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Products of Mixed Numbers

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Remember the bake sale from the Identify and Apply the Inverse Property of Addition in Fraction Operations Concept? Take a look at this dilemma.

There are 24 students in Mrs. Carroll’s seventh grade homeroom. Of the 24 students, three-fourths of the students participated in making food for the bake sale. The other students helped with signs and with actually selling the products at the bake sale. A few of them also brought in juice boxes to sell.

Of the three-fourths that baked, one-half of them made cookies. Michelle is trying to keep track of who did what for the bake sale. She has created a list and is writing down what each student’s participation has been.

This is where she is puzzled. Michelle wants to figure out three different things. How many students baked for the bake sale, how many students baked cookies, and what fraction of the class baked cookies?

To figure these things out, Michelle will need to know how to multiply fractions. In this Concept, you will learn all about multiplying fractions.

Guidance

Previously we worked on how to add and subtract fractions, but when you have a fraction and you want to figure out a part of that fraction, you need to multiply. Remember, that a fraction is a part of a whole. Sometimes it is tricky to figure out when to multiply fractions when you are faced with a real-world problem. First, let’s learn how to actually multiply fractions and then we can look at applying this to some real-world problems.

Multiplying fractions is always at least a two-step process.

First, you line up two fractions next two each other, and then you are ready to start multiplying.

\frac{1}{2} \cdot \frac{4}{5}

Notice that we used a dot to show that we were multiplying.

You will multiply twice. First, multiply the numerators and write the product of the numerators above a fraction bar. Next, multiply the denominators and write that product underneath the fraction bar. You don’t have to find a common denominator. You do, however, have to reduce your answer to simplest terms. We usually think of multiplying as increasing , but don’t be surprised to get a product that is smaller than one of the factors that you are multiplying.

Let’s try this out.

\frac{1}{2} \cdot \frac{4}{5}=\frac{1 \times 4}{2 \times 5}=\frac{4}{10}

Now we have a fraction called \frac{4}{10} . What next?

That’s right, it isn’t. We can simplify the fraction four-tenths, but dividing the top and the bottom number by the greatest common factor. The greatest common factor of four and ten is two. We divide the numerator and the denominator by two.

\frac{4}{10}=\frac{4 \div 2}{10 \div 2}=\frac{2}{5}

Our final answer is \frac{2}{5} .

What about a fraction and a whole number?

When you multiply a fraction and a whole number, we have to make the whole number into a fraction. Then you can multiply across just as you normally would with two fractions and simplify your answer if possible.

5 \cdot \frac{1}{2}= \frac{5}{1} \cdot \frac{1}{2}=\frac{5}{2}=2 \frac{1}{2}

How do we multiply mixed numbers?

Because mixed numbers involve wholes and parts, multiplying mixed numbers requires an extra step. Remember improper fractions? It’s essential that you convert mixed numbers to improper fractions before you multiply. Once you have the mixed numbers in the improper fraction form, multiply the numerators together and then multiply the denominators together. If you have an improper fraction as your product, you can convert it back to a mixed number to write your final answer.

3 \frac{1}{2} \cdot 2 \frac{1}{3}

First, we convert each to an improper fraction.

3 \frac{1}{2} &= \frac{7}{2}\\ 2 \frac{1}{3} &= \frac{7}{3}

Next, we multiply the two improper fractions.

\frac{7}{2} \cdot \frac{7}{3}=\frac{49}{6}

Now we can convert this improper fraction to a mixed number.

\frac{49}{6}=8 \frac{1}{6}

Our final answer is 8 \frac{1}{6} .

Sometimes, when you multiply fractions or mixed numbers, you can end up with very large numbers. When this happens, you can simplify BEFORE multiplying. You simplify on the diagonals by using the greatest common factor of the numbers on the diagonals.

\frac{2}{9} \cdot \frac{18}{30}

If we look at the numbers on the diagonals, we can see that there are greatest common factors both ways. The greatest common factor of two and thirty is 2. We can divide both by two to simplify them. The greatest common factor of 9 and 18 is 9. We can divide both by 9. Let’s simplify on the diagonals now.

\xcancel{\frac{2}{9} \cdot \frac{18}{30}} = \frac{1}{1} \cdot \frac{2}{15}

Now we multiply across for our final answer.

The answer is \frac{2}{15}

Multiply. Be sure that your answer is in simplest form.

Example A

\frac{1}{3} \cdot \frac{5}{6}

Solution: \frac{5}{18}

Example B

\frac{18}{20} \cdot \frac{4}{9}

Solution: \frac{2}{5}

Example C

2 \frac{1}{5} \cdot 3 \frac{1}{2}

Solution: 7 \frac{7}{10}

Here is the original problem once again.

There are 24 students in Mrs. Carroll’s seventh grade homeroom. Of the 24 students, three-fourths of the students participated in making food for the bake sale. The other students helped with signs and with actually selling the products at the bake sale. A few of them also brought in juice boxes to sell.

Of the three-fourths that baked, one-half of them made cookies. Michelle is trying to keep track of who did what for the bake sale. She has created a list and is writing down what each student’s participation has been.

This is where she is puzzled. Michelle wants to figure out three different things. How many students baked for the bake sale, how many students baked cookies, and what fraction of the class baked cookies?

First, let’s figure out how many students baked for the bake sale. We need to figure out three-fourths of 24. Let’s write this as a multiplication problem.

\frac{3}{4} \cdot 24=\frac{3}{4} \cdot \frac{24}{1}

Next, we can simplify on the diagonals before multiplying. We can simplify four and twenty-four by dividing by four.

\frac{3}{1} \cdot \frac{6}{1}=18

18 students out of 24 students baked for the bake sale.

Next, we want to find out how many baked cookies out of the three-fourths. We need to find out how many is one-half of 18? We can set this up as a multiplication problem.

\frac{1}{2} \cdot \frac{18}{1}=\frac{18}{2}=9

Nine students made cookies.

Michelle’s final question is what fraction of the class made cookies. Here is our multiplication problem.

\frac{1}{2} \cdot \frac{3}{4}=\frac{3}{8}

Three-eighths of the students made cookies.

Vocabulary

Fraction
a part of a whole.
Greatest Common Factor
the largest number that will divide evenly into the numerator and the denominator.
Mixed Number
a whole number and a fraction
Improper Fraction
a fraction where the numerator is larger than the denominator.

Guided Practice

Here is one for you to try on your own.

Dierdre claims that it takes her only 6 \frac{3}{4} hours to complete her homework every night. Carlos thinks he can finish his homework in \frac{2}{3} that time. How long does Carlos think it will take him to complete his homework?

Answer

We want to know the length of time Carlos thinks he needs to complete his homework.

What’s the relationship of this length of time to the length of time Dierdre takes to finish her homework? If we let D = the amount of time it takes for Dierdre to complete her homework, then we would say that the length of time it takes Carlos to finish his homework is \frac{2}{3} \cdot D . That’s a simple multiplication problem. We solve 6 \frac{3}{4} \cdot \frac{2}{3} .

We convert all mixed numbers to improper fractions, 6 \frac{3}{4} = \frac{27}{4} . \frac{27}{4} \cdot \frac{2}{3} = 4 \frac{1}{2} .

Carlos thinks that he can complete his homework in 4 \frac{1}{2} hours.

Video Review

- This is a James Sousa video on multiplying fractions and simplifying products.

- This is a James Sousa video on multiplying mixed numbers.

Practice

Directions: Multiply.

1. \frac{1}{4} \cdot \frac{3}{7}

2. \frac{5}{6} \cdot \frac{2}{3}

3. \frac{3}{10} \cdot \frac{10}{12}

4. \frac{4}{7} \cdot \frac{2}{3}

5. \frac{1}{3} \cdot 2 \frac{2}{3}

6. 2 \frac{5}{7} \cdot 1 \frac{1}{5}

7. 2 \frac{3}{10} \cdot 2 \frac{1}{4}

8. 7 \frac{1}{5} \cdot \frac{1}{11}

9. 4 \frac{5}{8} \cdot 2

10. \frac{1}{7} \cdot \frac{1}{6}

11. 3 \frac{5}{6} \cdot 1 \frac{2}{3}

12. \frac{1}{5} \cdot \frac{7}{12}

Directions: Multiply.

13. \frac{2}{3} \cdot \frac{9}{12} \cdot \frac{6}{7}

14. \frac{1}{3} \cdot 1 \frac{4}{5} \cdot \frac{3}{4}

15. \left(\frac{4}{9} \cdot \frac{5}{8}\right) \cdot \frac{3}{7}

16. \frac{10}{12} \cdot \left(3 \frac{1}{5} \cdot \frac{7}{10}\right)

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