Use Models to Multiply Two Fractions

Key Concepts

• Multiply fractions.
• Use models to multiply two fractions.
• Interpret the product   a/b x q as a part of a partition of q into b equal parts; equivalently, as the result of a sequence of operations (a  q) ÷ b.
•  Standard multiplication of two fractions.
• Understand some of the concept distributive property with multiplying fractions.

Multiply two fractions using models

Multiplying fractions

Multiplying fractions can be a little tricky to understand.  

When adding fractions, you are finding the sum.  

When you subtract fractions, you are finding the difference.  

When multiplying a fraction by a whole number, you are finding the sum of a  

repeaters fraction or a repeated group. 

When you multiply two fractions, it means that you are looking for a part of a part. Here is a multiplication problem with two fractions.  

Example 1: 

There was a pan of lasagna left. Tom ate 1/3 of this amount. What fraction of the whole pan of lasagna did he eat? 

Solution: Find 1/3 of 1/4  to solve this problem 

One way 

Divide one whole part into fourths. 

Divide 1/4 into 3 equal parts.  

Divide the other 1/4 s into 3 equal parts. 

12 parts make one whole, so one part is 1/12

𝟏/𝟑 x 𝟏/𝟒 = 1/12 

∴ 1/3 of  1/4 = 𝟏/𝟏𝟐

Another way: 

Shade one of 3 columns yellow to represent 1/3 . 

Shade 1 of the 4 rows red to represent 1/4. 

 The orange overlap shows the product.  

1 out of 12 parts are shaded orange.  

1/3 x 1/4 =1 X 1 / 3 X 4  = 1/12

Tom ate 𝟏/𝟏𝟐

of the pan of lasagna. 

Important Note: 

You can’t always draw pictures to figure out a problem, so you can multiply fractions using a few simple steps. 

To multiply two fractions, multiply the numerator by the numerator and the denominator by the denominator.  

a/b × c/d = a × c / b × d

Multiplying two fractions by using number line 

Example 2: 

Find 

𝟐/𝟑 x 𝟑/𝟒  using a number line. 

Solution: 

𝟏/𝟑 means 1 of 3 equal parts,  

so 𝟏/𝟑 of 𝟑/𝟒 is𝟏/𝟒 

𝟐/𝟑 means 2 of 3 equal parts,  

so 2/3 of 3/𝟒  is 2 times 1/4

∴   𝟐/𝟑 x 𝟑/𝟒 = 𝟐/𝟒 or 𝟏/𝟐

Practice

1. Find  5/𝟔 x 𝟏/𝟐 . Shade the model to help solve. 

Solution: 

Shade 5 of the 6 columns red to represent

𝟓/𝟔. 

Shade 1 of the 2 rows to represent 1/2. 

Violet color represents the product.  𝟓/𝟔 x 𝟏/𝟐 = 𝟓/𝟏𝟐

2. Find 3/4 of 4/9. 

Solution: 

3/4 x 4/9 = 3  X  4/4  X 9

        = 12/36

       = 1/3

3/4 x  𝟒/𝟗  = 𝟏/𝟑

3. Find 1/2 of 3/4. 

Solution: 

= 1/2 x 3/4 = 1  X  3 / 2  X  4

= 3/8

= 1/2 x 3/4  = 3/8

4. A scientist had 3/4 of a bottle of a solution. She used 1/6 of the solution in an experiment. How much of the bottle did she use? 

Solution: 

Given that, 

Total solution that the scientist had = 3/4

Solution used by scientist = 1/6

Then, 

1/6 x 3/4 = 1  X  3 / 6  X  4

= 3 / 24

=1/8

∴ She used 1/8 of the solution for the experiment. 

Standard multiplication of two fractions

Example 1: 

On dan’s Reader, 2/3 of the books are fiction. Of the fiction 4/5 are mysterious. What fraction of the books on Dan’s eReader are mysterious? Solve this problem any way you choose. 

Solution: 

Given that, 

No. of fiction books =  2/3

No. of books that are mysterious of fiction = 4/5

Then, 

2/3 x 4/5 = 2  X  4 / 3  X 

= 8/15

= 2 / 3 x 4/5  = 8/15

∴  8/15 books on Dans eReader are mysterious. 

Example 2: 

Amelia takes pictures with her smartphone. Of the pictures,5/6 are of animals. 3/4 of her animal photos are of dogs. What fraction of her pictures are of dogs?  

Solution: 

Step 1 

Estimate 3/4 x 5/6. Since both fractions are less than 1,Since both fractions are less than 1, the product will be less than 1the product will be less than 1. 

Step 2 

Multiply the numerators. Then multiply the denominators. 

3/4×5/6  = 3 × 54 × 63 × 54 × 6

=15/24 (∵15/24 < 1 ) 

= 5/8

The answer is reasonable.  

So, 5/8 of all Animal’s pictures have dogs in them. 

Example 3: 

Is the product of  3/6 x 5/4 equal to the product of 3/4×5/6 ? Explain how you know? 

Solution: 

Case 1 

3/6×5/4 = 3 × 5/6  ×  4

= 15/24

∴ 3/6 × 5/4 = 5/8

Case 2 

3/4× 5/6  = 3 × 5/4  ×  6

= 15/24

∴ 3/4× 5/6  = 5/8

∴   3/6× 5/4  = 3/4 × 5/6

Practice

1. Find  9/10 × 1/2 

Solution: 

9/10 × 1/2 = 9  × 1/10 × 29

= 9/20

∴ 9/10 ×1/2  

=9/20

2. Find  5/6× 1/3 

Solution: 

5/6×1/3  = 5  ×  1/6 × 3  

= 5/18

∴ 5/6 × 1/3 = 5/18

3. Find  4/7 of  7/9 

Solution: 

4/7× 7/9 = 4 × 7/7 × 9

= 28/63

∴ 4/7× 7/9  = 4/9

4. Find ( 1/6 + 1/6 ) × 3/4  

Solution: 

(1/6+1/6 ) × 3/4 = (1+1 / 6) × 3/4

= (2/6) × 3/4

= 2/6 x 3/4

= 2 x 3/6 x 4

∴ (1/6+1/6 ) x 3/4 = 6/24

5. Find (9/10 – 3/10 ) × 1/4 

Solution: 

(9/10-3/10 ) × 1/4  = (9 −3/10) × 1/4

= (6/10) × 1/4

= 6/10 × 1/4

= 6 × 1/ 10 × 4

= 6/40

∴ (9/10-3/10 ) x 1/4 = 3/20

6. Edurado runs 6 laps around the track at Lincoln Park school. Then he runs 3/12 miles to get home. How far will he run in all. Show your work. 

Solution: 

Given that, 

Distance covered in one lap = 1/4miles 

No of laps around the track = 6 

Distance covered to get to home = 3 1/2 miles =7/2

Total distance covered by Edurado = (6 × 1/4) + 7/2

= 6 × 1 + 7 × 2 / 4

= 6 + 14 / 4

= 20/4

= 5 miles

What have we learned

• Multiply fractions. 
• Use models to multiply two fractions. 
• Interpret the product ab × q as a part of a partition of q into b equal parts; equivalently as the result of a sequence of operations (a × q) ÷ b. 
• Understand standard multiplication of two fractions. 
• Understand some of the concepts of distributive property with multiplying fractions.