1. Multiplying Rational Numbers
3-3
Multiplying Rational Numbers
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra

2. Multiplying Rational Numbers
Pre-Algebra
3-3
Multiplying Rational Numbers
Warm Up
Write each number as an improper fraction. 1 7 3 7 15 8 2 17 5 1. 2 2. 1 3. 3 3 8 5 2 20 3 3 43 8 4. 6 5. 5 3 8

3. Problem of the Day
The sum of three consecutive integers is 168. What are the three integers?
55, 56, and 57

4. Learn to multiply fractions, decimals, and mixed numbers.

5. Kendall invited 36 people to a party
Kendall invited 36 people to a party. She needs to triple the recipe for a dip, or multiply the amount of each ingredient by 3. Remember that multiplication by a whole number can be written as repeated addition.
1 4
3 4 + =
Repeated addition 3 4 =
Multiplication
3 • 1 1
Notice that multiplying a fraction by a whole number is the same as multiplying the whole number by just the numerator of the fraction and keeping the same denominator.

6. RULES FOR MULTIPLYING TWO RATIONAL NUMBERS
If the signs of the factors are the same, the product is positive.
(+) • (+) = (+) or (–) • (–) = (+)
If the signs of the factors are different, the product is negative.
(+) • (–) = (–) or (–) • (+) = (–)

7. Additional Example 1A: Multiplying a Fraction and an Integer
Multiply. Write the answer in simplest form.
6 7
To write as a mixed number, divide:
Helpful Hint
12 5 12 5
= 2 R2
2 5
= 2 A. –8
–8 • 6 7 =
–48 7 =
Multiply –6
6 7 =
Simplify

8. Additional Example 1B: Multiplying a Fraction and an Integer
Multiply. Write the answer in simplest form.
1 3 B. 2 5
16 3 = 2
5(3) + 1 3 =
1 3 5
16 3
32 3 =
Multiply 10
2 3 =
Simplify

9. 5 8 A. –3 –3 • 5 8 = –15 8 = –1 7 8 = Multiply Simplify
Try This: Example 1A
Multiply. Write the answer in simplest form.
5 8 A. –3
–3 • 5 8 =
–15 8 =
Multiply –1
7 8 =
Simplify

10. Try This: Example 1B
Multiply. Write the answer in simplest form.
2 5 B. 4 9
47 5 = 4
9(5) =
2 5 9
47 5
188 5 =
Multiply 37
3 5 =
Simplify

11. 3 5
2 3
A model of is shown. Notice that to multiply fractions, you multiply the numerators and multiply the denominators. • 35 23
6 15 • =
If you place the first rectangle on top of the second, the number of green squares represents the numerator, and the number of total squares represents the denominator.

12. To simplify the product, rearrange the six green squares into the first two columns. You can see that this is .
2 5
2 5 =
6 15
A fraction is in lowest terms, or simplest form, when the numerator and denominator have no common factors.
Helpful Hint

13. 1 8 6 7 1(6) 8(7) = 1(6) 8(7) = 3 28 = Multiply numerators.
Additional Example 2A: Multiplying Fractions
Multiply. Write the answer in simplest form.
1 8
6 7 A. =
1(6) 8(7)
Multiply numerators.
Multiply denominators. 3 4
1(6) 8(7) =
Look for common factors: 2.
3 28 =
Simplest form

14. Additional Example 2B: Multiplying Fractions
Multiply. Write the answer in simplest form.
2 3
9 2 B. – =
–2(9) 3(2)
Multiply numerators.
Multiply denominators. 1 –1 3 1
–2(9) 3(2) =
Look for common factors: 2, 3. –3 =
Simplest form

15. Additional Example 2C: Multiplying Fractions
Multiply. Write the answer in simplest form.
3 7
1 2 C. 4
1 2
3 7 4 =
31 1
7 2
Write as an improper fraction. =
31(1) 7(2)
Multiply numerators. Multiply denominators.
= or 2
31 14
3 14
31 ÷ 14 = 2 R3

16. 3 5 5 8 A. 3(5) 5(8) = 3(5) 5(8) = 3 8 = Multiply numerators.
Try This: Example 2A
Multiply. Write the answer in simplest form.
3 5
5 8 A. =
3(5) 5(8)
Multiply numerators.
Multiply denominators. 1
3(5) 5(8) =
Look for common factors: 5.
3 8 =
Simplest form

17. 7 8 4 7 B. – = –7(4) 8(7) –1 –7(4) 8(7) = 1 2 = – Multiply numerators.
Try This: Example 2B
Multiply. Write the answer in simplest form.
7 8
4 7 B. – =
–7(4) 8(7)
Multiply numerators.
Multiply denominators. 1 –1 1 2
–7(4) 8(7) =
Look for common factors: 4, 7.
1 2
= –
Simplest form

18. C. 2 2 = = = or 2 Write as an improper fraction. Multiply numerators.
Try This: Example 2C
Multiply. Write the answer in simplest form.
3 5
7 9 C. 2
7 9
3 5 2 =
13 7
5 9
Write as an improper fraction. =
13(7) 5(9)
Multiply numerators.
Multiply denominators.
= or 2
91 45
1 45
91 ÷ 45 = 2 R 1

19. Additional Example 3: Multiplying Decimals
2(–0.51)
Product is negative with 2 decimal places.
2 • (–0.51) = –1.02 B.
(–0.4)(–3.75)
Product is positive with 3 decimal places.
(–0.4) • (–3.75) = 1.500
= 1.5 00
You can drop the zeros after the decimal point.

20. A. 3.1 (0.28) 3.1 • (0.28) = 0.868 B. Multiply. (–0.4)(–2.53)
Try This: Example 3
Multiply. A.
3.1 (0.28)
3.1 • (0.28) = 0.868
Product is positive with 3 decimal places. B.
(–0.4)(–2.53)
Product is positive with 3 decimal places.
(–0.4) • (–2.53) = 1.012

21. Additional Example 4A: Evaluating Expressions with Rational Numbers
Evaluate –3 x for the value of x.
1 8
A. x = 5
–3 x 18
–3 (5) 18 =
Substitute 5 for x.
–25 8 =
(5)
Write as an improper fraction.
–125 8 =
= –15
5 8
–125 ÷ 8 = –15 R5

22. 2 B. x = –3 x 7 = –3 = = = – Evaluate –3 x for the value of x.
Additional Example 4B: Evaluating Expressions with Rational Numbers Continued
Evaluate –3 x for the value of x.
1 8 2
B. x =
–3 x 18 7 –3 = 18 27
Substitute for x.
2 7 27
–25 8 =
Write as an improper fraction.
–25 • 2
8 • 7 = 1
Look for common factors: 2. 4
= –

23. A. y = –5 y –5 = = = = – , or – 4 Evaluate –5 y for the value of y.
Try This: Example 4A
Evaluate –5 y for the value of y.
3 5 67 A.
y =
–5 y 35
Substitute for x.
6 7 35 –5 = 67
Write as an improper fraction.
–28 5 = 67 –4
–28 • 6
5 • 7 =
Look for common factors: 7. 1
= –
24 5
, or – 4 45

24. B. y = 3 –5 y –5 (3) = = (3) = = –16 Evaluate –5 y for the value of y.
Try This: Example 4B
Evaluate –5 y for the value of y.
3 5
B. y = 3
–5 y 35 35
–5 (3) =
Substitute 3 for y.
–28 5 =
(3)
Write as an improper fraction.
–84 5 =
= –16
4 5
–84 ÷ 5 = –16 R4

25. Lesson Quiz: Part 1
Multiply.
1 7 1
2 7 1. 9
2 3
5 8 2. – –
5 12
3. –0.47(2.2)
–1.034
1 2 45
4. Evaluate 2 (x) for x = . 2

26. Lesson Quiz: Part 2 5.
Teri is shopping for new shoes. Her mom has agreed to pay half the cost (and all the sales tax). The shoes that Teri likes are normally $30 a pair but are on sale for off. How much money does Teri need to buy the shoes?
1 3
$10