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
- 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.
