NAME ______________________________________________ DATE ______________ PERIOD _____
3-1
Study Guide and Intervention
The Distributive Property
The expressions 2(1 ⫹ 5) and 2 ⭈ 1 ⫹ 2 ⭈ 5 are equivalent expressions because they have the same
value, 12. The Distributive Property combines addition and multiplication.
Symbols
Model b
a(b ⫹ c) ⫽ ab ⫹ ac
a
b
c
c
a
a
The Distributive Property also combines subtraction and multiplication. Rewrite subtraction expressions
as addition expressions.
Example 1
a.
Use the Distributive Property to write each expression as an
equivalent expression. Then evaluate the expression.
2(6 ⫹ 3)
2(6 ⫹ 3) ⫽ 2 ⭈ 6 ⫹ 2 ⭈ 3
⫽ 12 ⫹ 6
⫽ 18
b. 5(9 ⫺ 3)
5(9 – 3) ⫽ 5 [9 ⫹ (–3)]
⫽ 5 ⭈ 9 ⫹ 5 ⭈ (⫺3)
⫽ 45 ⫹ (⫺15)
⫽ 30
The Distributive Property can also be used with algebraic expressions containing variables.
Example 2
a.
Use the Distributive Property to write each expression as an
equivalent algebraic expression.
7(m ⫹ 5)
7(m ⫹ 5) ⫽ 7m ⫹ 7 ⭈ 5
⫽ 7m ⫹ 35
b. ⫺3(n ⫺ 8)
⫺3(n ⫺ 8) ⫽ ⫺3[n ⫹ (⫺8)]
⫽ ⫺3 ⭈ n ⫹ (⫺3)(⫺8)
⫽ ⫺3n ⫹ 24
Exercises
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate the expression, if possible.
1. 3(8 ⫹ 2)
3 ⭈ 8 ⫹ 3 ⭈ 2; 30
4. ⫺6(3 ⫹ 14)
⫺6 ⭈ 3 ⫹ (⫺6)(14);⫺102
7. 3(d ⫹ 4)
3d ⫹ 3 ⭈ 4; 3d ⫹ 12
2. 2(9 ⫹ 11)
2 ⭈ 9 ⫹ 2 ⭈ 11; 40
5. (17 ⫺ 4)3
17 ⭈ 3 ⫹ (⫺4) ⭈ 3; 39
8. (w ⫺ 5)4
3. 5(19 ⫺ 6)
5 ⭈ 19 ⫹ 5 ⭈ (⫺6); 65
6. (5 ⫹ 3)7
5 ⭈ 7 ⫹ 3 ⭈ 7; 56
9. ⫺2(c ⫹ 7)
w ⭈ 4 ⫹ (⫺5) ⭈ 4; 4w ⫺ 20 ⫺2c ⫹ (⫺2)(7); ⫺2c ⫺
14
©
Glencoe/McGraw-Hill
107
Glencoe Pre-Algebra
Lesson 3-1
(b ⫹ c)a ⫽ ba ⫹ ca
NAME ______________________________________________ DATE ______________ PERIOD _____
3-1
Skills Practice
The Distributive Property
Use the Distributive Property to write each expression as an equivalent expression.
Then evaluate the expression.
1. 8(50 ⫹ 4)
2. (20 ⫹ 9)5
400 ⫹ 32; 432
100 ⫹ 45; 145
5. 4(400 ⫺ 2)
⫺21 ⫹ 33; 12
13. ⫺2(1 ⫺ 6)
⫺2 ⫹ 12; 10
280 ⫺ 14; 266
7. ⫺8(4 ⫹ 1)
⫺64 ⫺ 20; ⫺84
8. 9(24 ⫺ 19)
⫺32 ⫺ 8; ⫺40
10. ⫺10(12 ⫺ 4)
11. (21 ⫹ 9)(⫺5)
⫺120 ⫹ 40; ⫺80
14. 4(15 ⫹ 25)
4. 7(40 ⫺ 2)
120 ⫹ 8; 128
6. ⫺4(16 ⫹ 5)
1600 ⫺ 8; 1592
9. ⫺3(7 ⫺ 11)
3. 2(60 ⫹ 4)
216 ⫺ 171; 45
12. ⫺7(1 ⫺ 10)
⫺105 ⫺ 45; ⫺150
15. 15(100 ⫹ 6)
60 ⫹ 100; 160
1500 ⫹ 90; 1590
⫺7 ⫹ 70; 63
16. 12(22 ⫺ 52)
264 ⫺ 624; ⫺360
Use the Distributive Property to write each expression as an equivalent algebraic
expression.
17. 4(d ⫹ 2)
4d ⫹ 8
21. 3(x ⫺ 7)
3x ⫺ 21
25. ⫺1(2 ⫺ y)
⫺2 ⫹ y
18. 1(u ⫺ 3)
19. ⫺6(f ⫹ 5)
u⫺3
22. 8(⫺b ⫹ 4)
⫺6f ⫺ 30
23. (9 ⫺ h)5
⫺8b ⫹ 32
26. ⫺7(a ⫹ 1)
45 ⫺ 5h
27. 11(k ⫺ 20)
⫺7a ⫺ 7
11k ⫺ 220
20. ⫺2(g ⫺ 3)
⫺2g ⫹ 6
24. (c ⫹ 1)(⫺4)
⫺4c ⫺ 4
28. ⫺9(r ⫺ 1)
⫺9r ⫹ 9
29. 5(1 ⫺ b)
30. 8(x ⫹ 12)
31. ⫺6(p ⫹ 15)
32. 4(h ⫺ 16)
5 ⫺ 5b
8x ⫹ 96
⫺6p ⫺ 90
4h ⫺ 64
33. ⫺3(w ⫺ 10)
⫺3w ⫹ 30
37. 12(n ⫹ 2)
12n ⫹ 24
41. 6(r ⫺ 20)
6r ⫺ 120
45. 5(q ⫺ 16)
5q ⫺ 80
©
Glencoe/McGraw-Hill
34. ⫺10(c ⫹ 9)
35. 2(11 ⫺ q)
36. ⫺4(12 ⫺ f )
22 ⫺ 2q
⫺48 ⫹ 4f
⫺10c ⫺ 90
38. 16( g ⫹ 1)
39. ⫺8(9 ⫹ b)
16g ⫹ 16
42. 7(2 ⫺ j)
⫺72 ⫺ 8b
43. ⫺1(m ⫹ 1)
14 ⫺ 7j
46. ⫺10(c ⫺ 7)
⫺m ⫺ 1
47. ⫺3(⫺x ⫺ 1)
⫺10c ⫹ 70
3x ⫹ 3
108
40. ⫺5(z ⫺ 4)
⫺5z ⫹ 20
44. ⫺2(v ⫺ 8)
⫺2v ⫹ 16
48. (9 ⫺ h)(⫺2)
⫺18 ⫹ 2h
Glencoe Pre-Algebra
NAME ______________________________________________ DATE ______________ PERIOD _____
3-1
Practice
The Distributive Property
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate the expression.
480 ⫹ 6; 486
5. 5(400 ⫺ 90)
2. 7(70 ⫺ 4)
3. (300 ⫹ 6)4
490 ⫺ 28; 462
1200 ⫹ 24; 1224
6. ⫺8(700 ⫺ 3)
7. 4(20 ⫺ 9)
2000 ⫺ 450; 1550 ⫺5600 ⫹ 24; ⫺5576
9. ⫺1(75 ⫺ 9)
⫺75 ⫹ 9; ⫺66
10. 14(21 ⫺ 11)
80 ⫺ 36; 44
11. ⫺25(80 ⫹ 2)
4. (100 ⫹ 10)9
900 ⫹ 90; 990
8. (100 ⫺ 3)(⫺7)
⫺700 ⫹ 21; ⫺679
12. 31(450 ⫺ 18)
294 ⫺ 154; 140 ⫺2000 ⫺ 50; ⫺2050 13,950 ⫺ 558; 13,392
Use the Distributive Property to write each expression as an equivalent algebraic
expression.
13. 7( y ⫹ 11)
14. ⫺6(t ⫺ 1)
7y ⫹ 77
⫺6t ⫹ 6
17. ⫺1(⫺h ⫹ 5)
h⫺5
21. ⫺2(d ⫺ 5)
⫺2d ⫹ 10
25. 17( p ⫹ 4)
17p ⫹ 68
29. ⫺20(19 ⫺ a)
⫺380 ⫹ 20a
33. ⫺19(v ⫺ 1)
⫺19v ⫹ 19
15. ⫺8(u ⫺ 2)
⫺8u ⫹ 16
18. ⫺2( f ⫹ 3)
19. ⫺4(b ⫺ 1)
⫺2f ⫺ 6
⫺4b ⫹ 4
22. 22(n ⫹ 10)
23. ⫺50(z ⫺ 1)
22n ⫹ 220
26. (k ⫺ 21)(⫺8)
⫺50z ⫹ 50
27. (⫺32 ⫺ s)(⫺9)
⫺8k ⫹ 168
30. 33(d ⫹ 4)
288 ⫹ 9s
31. ⫺18(⫺q ⫺ 5)
33d ⫹ 132
34. ⫺1(r ⫹ 27)
18q ⫹ 90
35. 53(x ⫹ 11)
⫺r ⫺ 27
53x ⫹ 583
16. (r ⫹ 9)(⫺4)
⫺4r ⫺ 36
20. 1(7 ⫺ v)
7⫺v
24. ⫺12( g ⫹ 12)
⫺12g ⫺ 144
28. ⫺28(a ⫺ 5)
⫺28a ⫹ 140
32. ⫺16(c ⫹ 45)
⫺16c ⫺ 720
36. ⫺17(⫺n ⫹ 1)
17n ⫺ 17
37. PLANTS A planter weighs 2 pounds and holds 3 pounds of soil. Write two equivalent
expressions for the total weight of nine planters. Then find the weight.
9(2 ⫹ 3), 9(2) ⫹ 9(3); 45 pounds
38. UNIFORMS A uniform costs $42 for the sweater and $29 for the slacks. Write two
equivalent expressions for the total cost of six uniforms. Then find the cost.
6(42 ⫹ 29), 6(42) ⫹ 6(29); $426
©
Glencoe/McGraw-Hill
109
Glencoe Pre-Algebra
Lesson 3-1
1. 6(80 ⫹ 1)
©
The Distributive Property
Study Guide and Intervention
Glencoe/McGraw-Hill
a
Model b
c
a
b
A2
3(n 8)
2 3(n 2 8) 5 2 3[n 1 (2 8)]
5 2 3 ?n 1 (2 3)(2 8)
5 2 3n 1 24
3 4; 3d
Glencoe/McGraw-Hill
3d
7. 3(d 1 4)
12
w 4
107
20
( 4) 3; 39
( 5) 4; 4w
8. (w 2 5)4
17 3
6 3
( 6)(14); 102
5. (17 2 4)3
4. 2 6(3 1 14)
2 11; 40
2 9
3 8
3 2; 30
2. 2(9 1 11)
1. 3(8 1 2)
c
2c
2c
Glencoe Pre-Algebra
( 2)(7);
14
5 ( 6); 65
3 7; 56
9. 2 2(c 1 7)
5 7
6. (5 1 3)7
5 19
3. 5(19 2 6)
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate the expression, if possible.
b.
Use the Distributive Property to write each expression as an
equivalent algebraic expression.
7(m 5)
7(m 1 5) 5 7m 1 7 ?5
5 7m 1 35
Exercises
a.
Example 2
The Distributive Property can also be used with algebraic expressions containing variables.
b. 5(9 3)
5(9 – 3) 5 5 [9 1 (–3)]
5 5 ?9 1 5 ?(2 3)
5 45 1 (2 15)
5 30
Use the Distributive Property to write each expression as an
equivalent expression. Then evaluate the expression.
2(6 3)
2(6 1 3) 5 2 ?6 1 2 ?3
5 12 1 6
5 18
Example 1
a.
©
a
The Distributive Property also combines subtraction and multiplication. Rewrite subtraction expressions
as addition expressions.
(b 1 c)a 5 ba 1 ca
a(b 1 c) 5 ab 1 ac
Symbols
The expressions 2(1 1 5) and 2 ?1 1 2 ?5 are equivalent expressions because they have the same
value, 12. The Distributive Property combines addition and multiplication.
3-1
NAME ______________________________________________ DATE ______________ PERIOD _____
The Distributive Property
Skills Practice
32; 432
21
33; 12
12; 10
20;
120
40;
60
100; 160
14. 4(15 1 25)
80
32
8;
45;
1500
14; 266
216
7
70; 63
264
624;
16. 12(22 2 52)
360
171; 45
8. 9(24 2 19)
280
4. 7(40 2 2)
12. 2 7(1 2 10)
150
90; 1590
15. 15(100 1 6)
105
11. (21 1 9)(2 5)
40
8; 128
7. 2 8(4 1 1)
120
3. 2(60 1 4)
7a
5b
30
24
120
©
80
Glencoe/McGraw-Hill
5q
45. 5(q 2 16)
6r
41. 6(r 2 20)
12n
37. 12(n 1 2)
3w
33. 2 3(w 2 10)
5
29. 5(1 2 b)
96
7j
16
90
10c
70
46. 2 10(c 2 7)
14
42. 7(2 2 j)
16g
38. 16( g 1 1)
10c
34. 2 10(c 1 9)
8x
30. 8(x 1 12)
7
26. 2 7(a 1 1)
y
2
32
25. 2 1(2 2 y)
21
8b
3
22. 8(2 b 1 4)
u
18. 1(u 2 3)
3x
8
21. 3(x 2 7)
4d
17. 4(d 1 2)
30
5h
220
90
2q
8b
1
108
3x
3
47. 2 3(2 x 2 1)
m
43. 2 1(m 1 1)
72
39. 2 8(9 1 b)
22
35. 2(11 2 q)
6p
31. 2 6(p 1 15)
11k
27. 11(k 2 20)
45
23. (9 2 h)5
6f
19. 2 6(f 1 5)
6
4
9
64
4f
20
16
2h
Glencoe Pre-Algebra
18
48. (9 2 h)(2 2)
2v
44. 2 2(v 2 8)
5z
40. 2 5(z 2 4)
48
36. 2 4(12 2 f )
4h
32. 4(h 2 16)
9r
28. 2 9(r 2 1)
4c
24. (c 1 1)(2 4)
2g
20. 2 2(g 2 3)
Use the Distributive Property to write each expression as an equivalent algebraic
expression.
2
13. 2 2(1 2 6)
64
84
45; 145
6. 2 4(16 1 5)
100
2. (20 1 9)5
10. 2 10(12 2 4)
8; 1592
9. 2 3(7 2 11)
1600
5. 4(400 2 2)
400
1. 8(50 1 4)
Use the Distributive Property to write each expression as an equivalent expression.
Then evaluate the expression.
3-1
NAME ______________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 3-1)
Glencoe Pre-Algebra
Lesson 3-1
©
The Distributive Property
Practice
Glencoe/McGraw-Hill
6; 486
9;
66
24;
154; 140
36; 44
24; 1224
2000
50;
90; 990
21;
679
558; 13,392
12. 31(450 2 18)
700
8. (100 2 3)(2 7)
900
4. (100 1 10)9
2050 13,950
11. 2 25(80 1 2)
80
7. 4(20 2 9)
1200
3. (300 1 6)4
77
A3
5
27
r
19v
4
50
9s
90
53x
583
35. 53(x 1 11)
18q
31. 2 18(2 q 2 5)
288
27. (2 32 2 s)(2 9)
50z
23. 2 50(z 2 1)
v
36
144
140
720
17n
17
36. 2 17(2 n 1 1)
16c
32. 2 16(c 1 45)
28a
28. 2 28(a 2 5)
12g
24. 2 12( g 1 12)
7
20. 1(7 2 v)
4r
16. (r 1 9)(2 4)
3), 9(2)
9(3); 45 pounds
©
29), 6(42)
Glencoe/McGraw-Hill
6(42
6(29); $426
109
Glencoe Pre-Algebra
Answers
Glencoe Pre-Algebra
38. UNIFORMS A uniform costs $42 for the sweater and $29 for the slacks. Write two
equivalent expressions for the total cost of six uniforms. Then find the cost.
9(2
37. PLANTS A planter weighs 2 pounds and holds 3 pounds of soil. Write two equivalent
expressions for the total weight of nine planters. Then find the weight.
19
34. 2 1(r 1 27)
33. 2 19(v 2 1)
132
33d
380
20a
30. 33(d 1 4)
168
29. 2 20(19 2 a)
68
8k
220
17p
22n
26. (k 2 21)(2 8)
10
22. 22(n 1 10)
6
4b
16
2f
8u
19. 2 4(b 2 1)
6
15. 2 8(u 2 2)
18. 2 2( f 1 3)
6t
14. 2 6(t 2 1)
25. 17( p 1 4)
2d
21. 2 2(d 2 5)
h
17. 2 1(2 h 1 5)
7y
13. 7( y 1 11)
Use the Distributive Property to write each expression as an equivalent algebraic
expression.
75
5600
10. 14(21 2 11)
5576
28; 462
6. 2 8(700 2 3)
490
2. 7(70 2 4)
294
450; 1550
9. 2 1(75 2 9)
2000
5. 5(400 2 90)
480
1. 6(80 1 1)
Use the Distributive Property to write each expression as an equivalent
expression. Then evaluate the expression.
3-1
NAME ______________________________________________ DATE ______________ PERIOD _____
The Distributive Property
R eading to Learn Mathematics
2 4
2
10
5
2
8 or 18;
4
2(5
4)
2(9) or 18
4 1
4
16
4
4 or 20;
4
4(4
1
1)
4(5) or 20
3 2
12
3
2
3
2
Vocabulary
equivalent
expressions
Definition
See students’ work.
Example
©
Glencoe/McGraw-Hill
110
“distributed” to each number inside.
Glencoe Pre-Algebra
b. Explain how the English definition can help you remember how the word distributive
relates to mathematics. The number outside the parentheses is
to give out or deliver especially to members of a group
a. Find the definition of distribute in a dictionary. Write the definition.
3. Distribute is a word that is used frequently in the English language.
Helping You Remember
2. In rewriting 3(x 1 2), which term is “distributed” to the other terms in the expression? 3
1.
Write a definition and give an example of the new vocabulary term.
Reading the Lesson
The areas are equal.
d. What did you notice about the total area in each case?
3 2
c. Draw any two rectangles that have the same width. Find the total
area in two ways. Sample answer: 3(2
2) 12;
4 4
b. Draw a 4-by-4 and a 4-by-1 rectangle. Find the total area in two ways.
2 5
a. Draw a 2-by-5 and a 2-by-4 rectangle. Find the total area in two ways.
Do the activity at the top of page 98 in your textbook. Write your
answers below.
Pre-Activity How are rectangles related to the Distributive Property?
3-1
NAME ______________________________________________ DATE ______________ PERIOD _____
Answers (Lesson 3-1)
Lesson 3-1