Unit 3: Linear Equations
Name _____________________________________
Multiplying and Dividing Fractions 2
Directions: Solve.
73
Appendix B: Answer Keys
Transparency/Guided Practice Book Answers
54
Unit 2: Linear Equations
Name _____________________________________
Multiplying and Dividing Fractions
Directions: Calculate.
31
Appendix B: Answer Keys
Guided Practice Book Answers
68
(cont.)
4
How to
• • • • • • • • • • • • • • • • • • • Divide Fractions
Facts to Know
The key to dividing with fractions is understanding reciprocals. A reciprocal is two numbers that have
the product of 1. Here are some examples:
Samples
2– and 3– are reciprocals because
3
2
4– and 1– are reciprocals because
1
4
2– x 3– = 6– or 1
3
2
6
4– x 1– = 4– or 1
1
4
4
Knowing what a reciprocal is will be your key to understanding how to divide with fractions.
Dividing Fractions by Fractions
Dividing a number by 2 gives the same answer as multiplying by 1–. For example,
2
6 ÷ 2 = 3 just as 6 x 1– = 3.
2
The answers are the same. The reciprocal of 2– is 1–. Turning a fraction upside down is called inverting.
1 2
To divide by a fraction, invert the fraction you are dividing by and then multiply.
Sample: Let’s say you’re building a kite, and you have a piece of string
3– yard long. Into how many 3– yard pieces can you snip the string?
4
8
The problem looks like this:
3– ÷ 3– = ?
4 8
3– inverted is 8–.
8
3
Step 1
Invert the fraction you are dividing by.
Step 2
Multiply across. 3– ÷ 3– = 3– x 8– = 24
—
4
8
4
3
12
Step 3
Reduce or change an improper fraction to a mixed or
whole number, if possible. 24
— = 2 pieces of string
12
The fraction to invert is always the fraction to the right of the division sign.
17
.
4
How to • • • • • • • • • • • • • • • • • • • Divide Fractions
Facts to Know (cont.)
Canceling in Dividing Fractions
Sometimes you can cancel in division. Invert before you cancel and multiply after you cancel.
Sample: 1
– ÷ 1– = ?
3 6
Invert and change to multiplication. 1– ÷ 1– = 1– x 6–
3
6
3
1
Step 2
3 is a factor of 3 and 6. Cancel and
divide the numerator and denominator by 3. 1– x 6–
3
1
2
1
6
Step 3
Multiply. – x – = 2
– = 2
3
1
1
1
Dividing a Fraction by a Whole Number
When dividing a fraction by a whole number, first change the whole number to a fraction with a
denominator of 1.
Sample: Suppose a recipe calls for dividing 1– pound of butter into four equal parts.
2
1
Hint: You must divide – by 4.
2
Step 1
Change the whole number, 4, to a fraction. 4 = 4–
1
4
1
Step 2
Invert the divisor. – inverted is –
1
4
1
1
1
Step 3
Multiply. – x – = – pound of butter.
2 4 8
Step 1
Dividing a Whole Number by a Fraction
Sometimes the whole number is not the number you are dividing by—it is the number being divided.
Sample: Divide 4 pounds of candy into bags holding 1– pound each.
2
1
Hint: You must divide 4 by –.
2
Step 1
Change the whole number to a fraction. 4 = 4–
1
Step 2
Invert the divisor. (Invert the number to the right of the division sign.) 1– inverted is
2
2–
1
Step 3
Multiply. 4 ÷ 1– = 4– x 2– = 8– or 8 bags
2
1
1
1
Dividing and Mixed Numbers
Before dividing, change any mixed numbers to improper fractions.
Sample: 1 1– ÷ 2– = ?
4
3
Step 1
Change the mixed number to an improper fraction. 1 1– = 5–
4 4
5
3
15
7
Step 2
Invert and multiply. – x – = — = 1 –
4 2 8
8
18
4
Practice
• • • • • • • • • • • • • • • • • Dividing Fractions
Directions: Divide the fractions below.
1.
3 2– ÷ 4 2– =
3
3
11.
13
3 =
—÷—
20 10
21.
5– ÷ 3 4– =
7
5
2.
4 3– ÷ 2 4– =
7
7
12.
2– ÷ 2– =
7 3
22.
1– ÷ 3 1– =
4
5
3.
2 1– ÷ 3 1– =
3
6
13.
1– ÷ 2– =
6 5
23.
2 3– ÷ 4– =
8 5
4.
3 =
5 6– ÷ 6 —
7
14
14.
4– ÷ 1– =
9 2
24.
3 3– ÷ 1– =
4 9
5.
3 3– ÷ 5 4– =
5
5
15.
8– ÷ 3– =
9 4
25.
2– ÷ 3 8– =
5
9
6.
5– ÷ 2– =
6 3
16.
3– ÷ 1 2– =
6
3
26.
3– ÷ 12 =
8
7.
1– ÷ 3– =
4 8
17.
2 1– ÷ 3– =
3 4
27.
1=
15 ÷ —
10
8.
5 ÷—
2 =
—
12 24
18.
3 3– ÷ 3– =
4 5
28.
2 3– ÷ 1– =
6 3
9.
3 ÷ 12
—
—=
10 30
19.
6 1– ÷ 2– =
2 3
29.
9 =
3 2– ÷ —
3 10
10.
1– ÷ —
4 =
5 15
20.
1 1– ÷ 6 =
2
30.
2 1– ÷ 1– =
2 4
19
.
4
Practice • • • • • • • • • • • • • • • • • Dividing Fractions
Directions: Divide the fractions, whole numbers, or mixed numbers. Reduce when possible.
1.
1– ÷ 3– =
2 5
10.
7– ÷ 1– =
9 3
19.
8 ÷ 4– =
5
28.
2 1– ÷ 3– =
2 4
2.
1– ÷ 3– =
3 4
11.
2– ÷ 3 =
3
20.
12 ÷ 2– =
3
29.
1 1– ÷ 1– =
2 5
3.
3– ÷ 4– =
4 5
12.
4– ÷ 3 =
5
21.
9 ÷ 6– =
7
30.
2– ÷ 1 1– =
7
5
4.
6– ÷ 1– =
7 3
13.
2 ÷ 1– =
2
22.
7 ÷ 3– =
4
31.
3 1– ÷ 3– =
2 4
5.
1– ÷ 1– =
5 2
14.
1– ÷ 2 =
6
23.
10 ÷ 5– =
6
32.
2 1– ÷ 1– =
4 4
6.
1– ÷ 1– =
2 8
15.
7– ÷ 2 =
8
24.
6 ÷ 4–
9
33.
5 1– ÷ 3 =
2
7.
3– ÷ 3– =
8 4
16.
4 ÷ 3– =
5
25.
4 ÷ 3– =
8
34.
3 1– ÷ 2 1– =
8
2
8.
1– ÷ 1– =
4 4
17.
2– ÷ 5 =
3
26.
1 3– ÷ 1– =
4 4
35.
3 3– ÷ 3 3– =
4
4
9.
4– ÷ 1– =
5 5
18.
4 ÷ 2– =
3
27.
8 ÷ 1 2– =
3
20
3
How to
• • • • • • • • • • • • • • • • • Multiply Fractions
Facts to Know
All fractions can be multiplied by multiplying the numerators and then multiplying the denominators.
You don’t need common denominators to multiply.
Sample: Andrew can jog a mile in 1– of an hour. Lauren can run the same distance in 1– that time.
4
2
How long would it take Lauren to run a mile?
To find out, you need to multiply 1– by 1–.
2
4
Step 1
Multiply the numerators.
1x1=1
Step 2
Multiply the denominators.
2x4=8
1
– x 1– = 1– x 1– = 1–
Step 3
Write the answer as a fraction.
2 4
2x4
8
Sometimes the answer to a multiplication problem can be reduced.
Sample: 3– x 1– = ?
4 6
Step 1
Multiply the fractions.
Step 2
3– x 1– = —
3
4 6 24
Write the fraction in simplest form by dividing the
numerator and denominator by the greatest
common factor, which in this case is 3.
3 ÷ 3– = 1–
—
24 ÷ 3
8
Multiplying Three Fractions
Now and then you will have to multiply three fractions together.
Sample: 2– x 1– x 1– = ?
3 5 3
Step 1
Multiply the numerators of the first two fractions. Then multiply that answer by the
numerator of the third fraction.
Step 2
Multiply the denominators of the first two
fractions. Then multiply that answer by the
denominator of the third fraction.
Step 3
Write fraction in simplest form, if possible.
2 is reduced as far as it can be.
The fraction —
45
13
2– x 1– = —
2 x 1– = —
2
3 x 5 15 x 3 45
.
3
How to
• • • • • • • • • • • • • • • • • Multiply Fractions
Facts to Know (cont.)
Canceling as a Shortcut
Sometimes the fractions in a multiplication problem can be canceled. Canceling is a shortcut to
reducing. To cancel find a number that divides evenly into both the numerator and denominator of the
problem.
7 =?
Sample: 3
–x—
4 24
Step 1
Step 2
Step 3
Step 4
Find a number that will divide evenly. The number 3 will divide evenly into 3 and
24.
Cancel 3 and 24 by 3. 3 ÷ 3 = 1
24 ÷ 3 = 8
1
Multiply across by the new numbers.
3– x —
7 =—
7
4
24
32
Reduce if possible, but by canceling you already did some of the
8
work of reducing.
Multiplying Fractions and Whole Numbers
When multiplying a fraction by a whole number, turn the whole number into a fraction with a
denominator of 1.
Sample: Andrew plans to sprint 3– of a mile each day for four days in a row as part of his regular
5
workout. How many miles will this total?
You will need to multiply 3– by 4.
5
Step 1
Write 4 as an improper fraction with a denominator of 1.
4 = 4–
1
Step 2
Multiply across.
Step 3
Change the improper fraction to a whole or mixed number.3– x 4– = 12
— = 2 2–
5
1
5
5
Multiplying Mixed Numbers
When multiplying mixed numbers, it is usually helpful to change the mixed number to an improper
fraction.
Sample: 3 2– x 4 1– = ?
5
3
Step 1
Change both mixed numbers to improper fractions.
3 2– = (3 x 5) + 2 = 17
—
5
5
4 1– = (4 x 3) + 1 = 13
—
3
3
17
13
221
Step 2
Multiply across the new numbers.
— x —= —
—
5
3 15
Step 3
Change the improper fraction to a whole or
mixed number.
14
221
11
—
— = 14 —
15
15
3
Practice
• • • • • • • • • • • • • • • Multiplying Fractions
Directions: Multiply the fractions. Remember to write the answer in simplest form when possible.
1.
1– x 3– =
2 4
12.
2– x 3– =
9 4
23.
20
9 x 4– =
—x—
21 16 5
2.
2– x 1– =
3 7
13.
3– x 2– =
8 5
24.
3– x 5– x —
2 =
4 7 11
3.
3– x 3– =
8 5
14.
5– x 7– =
8 9
25.
11 x 3– =
—
12 4
4.
1– x 6– =
5 7
15.
1– x 1– x 1– =
2 2 2
26.
7– x —
2 =
8 14
5.
1– x 1– =
2 3
16.
1– x 1– x 4– =
2 4 5
27.
5 =
4 x—
—
15 13
6.
2– x 1– =
3 4
17.
1– x 2– x 3– =
2 3 5
28.
3– x 10
— =
5 21
7.
1– x 6– =
3 7
18.
5– x 3– x —
14 =
9 7 15
29.
121
—
—— x 10
—=
300 11
8.
4– x 1– =
9 2
19.
6– x 7– x 4– =
7 8 5
30.
125
320
—
—
— x—
—
—
—=
470 1000
9.
1– x 1– =
2 2
20.
11 x 10
—
— x 3– =
15 11 4
31.
289
2 =
—
—
—
— x—
1222 17
10.
1– x 3– =
2 2
21.
32.
14
15
—
—
—x—
—
—=
525 320
11.
1– x 3– =
3 4
22.
9 x 1– x 8– =
—
10 4 9
5– x —
14 x —
2 =
6 15 21
15
.
3
Practice • • • • • • • • • • • • • • • Multiplying Fractions
Directions: Multiply the fractions and whole numbers. Use canceling whenever you can.
1.
1– x 5 =
4
5.
2– x 6 =
7
9.
2– x 4 =
9
13.
8 x 2– =
3
2.
4 x 2– =
3
6.
1 x3=
—
10
10.
3– x 3 =
8
14.
7 x 2– =
3
3.
1– x 7 =
6
7.
7– x 7 =
8
11.
5 x 2– =
3
15.
4 x 3– =
5
4.
6 x 3– =
5
8.
3 x 4– =
5
12.
2– x 5 =
3
16.
7– x 1– =
8 2
Directions: Multiply the mixed numbers. Use canceling whenever you can.
17.
3 3– x 2– =
4 3
21.
2 4– x 3 1– =
7
9
25.
5 x 2 6– x 2 2– =
1—
16
7
5
18.
3 =
3 3– x —
7 10
22.
9 x 1 1– =
1—
11
6
26.
1 7– x 3– x 2 2– =
8 4
9
19.
5– x 2 2– =
8
9
23.
1 5– x 1 5– =
9
7
27.
2 5– x 1– x 4 1– =
6 7
4
20.
9 =
8 x 1—
—
16
15
24.
2 =
2 5– x 2 —
8
15
28.
2 x 1– =
8 2– x —
8 13 7
16
1– .50
50%
2
Page 8
1. 1 3/4
2. 1 4/5
3. 1 1/3
4. 1 3/5
5. 2 1/5
6. 1 3/4
7. 2 1/7
8. 2 1/5
9. 2 1/8
10. 5
11. 7/4
12. 8/5
13. 9/4
14. 23/8
15. 17/5
16. 13/3
17. 17/3
18. 23/2
19. 41/8
20. 53/12
21. 1/2
22. 2/3
23. 1/4
24. 2/3
25. 1/3
26. 6/13
27. 1/2
28. 1/3
29. 2/3
30. 1/2
31. 3/15
32. 9/12
33. 4/16
34. 6/40
35. 25/35
36. 6/36
37. 12/18
38. 10/45
39. 3/4
40. 5/7
41. 1
42 3 1/3
43. 1 1/7
44. 4
45. 2 1/2
46. 7 2/3
47. 1 3/8
48. 25/28
49. 31/36
50. 14 1/12
51. 2
52. 1 4/5
53. 11 11/24
54. 14 16/35
55. 1 7/10
Page 11
1. 2/5
2. 1/2
3. 1/12
4. 4/7
5. 6/23
6. 1/7
7. 1/2
8. 1/15
9. 1/3
10. 5/8
11. 1/2
12. 1/8
13. 5/9
14. 7/10
15. 1/4
16. 19/30
17. 4 1/4
18. 6 15/16
19. 3 5/8
20. 9 2/7
21. 11 13/25
22. 7 1/4
23. 4 3/4
24. 24 3/17
25. 12 1/2
26. 4 2/7
Page 12
1. 3 1/4
2. 8 1/5
3. 6 7/12
4. 6 2/3
5. 3 1/2
6. 3 3/7
7. 3/10
8. 6 5/8
9. 3 7/11
10. 1 12/13
11. 2 11/12
12. 1 17/20
13. 2 13/18
14. 8 5/12
15. 10 3/4
16. 7 13/15
17. 8 5/6
18. 7 27/40
Page 15
1. 3/8
2. 2/21
3. 9/40
• • • • • • • • • • • • • • • • • • • • • • Answer Key
4. 6/35
5. 1/6
6. 1/6
7. 2/7
8. 2/9
9. 1/4
10. 3/4
11. 1/4
12. 1/6
13. 3/20
14. 35/72
15. 1/8
16. 1/10
17. 1/5
18. 2/9
19. 3/5
20. 1/2
21. 1/5
22. 2/27
23. 3/7
24. 15/154
25. 11/16
26. 1/8
27. 4/39
28. 2/7
29. 11/30
30. 4/47
31. 17/611
32. 7/5,600
Page 16
1. 1 1/4
2. 2 2/3
3. 1 1/6
4. 3 3/5
5. 1 5/7
6. 3/10
7. 6 1/8
8. 2 2/5
9. 8/9
10. 1 1/8
11. 3 1/3
12. 3 1/3
13. 5 1/3
14. 4 2/3
15. 2 2/5
16. 7/16
17. 2 1/2
18. 1 1/35
19. 1 7/18
20. 5/6
21. 8
22. 2 4/33
23. 2 2/3
47
24. 5 3/5
25. 9
26. 3 1/8
27. 9 5/7
28. 4/21
Page 19
1. 11/14
2. 1 13/18
3. 14/19
4. 82/87
5. 18/29
6. 1 1/4
7. 2/3
8. 5
9. 3/4
10. 3/4
11. 2 1/6
12. 3/7
13. 5/12
14. 8/9
15. 1 5/27
16. 3/10
17. 3 1/9
18. 6 1/4
19. 9 3/4
20. 1/4
21. 25/133
22. 5/64
23. 2 31/32
24. 33 3/4
25. 18/175
26. 1/32
27. 150
28. 7 1/2
29. 4 2/27
30. 10
Page 20
1. 5/6
2. 4/9
3. 15/16
4. 2 4/7
5. 2/5
6. 4
7. 1/2
8. 1
9. 4
10. 2 1/3
11. 2/9
12. 4/15
13. 4
14. 1/12
15. 7/16
16. 6 2/3
17. 2/15
18. 6
19. 10
20. 18
21. 10 1/2
22. 9 1/3
23. 12
24. 13 1/2
25. 10 2/3
26. 7
27. 4 4/5
28. 3 1/3
29. 7 1/2
30. 7/20
31. 4 2/3
32. 9
33. 1 5/6
34. 1 1/4
35. 1
Page 24
1. nine tenths
2. three hundred six
thousandths
3. forty-two
thousandths
4. six and three
hundredths
5. eighty and seven
tenths
6. two hundred
thirty-four and six
hundred twelve
thousandths
7. sixty-eight and
thirty-five ten
thousandths
8. one thousand two
hundred thirtyfour ten
thousandths
9. one and two
hundred thirtyfour thousandths
10. twelve and thirtyfour hundredths
11. .43
12. 40.03
13. .017
14. 86.6
15. .0508
16. 5.04
17. 12.140; 12.404;
12.444; 12,400
18. 0.96; 0.9666;
10.96; 109.6
19. 0.055; 0.5; 0.505;
0.55
.