# Multiplying fractions by fractions

In this 5th grade lesson students first notice a shortcut for multiplying fractions of the type 1/n (such as 1/3 x 1/4). From that, we arrive at the common shortcut or rule for fraction multiplication. The lesson also contains many word problems.

In the video below, I first explain how that (1/2) x (1/3) means 1/2 of 1/3, and we find that visually. Next, we find what is 2/3 of 1/4. First, we find 1/3 of 1/4 as being 1/12. Therefore, 2/3 has to be double that much, or 2/12. After introducing the shortcut for fraction multiplication (multiply the numerators, multiply the denominators), I solve a few simple problems and a word problem. Lastly in the video, I justify the common rule for fraction division.

We have studied how to find a fractional part of a whole number using multiplication.

 For example, 3 5 of 80 is written as a multiplication: 3 5 × 80 = 240 5 = 48.

NOTE: The word OF translates here into MULTIPLICATION.

We can use the same idea to find a fractional part of a fraction!
 One-half of is .
 As a multiplication, 1 2 × 1 3 = 1 6 .
 One-fourth of is .
 As a multiplication, 1 4 × 1 3 = 1 12 .

1. Find a fractional part of the given fraction. You can think of a leftover pizza piece, which you must
share equally with one, two, or three other people. Write a multiplication sentence.

 a. Find 1 2 of
 1 2 × 1 4 =
 b. Find 1 2 of
 × =
 c. Find 1 2 of
 × =
 d. Find 1 3 of
 e. Find 1 3 of
 f. Find 1 3 of
 g. Find 1 4 of
 h. Find 1 4 of
 i. Find 1 4 of
 Did you notice a shortcut?           If so, calculate 1 5 × 1 6 =

Shortcut: multiplying fractions of the type 1/n

To multiply fractions of the form 1/n where

n is a whole number, simply multiply the
denominators to get the new denominator  →

 1 4 × 1 5 = 1 20 or 1 2 × 1 6 = 1 12

2. Multiply.

 a. 1 9 × 1 2
 b. 1 13 × 1 3
 c. 1 5 × 1 20

We have now studied how to find 1/2 or 1/3 or 1/5 of some fractions. What about finding some
other kind of fractional part? Let’s again compare this to finding fractional parts of whole numbers.

 Review: To find 3 4 of 16, or in other words 3 4 × 16, you can first find 1 4 of 16, which is 4.
 Then just take that three times, which is 12. In other words, 3 4 × 16 = 12.

We can use the same idea when finding a fractional part of another fraction.

 Example. Find 2 3 of 1 4 .    First, we find 1 3 of 1 4 , which is 1 12 .
 Then, 2 3 of 1 4 is double that much, or 2 12 .
 2 3 of =
 Example. Find 4 5 of 1 7 .
 First, we find 1 5 of 1 7 , which is 1 35 .  Then, 4 5 of 1 7 is four times that much, or 4 35 .
Multiplying a fraction by a fraction means taking that fractional part of the fraction.
It is just like taking a certain part of the leftovers, when what is left over is a fraction.

3. The pictures show how much pizza is left, and you get a certain part of the leftovers. How much will
you get? Color in a picture to show the answer.

 a. 3 4 × =
 b. 2 3 × =
 c. 3 4 × =
 d. 2 3 × =
 e. 2 5 × =
 f. 4 5 × =

4. Solve the multiplications by using two helping multiplications. Lastly, simplify if possible.

 a. 2 3 × 1 8 =

First find 1/3 of 1/8, then multiply the result by 2.

 1 3 × 1 8 = 1 24 and 1 24 × 2 = =
 b. 3 4 × 1 10 =

First find 1/4 of 1/10, then multiply the result by 3.

 1 4 × 1 10 = and × 3 =
 c. 3 5 × 1 6 =

First find 1/5 of 1/6, then multiply the result by 3.

 1 5 × 1 6 = and × 3 = =
 d. 5 6 × 1 9 =

First find 1/6 of 1/9, then multiply the result by 5.

 1 6 × 1 9 = and × 5 =
 e. 2 3 × 1 7 =

 f. 3 8 × 1 4 =

A shortcut for multiplying fractions

Multiply the numerators to get the numerator for the answer.
Multiply the denominators to get the denominator for the answer.

Study the examples on the right.

Remember always to give your final
answer as a mixed number and in
lowest terms (simplified).

 3 7 × 4 9 = 3 × 4 7 × 9 = 12 63 = 4 21
 4 5 × 11 8 = 4 × 11 5 × 8 = 44 40 = 11 10 =  1 1 10

5. Multiply. Give your answers in the lowest terms (simplified) and as mixed numbers, if possible.

 a. 3 9 × 2 9
 b. 11 12 × 1 6
 c. 1 3 × 3 13
 d. 9 × 2 3
 e. 2 9 × 6 7
 f. 10 × 5 7

COMPARE
 5 6 × 1 2 = ?

First find 1/6 of 1/2, then multiply the result by 5.

 1 6 × 1 2 = 1 12 and 1 12 × 5 = 5 12
 5 6 × 1 2 = 5 × 1 6 × 2 = 5 12
 2 8 × 3 5 = ?

Find 1/8 of 3/5, then multiply that result by 2. And to find
1/8 of 3/5, first find 1/8 of 1/5, and then multiply that by 3.

 1 8 × 1 5 = 1 40 . That multiplied by 3 is 1 40 × 3 = 3 40 .
 Then, that multiplied by 2 is 3 40 × 2 = 6 40 = 3 20 .
 2 8 × 3 5 = 2 × 3 8 × 5 = 6 40 = 3 20
In the “roundabout way,” we do each multiplication separately.
In the shortcut, we can just do them all at once.

6. Multiply. Give your answers in the lowest terms (simplified) and as mixed numbers, if possible.

 a. 3 4 × 7 8 =
 b. 7 10 × 8 5 =
 c. 9 20 × 4 5 =
 d. 2 5 × 1 3 =
 e. 1 4 × 2 7 =
 f. 9 4 × 1 3 =
 g. 2 3 × 11 8 =
 h. 2 9 × 3 10 =

7. There was 1/4 of the pizza left. Marie ate 2/3 of that.

a. What part of the original pizza did she eat?

b. What part of the original pizza is left now?

8. Theresa has painted 5/8 of the room.

a. What part is still left to paint?

b. Now, Theresa has painted half of what was still left.
Draw a bar model of the situation.
What part of the room is still left to paint?

9. Ted has completed 2/3 of a job that his boss gave him.

a. What part is still left to do?

b. Now Ted has completed a third of what was still left to do.
Draw a bar model of the situation.
What (fractional) part of the original job is still left undone?

What part is completed?

10. Sally wants to make 1/3 of the recipe on the right.
How much does she need of each ingredient?
 Carob Brownies 3 cups sweetened carob chips 8 tablespoons extra virgin olive oil 2 eggs 1/2 cup honey 1 teaspoon vanilla 3/4 cup whole wheat flour 3/4 teaspoon baking powder 1 cup walnuts or other nuts

11. For an upcoming get-together, Alison needs to multiply the coffee
recipe. Assume that half of the guests drink one serving, and the other
half drink two servings. Find how much coffee she will need, if she has:

a.  30 guests

b.  50 guests

c.  80 guests.

 Coffee (5 servings) 3 1/2 cups water 1/4 cup coffee

 Find the missing factors.
 a.        × 6 7 = 1 7
 b.       × 1 4 = 5 16
 c.        × 3 8 = 1 16
 d.       × 2 5 = 3 10

You can find free, printable worksheets for multiplying fractions by fractions here.

This lesson is taken from Maria Miller's book Math Mammoth Fractions 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.