1. Multiplying Fractions
Lesson 2.3.3

2. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
California Standard:
Number Sense 1.2
Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
What it means for you:
You’ll practice multiplying fractions, and you’ll extend this to multiplying fractions by integers and mixed numbers.
Key words:
area model
mixed numbers

3. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
You’ve multiplied fractions in other grades — but it’s still not an easy topic.
In this Lesson you’ll get plenty more practice at it.

4. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Area Models Show Fraction Multiplication
Multiplying a fraction by another fraction means working out parts of a part. 1 5 2 3
For example, × means “one-fifth of two-thirds.”
You can show this graphically using an area model.

5. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
To use an area model, you start by drawing a rectangle.
Shade in of the rectangle in one direction: 1 5 1 5 2 3
Then shade in of it in the other direction,
using a different color: 2 3
The part showing × is the part that represents one-fifth of two-thirds — this is the part that’s shaded in both colors. 1 5 2 3
There are 2 squares shaded out of a total of 15, so × = . 2 15 1 5 3

6. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Example 1
Calculate × using the area model method. 3 4 1
Solution
You need to work out three-quarters of one-third — so shade in three-quarters of the rectangle in one direction, and one-third in the other direction: 1 3
There are 3 out of 12 squares shaded in both colors, so × = . 3 4 1 12 3 4
Solution follows…

7. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Guided Practice
Calculate these fraction multiplications by drawing area models:
1. ×
2. ×
3. × 1 3 2 5 4 6 2 15 3 20 3 24
Solution follows…

8. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
You Can Multiply Fractions Without Drawing Diagrams
When you draw an area model, the total number of squares is always the same as the product of the denominators of the fractions you’re multiplying.
You’ve already seen the area model for × : 1 5 2 3 2 3 1 5
Multiply the denominators: The total number of squares is 5 × 3 = 15.

9. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Also, the number of squares shaded in both colors is always the same as the product of the numerators. 2 3 1 5
Multiply the numerators: The total number of squares shaded in both colors is 1 × 2 = 2.
That means you can work the product out without drawing the area model.

10. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Example 2
Calculate × without drawing a diagram. 3 4 1
Solution
Multiply the numerators:
3 × 1 = 3
Multiply the denominators:
4 × 3 = 12
Now write this as a fraction: 3 12
numerator
denominator
So × = . 3 4 1 12
Solution follows…

11. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
The solution to Example 2 could be simplified a bit more.
Simplifying just means writing the solution using smaller numbers, but so that the fraction still means the same thing. 1 4 ÷3
The numerator and denominator can both be divided by 3… 3 12
…so represents
the same amount, but simplified 1 4
That means you could write × = . 3 4 1

12. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Guided Practice
Calculate these fraction multiplications without drawing area models. Simplify your answer where possible.
4. × × ×
7. × × × 5 6 1 2 3 7 8 4 11 12 5 12 2 9 2 7 3 32 3 7 11 14
Solution follows…

13. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
First Convert Whole or Mixed Numbers to Fractions
To multiply fractions by mixed numbers, you can just write out the mixed numbers as a single fraction and carry on multiplying as normal.
The same is true if you need to multiply a fraction by an integer — you can write the integer as a fraction and use the multiplication method from before.

14. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Example 3 1 2 1 4 4 5
Calculate: (i) 3 × , (ii) × 8
Solution
(i) Convert 3 to a fraction: 1 2
3 = = 1 2 7
(3 × 2) + 1
Then just multiply out the fractions as normal:
3 × = × = 1 2 4 7 8
Solution continues…
Solution follows…

15. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Example 3 1 2 1 4 4 5
Calculate: (i) 3 × , (ii) × 8
Solution (continued)
(ii) The integer 8 can be written as . 8 1
So you can multiply as normal:
× 8 = × = 4 5 8 1 32

16. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Guided Practice
Calculate the following. Simplify your solutions where possible.
× × 2
× ×
× × 1 1 3 2 5 7 4 4 15 7 9 10 9 1
or 1 6 7 1 4 18 7 4
or 2
Solution follows…

17. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Independent Practice
Find the product and simplify each calculation in Exercises 1–3.
1. ×
2. – ×
3. – × 2 2 3 7 10 7 15 4 9 3 15 4 45 – 5 12 1 3 35 36 –
Solution follows…

18. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Independent Practice
4. A positive whole number is multiplied by a positive fraction smaller than one. Explain how the size of the product compares to the original whole number.
5. A rectangular patio measures 8 feet wide and 12 feet long. What is the area of the patio?
6. A recipe for 12 muffins calls for 3 cups of flour. How many cups of flour are needed to make 42 muffins?
The product will always be smaller than the original whole number.
square feet 1 8
11 cups 3 8
Solution follows…

19. Multiplying Fractions
Lesson
2.3.3
Multiplying Fractions
Round Up
Multiplying fractions is OK because you don’t need to put each fraction over the same denominator.
If you need to multiply by integers or mixed numbers, just turn them into fractions too.