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! 
Onehalf of 

is 
. 
As a multiplication, 
1
2 
× 
1
3 
= 
1
6 
. 

Onefourth 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 
. 



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.
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.

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

d. 
5
6 
× 
1
9 
= 
First find
1/6 of 1/9, then multiply the result by 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 
The roundabout way 
The shortcut 
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 gettogether, 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 

This lesson is taken from my book Math Mammoth Fractions 2.
A selfteaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.
Download ($5.75). Also available as a printed copy.
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