 # Multiplying Fractions and 2.2 Whole Numbers

```English
2.2
Spanish
Multiplying Fractions and
Whole Numbers
What does it mean when a whole
number is multiplied by a fraction? Will the product be greater than or
less than the whole number?
1
EXAMPLE: Multiplying a Fraction and a Whole Number
3
You have 3 gallons of paint. You use — of the paint. How many gallons
4
did you use?
to rewrite the question.
Words:
3
4
What is — of 3?
3
4
—×3=?
Numbers:
Here is one way to get the answer.
●
Draw a length of 3.
3
3
Because you want to find 4 of the
length, divide it into 4 equal sections.
+
?
+
?
+
?
?
= 3
Now, you need to think of a way to divide 3 into 4 equal parts.
●
Rewrite the number 3 as a fraction whose numerator is divisible by 4.
Because the length is divided into 4
equal sections, multiply the numerator
and denominator by 4.
3 3 × 4 12
=
=
1 1×4
4
0
12
4
3
4
In this form, you see that — can be divided into four equal parts of —.
●
3
4
Each part is — gallon and you
used three of them. Written as
multiplication, you have
3
4
9
4
— × 3 = —.
9
4
So, you used — gallons of paint.
50
Chapter 2
Multiplying and Dividing Fractions
0
3
4
+
3
4
6
4
+
3
4
9
4
+
3
4
=
12
4
9
4
English
Spanish
2
EXAMPLE: Multiplying a Whole Number and a Fraction
2
5
Find 4 × —.
2
+
2
5
+
2
5
+
2
5
+
2
5
8
End at 5 .
Start at 0.
1
5
0
2
5
8
5
2
5
3
5
4
5
1
6
5
7
5
8
5
9
5
2
3
5
So, 4 × — = —, or 1 —.
Inductive Reasoning
Work with a partner. Complete the table using a number line.
Exercise
3
4
3
4
1
3. — × 3
2
4. 4 × —
3
4
3
4
9
4
—+—+—=—
2
5
2
5
2
5
2
5
2
5
8
5
—+—+—+—=—
7
6
5. — × 5
9
5
6. 3 × —
1
3
7. — × 12
2
3
8. a. Write a real-life problem that is related to the product — × 5.
2
3
b. Write a different real-life problem that is related to the product 5 × —.
c. Are the two products equal? How is your answer related to the
Commutative Property of Multiplication?
9. IN YOUR OWN WORDS What does it mean when a whole number is
multiplied by a fraction? Will the product be greater than or less than
the whole number?
10. Write a general rule for multiplying fractions and whole numbers.
Use what you learned about multiplying fractions and whole
numbers to complete Exercises 4 –11 on page 54.
Section 2.2
Multiplying Fractions and Whole Numbers
51
English
2.2
Spanish
Lesson
Lesson Tutorials
Multiplying Fractions and Whole Numbers
Multiply the numerator of the fraction by the whole number.
Then write the product over the denominator.
Words
1
8
9
⋅
⋅ bc
a b
c
a — = —, where c ≠ 0
Algebra
EXAMPLE
2×4
9
4
9
2× —= —= —
Numbers
Multiplying a Whole Number and a Fraction
7
8
Find 3 × —.
Estimate 3 × 1 = 3
Multiply the numerator and whole number.
7 3×7
3×—=—
8
8
Write the product over the denominator.
21
8
5
8
= —, or 2 —
Simplify.
5
8
So, the product is 2 —.
EXAMPLE
2
✓
Multiplying a Fraction and a Whole Number
11
12
Find — × 6.
Remember
A common factor is
a factor that is shared
by two or more whole
numbers. For example,
3 and 9 share a
common factor of 3.
5
8
Reasonable? 2 — ≈ 3
Estimate 1 × 6 = 6
Multiply the numerator and whole number.
11 × 6
12
11
12
—×6=—
Write the product over the denominator.
1
11 × 6
12
=—
Divide out the common factor, 6, from 6 and 12.
2
11
2
1
2
= —, or 5 —
Simplify.
1
2
So, the product is 5 —.
1
2
Reasonable? 5 — ≈ 6
✓
Multiply. Write the answer in simplest form.
Exercises 4 –19
52
Chapter 2
1
5
1. 4 × —
3
8
2. — × 7
Multiplying and Dividing Fractions
2
3
3. 9 × —
7
10
4. — × 5
English
Spanish
EXAMPLE
3
Standardized Test Practice
⋅
1
3
2
9
What is the value of — + x y when x = 3 and y = — ?
A
○
2
3
—
B
○
1
3
—+x
Remember
Be sure to use the order
of operations when
evaluating numerical
expressions.
20
27
—
⋅ y = —13 + 3 ⋅ —29
1
3
2
9
Substitute 3 for x and — for y.
⋅
Multiply 3 and —.
⋅
Divide out the common factor 3.
3 2
9
2
9
=—+—
1
2
3
D 1—
○
C 1
○
1 3 2
=—+—
3
9
3
1
3
2
3
=—+—
Simplify.
=1
C .
EXAMPLE
4
Real-Life Application
9
10
About — of the weight of a watermelon is water. How many pounds
of water are in the watermelon?
9
10
Estimate 1 × 20 = 20
To find — of 20, multiply.
9
10
9 × 20
10
— × 20 = —
9 × 20
10
2
=—
Divide out the common factor 10.
1
= 18
So, 18 pounds of water are in the watermelon.
Reasonable? 18 ≈ 20
Exercises 32–39
✓
⋅
1
2
5
8
5. Find the value of a b − — when a = — and b = 4.
6. WHAT IF? In Example 4, a watermelon weighs 15 pounds. How
many pounds of water are in the watermelon?
Section 2.2
Multiplying Fractions and Whole Numbers
53
English
Spanish
Exercises
2.2
Help with Homework
1. WRITING Describe how to multiply a fraction by a whole number.
7
8
2. NUMBER SENSE Use repeated addition to find the product 6 × —.
1
3
3. NUMBER SENSE Without multiplying, which is greater, — × 24
1
4
or — × 24? Explain.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Multiply. Write the answer in simplest form.
1
2
4. 3 × —
1
7
5. — × 5
6. — × 7
3
4
7. 4 × —
8. — × 8
6
5
9. 9 × —
5
3
10. — × 8
5
6
11. 25 × —
2
3
13. 7 × —
3
10
14. 15 × —
3
20
17. — × 24
3
8
18. 9 × —
12. — × 5
16. — × 5
1
8
4
9
7
10
13
14
2
3
15. — × 28
13
24
3
4
19. 18 × —
ERROR ANALYSIS Describe and correct the error in finding the product.
✗
20.
21.
2
2+8
—×8=—
5
5
✗
3
7
3
63
9×—=—
10
5
1
21
=—=2
=—
3
22. CDS Your friend has 12 CDs and — of them are pop music. How many CDs
4
are pop music?
2
3
23. OATMEAL MUFFINS A batch of oatmeal muffins calls for — cup of oats. How
many cups of oats do you need to make four batches of muffins?
2
24. RAIN There are 365 days in a year and rain falls on — of the days. How many
5
days does it rain during the year?
Use compatible numbers to estimate the product.
1
3
25. — × 17
54
Chapter 2
1
8
26. 35 × —
Multiplying and Dividing Fractions
5
6
27. 22 × —
4
5
28. — × 7
English
Spanish
Tell how the Commutative and Associative Properties of Multiplication can help
find the product mentally. Then find the product.
2
5
29. 25 × 6 × —
30.
( )
3
7
5
9
8 × — × 18
31. — × 13 × 14
3
16
ALGEBRA Evaluate the expression when x = 6, y = —, and z = 20.
3
4
⋅
3 32. — x
2
5
36. —xz
⋅
7
10
⋅
33. 8 y
34. — z
37. xyz
38. — + xy
⋅ 49
35. x —
3
8
5
12
39. — + yz
40
40. AREA A rectangular picnic shelter is 75 feet long by
1
15
1
15
60 feet wide. A model of the shelter is — as wide and —
as long as the actual shelter. What is the area, in square
feet, of the model of the shelter?
1
3
1
2
41. REASONING You spend — of your money and your friend spends — of her
money shopping. Is it possible that you spend more money than your friend?
shown in the table. Each necklace has a clasp that is
Length
— centimeter long.
9
10
Bugle
— cm
a. How long is a necklace with 48 bugle beads and
Crow
— cm
b. How long is a necklace with 18 bugle beads,
Tube
— cm
c.
9
20
9
10
2
5
You want to make a necklace 38 centimeters or longer.
have enough beads? If not, what is the smallest number of bugle beads
Evaluate the expression. SKILLS REVIEW HANDBOOK
3×5
2×4
43. —
9×2
1×5
44. —
4×8
7 × 25
45. —
10 × 13
3 × 11
46. —
47. MULTIPLE CHOICE What is the area of a parallelogram with a base of
12 inches and a height of 4 inches?
SECTION 1.5
A 24 in.
○
B 48 in.
○
Section 2.2
C 24 in.2
○
D 48 in.2
○
Multiplying Fractions and Whole Numbers
55
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