\frac{1}{6}\text{s} there are in 12\frac{1}{2}. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three 16\frac{1}{6} tiles in 12\frac{1}{2}, so 12÷16=3\frac{1}{2}\div \frac{1}{6}=3. A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth. Doing the Manipulative Mathematics activity "Model Fraction Division" will help you develop a better understanding of dividing fractions.

Example

Model: 14÷18\frac{1}{4}\div \frac{1}{8} Solution: We want to determine how many 18s\frac{1}{8}\text{s} are in 14\frac{1}{4}. Start with one 14\frac{1}{4} tile. Line up 18\frac{1}{8} tiles underneath the 14\frac{1}{4} tile. A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth. There are two 18s\frac{1}{8}\text{s} in 14\frac{1}{4}. So, 14÷18=2\frac{1}{4}\div \frac{1}{8}=2.

Try It

Model: 13÷16\frac{1}{3}\div \frac{1}{6}

Answer: A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.

Model: 12÷14\frac{1}{2}\div \frac{1}{4}

Answer: A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.

**Don't delete these** #117916 [ohm_question height="270"]117916[/ohm_question]
The following video shows another way to model division of two fractions. https://youtu.be/pk-K5JF9iMo

Example

Model: 2÷142\div \frac{1}{4}

Answer: Solution: We are trying to determine how many 14s\frac{1}{4}\text{s} there are in 22. We can model this as shown. Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth. Because there are eight 14s\frac{1}{4}\text{s} in 2,2÷14=82,2\div \frac{1}{4}=8.

Try It

Model: 2÷132\div \frac{1}{3}

Answer: Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.

Model: 3÷123\div \frac{1}{2}

Answer: Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.

**Don't delete these. #117216 [ohm_question height="270"]117216[/ohm_question]
The next video shows more examples of how to divide a whole number by a fraction. https://youtu.be/JKsfdK1WT1s Let’s use money to model 2÷142\div \frac{1}{4} in another way. We often read 14\frac{1}{4} as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of 2÷142\div \frac{1}{4} as, "How many quarters are there in two dollars?" One dollar is 44 quarters, so 22 dollars would be 88 quarters. So again, 2÷14=82\div \frac{1}{4}=8. The U.S. coin called a quarter is worth one-fourth of a dollar. A picture of a United States quarter is shown. Using fraction tiles, we showed that 12÷16=3\frac{1}{2}\div \frac{1}{6}=3. Notice that 12⋅61=3\frac{1}{2}\cdot \frac{6}{1}=3 also. How are 16\frac{1}{6} and 61\frac{6}{1} related? They are reciprocals. This leads us to the procedure for fraction division.

Fraction Division

If a,b,c, and da,b,c,\text{ and }d are numbers where b≠0,c≠0, and d≠0b\ne 0,c\ne 0,\text{ and }d\ne 0, then ab÷cd=ab⋅dc\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. We need to say b≠0,c≠0 and d≠0b\ne 0,c\ne 0\text{ and }d\ne 0 to be sure we don’t divide by zero.

Example

Divide, and write the answer in simplified form: 25÷(−37)\frac{2}{5}\div \left(-\frac{3}{7}\right)

Answer: Solution:

25÷(−37)\frac{2}{5}\div \left(-\frac{3}{7}\right)
Multiply the first fraction by the reciprocal of the second. 25(−73)\frac{2}{5}\left(-\frac{7}{3}\right)
Multiply. The product is negative. −1415-\frac{14}{15}

Try It

#146066 [ohm_question height="270"]146066[/ohm_question] #146067 [ohm_question height="270"]146067[/ohm_question]
Watch this video for more examples of dividing fractions using a reciprocal. https://youtu.be/fnaRnEXlUvs

Example

Divide, and write the answer in simplified form: 23÷n5\frac{2}{3}\div \frac{n}{5}

Answer: Solution:

23÷n5\frac{2}{3}\div \frac{n}{5}
Multiply the first fraction by the reciprocal of the second. 23⋅5n\frac{2}{3}\cdot \frac{5}{n}
Multiply. 103n\frac{10}{3n}

Try It

#146089 [ohm_question height="270"]146089[/ohm_question]

Example

Divide, and write the answer in simplified form: −34÷(−78)-\frac{3}{4}\div \left(-\frac{7}{8}\right)

Answer: Solution:

−34÷(−78)-\frac{3}{4}\div \left(-\frac{7}{8}\right)
Multiply the first fraction by the reciprocal of the second. −34⋅(−87)-\frac{3}{4}\cdot \left(-\frac{8}{7}\right)
Multiply. Remember to determine the sign first. 3⋅84⋅7\frac{3\cdot 8}{4\cdot 7}
Rewrite to show common factors. 3⋅)4‾⋅2)4‾⋅7\frac{3\cdot \overline{)4}\cdot 2}{\overline{)4}\cdot 7}
Remove common factors and simplify. 67\frac{6}{7}

Try It

#146066 [ohm_question height="270"]146066[/ohm_question]
The following video shows more examples of dividing fractions that are negative. https://youtu.be/OPHdadhDJoI

Example

Divide, and write the answer in simplified form: 718÷1427\frac{7}{18}\div \frac{14}{27}

Answer: Solution:

718÷1427\frac{7}{18}\div \frac{14}{27}
Multiply the first fraction by the reciprocal of the second. 718⋅2714\frac{7}{18}\cdot \frac{27}{14}
Multiply. 7⋅2718⋅14\frac{7\cdot 27}{18\cdot 14}
Rewrite showing common factors. 7⋅9⋅39⋅2⋅7⋅2\frac{\color{red}{7}\cdot\color{blue}{9}\cdot3}{\color{blue}{9}\cdot2\cdot\color{red}{7}\cdot2}
Remove common factors. 32⋅2\frac{3}{2\cdot 2}
Simplify. 34\frac{3}{4}

Try It

#146091 [ohm_question height="270"]146091[/ohm_question]

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