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3.4: Multiplying Fractions

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Introduction

The Class Cookie Count

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 was.

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 lesson, you will learn all about multiplying fractions. Take notes on how this is done, because at the end of the lesson, you will see this problem again.

What You Will Learn

In this lesson, you will learn how to do the following:

  • Multiply fractions and mixed numbers
  • Estimate products of fractions and mixed numbers
  • Identify and apply the Commutative and Associative Properties of Multiplication in fraction operations, using numerical and variable expressions.
  • Model and solve real-world problems using simple equations involving products of fractions or mixed numbers.

Teaching Time

I. Multiply Fractions and Mixed Numbers

You have already learned 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 when you get a product that is smaller than one of the factors that you are multiplying.

Let’s try this out.

Example

\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, by 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 multiply across just as you would with two fractions and finally, simplify your answer if possible.

Example

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 and then multiply the denominators. If you have an improper fraction as your product, convert it back to a mixed number as your final answer.

Let’s look at an example.

Example

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.

Let’s look at an example.

Example

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

If we look at the numbers on the diagonals, we can see that there are 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}

3J. Lesson Exercises

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

  1. \frac{1}{3} \cdot \frac{5}{6}
  2. \frac{18}{20} \cdot \frac{4}{9}
  3. 2 \frac{1}{5} \cdot 3 \frac{1}{2}

Check your answers with a friend.

Make a few notes on how to multiply fractions and mixed numbers. Then continue with the lesson.

II. Estimate Products of Fractions and Mixed Numbers

By now you are very familiar with estimation as a tool for getting an approximate sense of the value of numbers, sums of numbers and differences of numbers. Now, we are going to add our knowledge of estimation to get approximate answers for products of fractions.

Just like rounding whole numbers, we can find approximate values of fractions by comparing the fractions to three benchmarks, 0, \frac{1}{2} and 1. Is the fraction closer to 0, \frac{1}{2} or 1? If it’s closest to one-half, we say that the value of the fraction is “about \frac{1}{2}.” We can use these approximate values of fractions to estimate the products of fractions and mixed numbers.

Next, we find the product of the approximate values. Even when you are asked to find an exact answer, estimation is a useful way to get an idea of a reasonable solution to a problem. Once you have finished solving for the exact answer of a problem, you can check your answer against the estimate. Refer back to the following steps if necessary.

Example

\frac{19}{20} \cdot \frac{6}{7}

First, we have to take each fraction and find its benchmark. Nineteen-twentieths is close to 1. Six-sevenths is also close to one.

1 \times 1 = 1

We can say that \frac{19}{20} \cdot \frac{6}{7} is approximately 1.

What about mixed numbers?

Mixed numbers work the same way except that your benchmarks will have whole numbers in them too. Let’s look at an example.

Example

3 \frac{6}{8} \cdot 5 \frac{1}{10}

Three and six-eighths is closest to 4.

Five and one-tenths is closest to 5.

4 \times 5 = 20

We can say that 3 \frac{6}{8} \cdot 5 \frac{1}{10} is approximately 20.

3K. Lesson Exercises

Estimate each product using benchmarks.

  1. \frac{1}{12} \cdot \frac{2}{11}
  2. \frac{9}{10} \cdot \frac{3}{6}
  3. 4 \frac{1}{8} \cdot 2 \frac{11}{13}

Take a few minutes to check your work with a peer.

III. Identify and Apply the Commutative and Associative Property of Multiplication in Fraction Operations, using Numerical and Variable Expressions

Do you remember the commutative and associative properties of addition? Knowing how the mechanism of addition works helped us solve more complicated addition problems involving fractions. The properties of multiplication are a lot like the properties of addition. In this lesson, we are going to discover how to use the commutative property of multiplication and the associative property of multiplication.

The Commutative Property of Multiplication states that the order that the factors are multiplied in does not change the product. Let’s test the property using simple whole numbers.

& 1 \cdot 2 \cdot 3 = 6 && 2 \cdot 1 \cdot 3 = 6 && 2 \cdot 3 \cdot 1 = 6\\& 3 \cdot 2 \cdot 1 = 6 && 3 \cdot 1 \cdot 2 = 6 && 1 \cdot 3 \cdot 2 = 6

As you can see, we can multiply the three factors (1, 2, and 3) in any orders. The Commutative Property of multiplication works also works for four, five, or even 100 factors. It works for fraction addends, too.

The Associative Property of Multiplication states that the way in which factors are grouped does not change the product. Notice that we use parentheses as the grouping symbol just as we did with addition. Once again, let’s test the property using simple whole numbers.

& (1 \cdot 3) \cdot 2 = 6 && (2 \cdot 3) \cdot 1 = 6 && (2 \cdot 1) \cdot 3 = 6

Clearly, the different way the factors are grouped has no effect on the final product.

These two properties are extremely useful when multiplying fractions. If you are multiplying three fractions and two of the fractions contain factors that you can cancel out, you can multiply those two fractions together and have a new fraction in simplest terms, then simply multiply your new simpler fraction with the third fraction.

Example

\frac{6}{8} \cdot \frac{1}{2} \cdot \frac{16}{18}

Here the easiest thing to do is to simplify the first and the third fraction. We can rearrange the fractions thanks to the Commutative Property to make our work simpler.

\frac{6}{8} \cdot \frac{16}{18} \cdot \frac{1}{2}

Now we can simplify on the diagonals of the first two fractions.

6 and 18 have the greatest common factor of 6.

8 and 16 have the greatest common factor of 8.

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

Now we multiply across to find our answer.

\frac{2}{6}

Finally we simplify.

Our final answer is \frac{1}{3}.

How do we apply this to variable expressions?

When you are working with variable expressions or expressions which contain an algebraic unknown (like x) you can use the commutative and associative properties of multiplication to simplify the expression. Let’s see how it works.

Example

\frac{2}{3} \cdot x \cdot \frac{7}{8}

We can use the Commutative Property of Multiplication to move the fractions together. Then we can find the product of the two fractions and then we will have simplified the expression. Notice that we can’t solve the expression because we don’t know the value of x.

\frac{2}{3} \cdot \frac{7}{8} \cdot x

We can simplify the two and the eight on the diagonals before we multiply.

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

Our simplified expressions is \frac{7}{12} \cdot x.

Here is an example where the Associative Property is very useful.

Example

\left(x \cdot \frac{1}{2}\right) \cdot \frac{3}{5}

Here we can move the grouping symbol or the parentheses to include the two fractions. Then we can multiply the two fractions and that will give us our simplified expression. Notice that we can’t solve this because we don’t know the value of the variable.

x \cdot \left(\frac{1}{2} \cdot \frac{3}{5}\right)

Our simplified expression is x \cdot \frac{3}{10}.

3L. Lesson Exercises

Use the Commutative Property and the Associative Property to simplify each expression.

  1. \left(x \cdot \frac{4}{5}\right) \cdot \frac{1}{2}
  2. \frac{6}{7} \cdot x \cdot \frac{1}{3}

Take a few minutes to check your answers with a friend.

IV. Model and Solve Real-World Problems Using Simple Equations Involving Products of Fractions or Mixed Numbers

“Let me have about a fourth of that.”

This is an example that would involve multiplying fractions. One of the key words that you will see when working with multiplication and real-world examples is the word “of”. Of is a key word that means multiplication. If you want \frac{1}{4} pound of turkey at the deli, you will ask the butcher to cut \frac{1}{4} times 1 pound \left[\frac{1}{4} \cdot 1\right].

Let’s look at some other real-world situations involving products of fractions and mixed numbers.

Example

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?

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 requires 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} which leads to \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.

Let’s think about the seventh graders and the bake sale. Now we have enough information to help Michelle with her list.

Real Life Example Completed

The Class Cookie Count

Here is the original problem once again. Reread it and underline any important information.

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 was.

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

Here are the vocabulary words that are found in this lesson.

Fraction
a part of a whole.
Greatest Common Factor
the largest number that will divide evenly into two numbers.
Mixed Number
a whole number and a fraction
Improper Fraction
a fraction where the numerator is larger than the denominator.
Variable Expression
an expression that uses numbers, operations and variables.
Commutative Property of Multiplication
the order that you multiply numbers does not affect the product.
Associative Property of Multiplication
the grouping of the numbers does not affect the product of those numbers.

Technology Integration

Khan Academy Multiplying Fractions

James Sousa, Multiplying Fractions

James Sousa, Example of Multiplying Fractions

Other Videos:

http://www.mathplayground.com/howto_fractionofanumber.html – This is a video where we find the fraction of a number.

Time to 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: Estimate the product.

13. \frac{1}{9} \cdot \frac{4}{5}

14. 12 \cdot \frac{6}{7}

15. \frac{18}{37} \cdot \frac{10}{11}

16. \frac{13}{15} \cdot \frac{4}{9}

17. 6 \frac{2}{3} \cdot 2 \frac{6}{11}

18. 5 \frac{27}{29} \cdot 3 \frac{1}{18}

19. 4 \frac{6}{7} \cdot 1 \frac{4}{7}

20. 4 \frac{15}{16} \cdot 7 \frac{2}{21}

Directions: Multiply.

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

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

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

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

Directions: Simplify the following expressions using the commutative and associative properties of multiplication.

25. \frac{7}{8} \cdot x \cdot \frac{4}{5}

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

27. \frac{5}{8} \cdot \left(1 \frac{2}{3} \cdot x \right)

28. Richard is baking 2 \frac{1}{2} casseroles for the archery club’s pot-luck dinner. Victoria does not think this will be enough food. She thinks he should bake at least 6 \frac{7}{8} times this amount. How many casseroles does Victoria think Richard needs to bake?

29. Crazy Sal’s is having a Delirious Discount Sale. He is selling everything in his store for \frac{3}{8} of the marked price. Rowena finds a t-shirt that is marked at $36. How much will she pay for the shirt at the discounted price?

30. Dan is cutting plywood for his science fair project. He cuts a board that is 3 \frac{1}{4} feet long. After he cuts it, he realizes that he really needs a piece about \frac{2}{3} of this length. How long will the new piece of wood that Dan cuts be?

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