# 2.8: Multiply and Divide Fractions and Mixed Numbers

Difficulty Level: At Grade Created by: CK-12
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Practice Multiplication of Fractions

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The students have ordered their supplies and on Monday they will have the grand opening of the school store. To attract students, they have decided to bake cookies in celebration of the grand opening day. Each student has decided to bake 5 batches of cookies. There are 24 cookies in a batch. So each student will be bringing 120 cookies in on Monday.

At his house, Trevor is actively working on baking his cookies. The problem is that he has found that he has only 612\begin{align*}6 \frac{1}{2}\end{align*} cups of flour. Each batch of cookies needs 112\begin{align*}1 \frac{1}{2}\end{align*} cups. Based on these numbers, Trevor will need to figure out how many batches he will need for the cookies.

Then, he will need to figure out how many cookies he will have baked with these supplies.

Trevor begins with some division.

Do you know why he is dividing? Division is a way of splitting things up. Trevor needs to split up the flour. To accomplish this task, you will need to understand how to divide and multiply fractions and mixed numbers. Pay close attention and you will know how to solve this dilemma at the end of the Concept.

### Guidance

Multiplying and dividing fractions is a lot less work then adding and subtracting them.

To multiply two fractions, simply multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product.

To divide two fractions, you first need to find the reciprocal of the divisor. That means that you need to flip the second fraction upside down. Then multiply the numerators and multiply the denominators.

Write these notes in your notebook.

Multiply: 27×35\begin{align*}\frac{2}{7} \times \frac{3}{5}\end{align*}

Multiply the numerators and multiply the denominators.

27×35=2×37×5=635\begin{align*}\frac{2}{7}\times\frac{3}{5}=\frac{2\times3}{7\times5}=\frac{6}{35}\end{align*}

Now let’s look at dividing fractions.

Divide: 4310÷12\begin{align*}4\frac{3}{10}\div \frac{1}{2}\end{align*}

Wow! This one has a mixed number and a fraction. Don’t let that throw you! You can work with mixed numbers quite easily. Just remember to convert them to improper fractions first.

First change the mixed number to an improper fraction.

4310=4×10+310=4310\begin{align*}4\frac{3}{10}=\frac{4\times10+3}{10}=\frac{43}{10}\end{align*}

Then flip the second fraction and multiply.

4310÷12=4310×21=8610\begin{align*}\frac{43}{10}\div\frac{1}{2}=\frac{43}{10}\times\frac{2}{1}=\frac{86}{10}\end{align*}

Finally, simplify the fraction.

8610=8610=835\begin{align*}\frac{86}{10}=8\frac{6}{10}=8\frac{3}{5}\end{align*}

#### Example A

914÷13\begin{align*}9\frac{1}{4} \div \frac{1}{3}\end{align*}

Solution:  2734\begin{align*}27 \frac{3}{4}\end{align*}

#### Example B

14×56\begin{align*}\frac{1}{4}\times\frac{5}{6}\end{align*}

Solution:  524\begin{align*}\frac{5}{24}\end{align*}

#### Example C

212÷13\begin{align*}2\frac{1}{2} \div \frac{1}{3}\end{align*}

Solution:  7 12\begin{align*}\frac{1}{2}\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

Remember, there are three parts to this problem.

First, we need to figure out how many batches of cookies Trevor can make with amount of flour. We begin with division.

612÷112=132÷32=132×23=133=413\begin{align*}6 \frac{1}{2}\div 1 \frac{1}{2}= \frac{13}{2} \div \frac{3}{2} =\frac{13}{2}\times \frac{2}{3} = \frac{13}{3} = 4 \frac{1}{3}\end{align*}

There are 24 cookies in a batch. Let’s multiply the number of batches times the number of cookies per batch.

413×24=133×241=104 cookies\begin{align*}4 \frac{1}{3} \times 24= \frac{13}{3} \times \frac{24}{1}=104 \ cookies\end{align*}

Trevor is short 23\begin{align*}\frac{2}{3}\end{align*} of a batch of cookies. Because of this, he will need to bake another batch of cookies. This way he will have a total of 129 cookies.

There will be 9 cookies for Trevor to eat with his family.

### Vocabulary

Greatest Common Factor
a number that will divide evenly into both the numerator and the denominator of a fraction.
Product
the answer in a multiplication problem.
Quotient
the answer in a division problem.
Fraction
a part of a whole
Mixed Number
a number with a whole number and a fraction.
Improper Fraction
a number that is greater than a whole with a larger top number and a smaller bottom number.

### Guided Practice

Here is one for you to try on your own.

23×46\begin{align*}\frac{2}{3}\times\frac{4}{6}\end{align*}

Solution

Multiply the numerators and the denominators.

23×46=818\begin{align*}\frac{2}{3}\times\frac{4}{6}=\frac{8}{18}\end{align*}

Now we need to simplify the product.

That’s alright. We can review it here.

When we simplify a fraction, we rewrite it as an equal fraction that is smaller than the fraction we have as our answer. We look for the greatest common factor that will divide into both the numerator and the denominator. The greatest common factor is the largest number that will divide into both the numerator and the denominator. This is how we will rewrite the fraction in simplest form.

The greatest common factor of 8 and 18 is 2. We divide both the numerator and denominator by 2.

818=8÷218÷2=49\begin{align*}\frac{8}{18}=\frac{8\div2}{18\div2}=\frac{4}{9}\end{align*}

### Practice

Directions: Multiply the following fractions. Be sure to simplify your answer when necessary.

1. 12×34=\begin{align*}\frac{1}{2} \times \frac{3}{4} =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. 34×56=\begin{align*}\frac{3}{4}\times\frac{5}{6}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. 16×13=\begin{align*}\frac{1}{6}\times\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. 56×1012=\begin{align*}\frac{5}{6}\times\frac{10}{12}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. 78×13=\begin{align*}\frac{7}{8}\times\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. 89×13=\begin{align*}\frac{8}{9}\times\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. 1011×25=\begin{align*}\frac{10}{11}\times\frac{2}{5}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. 910×46=\begin{align*}\frac{9}{10}\times\frac{4}{6}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. 47×12=\begin{align*}\frac{4}{7}\times\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Divide the following fractions. Be sure to convert any improper fractions to mixed numbers.

1. 34÷12=\begin{align*}\frac{3}{4} \div \frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
2. 56÷13=\begin{align*}\frac{5}{6} \div \frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. 89÷12=\begin{align*}\frac{8}{9} \div \frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. 1516÷12=\begin{align*}\frac{15}{16} \div \frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. 89÷13=\begin{align*}\frac{8}{9} \div \frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. 510÷12=\begin{align*}\frac{5}{10} \div \frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. 68÷34=\begin{align*}\frac{6}{8} \div \frac{3}{4}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. 67÷12=\begin{align*}\frac{6}{7} \div \frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. 1012÷13=\begin{align*}\frac{10}{12} \div \frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

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### Vocabulary Language: English

TermDefinition
fraction A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Greatest Common Factor The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
improper fraction An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
Mixed Number A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.
Product The product is the result after two amounts have been multiplied.
Quotient The quotient is the result after two amounts have been divided.

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