8
MATH
MATH
WORKBOOK
EMP4055i
6.25mm
hb
=
A
X
18 = 1
3
<
LCM
= 60
=
y
4
1
x
2
3
>
30x + 22y
831
-25
6
112
43
.625
I=PxRxT
3.1416
3
x
2
x
=2
2
1
22%
6
6
.
4
3
GC
F=
8
Math
WORKBOOK
8
Finally, a new workbook for eighth grade math!
Author
Beverly Nance
Basic skills are reviewed and expanded as students work through 58
pages of activities. Each page gives an example and step–by–step
solution of the problem presented. Some of the many skills covered
include a review of addition, subtraction, multiplication, and division,
plus challenges in decimal fractions, exponential and scientific
notation, primes, probability, percents, and basic geometric principles.
Six answer pages are provided.
Project Manager
E. Rohne Rudder
Typesetting
Pagecrafters, Inc.
Cover Art
E. Rohne Rudder
Managing Editor
Kathleen Hilmes
Copyright © 1993
Milliken Publishing Company
All rights reserved.
The purchase of this book entitles the individual purchaser to reproduce
copies by duplicating master or by any photocopy process for single classroom use. The reproduction of any part of this book for commercial resale or
for use by an entire school or school system is strictly prohibited. Storage of
any part of this book in any type of electronic retrieval system is prohibited
unless purchaser receives written authorization from the publisher.
Addition and subtraction of whole numbers
Find each sum.
19405
5281
324
+ 18200
43, 210
46 + 54 + 62 + 38 =
85 + 15 + 37 + 13 =
127 + 43 + 130 + 30 =
21300
– 8955
12,345
17 + 23 + 35 + 25 =
250 + 350 + 25 + 175 =
827
365
+ 942
724
599
+ 307
8245
3605
+ 2913
4537
2009
+ 5432
6331
5224
+ 8445
1234
5678
+ 9012
246
802
+ 468
13579
79135
+ 35793
20406
1030
+ 99999
1579
2631
+ 4682
Find each difference.
4829
– 3718
2056
–1264
8000
– 4654
7248
– 3585
5040
– 3999
6213
– 5876
4000
– 805
8219
– 7878
2467
– 908
3579
– 2468
12345
– 6789
9876
– 6789
50604
– 3080
7000
– 4778
48113
– 36251
© Milliken Publishing Company
1
MP4055
Multiplication and division of whole
numbers
17 R 24
358
32 568
32
× 49
248
3222
1432
224
17542
24
Find each product.
)
246
× 82
135
× 791
1350
× 405
2863
× 823
1792
× 544
8921
× 345
5628
× 500
5701
× 803
9287
× 123
8040
× 506
Find each quotient.
17 ) 561
358 )17542
© Milliken Publishing Company
29 ) 899
35 ) 875
256 ) 82176
2
48 ) 960
987 ) 42441
MP4055
Order of operations (multiplication
and division)
Simplify each expression.
8 + 12 ÷ 2 – 4 x 3 + 1 =
2+8÷4–2=
8 + 6 –12 + 1 =
14 – 12 + 1 =
2+1=
8–5÷5+5=
3
Find each sum.
6 x 3 + 2 ÷ 2=
9÷3+6–1=
17 + 2 x 5 ÷ 5 =
24 ÷ 8 + 3 – 2 =
18 – 5 x 3 + 6 =
17 + 5 x 2 – 2 =
8+8÷8–8=
25 – 6 ÷ 3 + 3 =
41 – 6 ÷ 6 + 5 =
36 ÷ 12 + 8 – 4 =
52 ÷ 4 x 2 + 1 =
12 x 3 – 4 + 5 =
2 + 3 (5 – 2) =
6 + 4 (8 + 2) =
5 (3 – 2) + 1 =
6 (8 ÷ 8) + 7 =
7 – 2 (5 – 5) =
18 + 3 (4 – 2) =
24 ÷ 2 (1 – 0) =
16 – 3 (8 ÷ 4) =
© Milliken Publishing Company
3
MP4055
More order of operations (inside out)
Simplify each expression.
2 + 3 [5 + 2 (8 – 5)] =
2 + 3 [5 + 2 (3)] =
5 + 4 [3 + 4(8 – 6)] =
2 + 3 [5 + 6] =
2 + 3 [11] =
2 + 33 =
35
16 – 2 [5 + 3 ÷ 3] =
18 + 7 [6 – 4 ÷ 4] =
37 – 3 [4 x 2 – 5] =
21 + 4[8 x 3 – 16] =
58 – 5 [6 + 8 ÷ 4] =
6 [5 + 3 x 2] – 12 =
2 + 3 [5 + 8 (6 – 2)] =
78 – 2 [24 – 2(8 – 5)] =
28 + 7[16 – 2 (4 + 3)] =
© Milliken Publishing Company
146 – 5 [8 (6 – 4) – 12] =
4
MP4055
Addition and subtraction of decimals
Find each sum.
2
3.5
.4
+ 6.02
11.92
2 + 4.8 + .2 + 3 =
3.4 + 2.6 + 5 + 2.4 =
8.40
– 2.35
6.05
6.5 + 8.2 + 2.3 + .5 =
9 + .4 + 8 + .3 + 2.3 =
2.85 + .05 + 2.1 + 4 =
0.456
1.234
+ 2.345
23.4
7.25
+ 18.07
2.35
4
+ 5.6
10.2
1.35
+ .4
7.2
5.68
+4
Find the difference.
20.4
– 8.6
16.23
– 4.5
8.4
– 2.39
12.25
– 6.47
2.803
– 1.04
42.56
– 13.24
6.984
– 4.279
8
– 4.35
8.35
–4
2.45
– 1.46
Simplify each expression.
21.4 – (3.2 + 8.4)
18.4 + (6.35 – 2.42)
17.56 + (18.42 – 6)
29.35 – (4.38 + 2.9)
(8.2 + 3.5) – (4.6 + 2.5) + 2.1
© Milliken Publishing Company
5
MP4055
Find each product.
Multiplication and division of decimals
3.45
x 1.2
690
345
4.140
3 x 1.5 =
5 x .25 =
12. 06
. 4 4. 824
)
2 x .37 =
10 x .04 =
8 x 1.11 =
2.8
x .7
.45
x .6
1.23
x .9
.456
x .2
24.3
x .3
2.9
x .17
.47
x 2.6
2.31
x .12
.123
x .04
12.04
x .05
Find the quotient.
. 4 )16.8
4 )16.8
.04 )16.8
.004 ).168
2.5 ) 6.25
.25 ) 6.25
25 ) 6.25
.025 ) 6.25
1.2 ) 9.6
.15 ) 7.5
.11).099
8 ).048
3.2 ) 960
.21) 8.4
14 )1.96
.35 ) 70
.123 ) 2. 46
2.04 ).612
311) 6.22
. 481).962
© Milliken Publishing Company
6
MP4055
Write <, >, or = to make a
true statement.
Comparing decimals
2.45
<
2.5
2.005
2.5
3
>
2.999
3.14
3.41
8
7.895
2.115
2.151
4.92
4.29
12.46
14.2
6.35
5.63
3.405
3.045
8.002
8.02
.88
.8
.119
.1109
3.801
3.108
.707
.7
.003
.02
16.16
17
18.05
15.88
2.3
.32
Rounding decimals
6.23 ≈ 6.2
6.28 ≈ 6.3
Round to nearest whole number.
7.4 ≈
8.6 ≈
17.11 ≈
21.51 ≈
1.45 ≈
.09 ≈
2.145 ≈
.96 ≈
2.43 ≈
.48 ≈
Round to nearest tenth.
.24 ≈
.17 ≈
Round to nearest hundreth.
2.456 ≈
.045 ≈
1.181 ≈
2.009 ≈
Round to nearest thousandth.
.0456 ≈
© Milliken Publishing Company
.1234 ≈
.2345 ≈
7
.0604 ≈
MP4055
Write in exponential notation.
Using exponents
3•3•3=
5
2•2•2•2•2=2
5 5 5 5=
•
•
•
2
3 =3•3=9
1•1•1•1•1=
2
4
•
3
5
4 = (4 • 4) (4 • 4 • 4) = 4
2•2•2=
(2.4) (2.4) (2.4) =
10 • 10 • 10 • 10 =
(3.2) ( 3.2) =
9•9•9=
7•7•7•7•7=
8•8•8•8=
6•6•6•6•6•6=
Evaluate each expression.
2
2
2 =
2
5 =
3
3
3 =
4
4
2
4
5
4
3
10 =
10
2 =
4
5 =
0
10 =
6
3 =
5 =
3=
1
10 =
3
8 =
1 =
4
10 =
9 =
3
2 =
2 =
2
1=
3
4 =
4 =
2
7 =
10 =
2
1 =
5
4 =
5 =
Write the missing exponent.
2
•
2 =2■
8
•
5■=5
4■ • 4 = 4
•
3 =3■
9
3
4
3
5
4
5
3
7
2
8 =8■
5
•
3
1
5
7 =7■
•
6 =6■
5
8
6
9■= 9
8
•
•
4
7
3
10
3
10 = 10 ■
4
•
(2.5) (2.5) (2.5) (2.5)(2.5) (2.5) = (2.5) ■
© Milliken Publishing Company
8
MP4055
Scientific notation
Evaluate.
2
2.4 x 10 =
3
5.23 x 10 = 5230
3
3.71 x 10 =
24,680 = 2.468 x 10
4
1
5.06 x 10 =
3
8.371 x 10 =
2
7.008 x 10 =
3
9.23 x 10 =
4
6.204 x 10 =
6
1.49 x 10 =
6
1
8.405 x 10 =
5
2.3 x 10 =
3
3.755 x 10 =
5
5 x 10 =
8.1 x 10 =
4
7.42 x 10 =
8
9 x 10 =
Write in scientific notation.
405 =
8400 =
2468 =
700,000 =
8,040,000 =
42,000 =
1005 =
27 =
550,000 =
4,000,000 =
80,808 =
1450 =
9000 =
6002 =
50,000 =
12345 =
18 =
246 =
908 =
325,000 =
© Milliken Publishing Company
9
MP4055
Use Prime Factor Trees to find GCF
Complete the Prime Factor Trees
to find each GCF.
8
12
2
20
2
18
6
3
2
2
3
9
2
2• 2•3
3
2•3 •3
GCF = 6
GCF =
81
54
3
3
175
5
GCF =
5
GCF =
120
48
2
225
3
90
72
2
2
GCF =
GCF =
220
340
170
2
2
2
130
2
GCF =
GCF =
© Milliken Publishing Company
10
MP4055
Find the LCM of 15 and 20
The following word problems
require you to find a least common
multiple (LCM). Use either method
(A) or method (B).
Method (A): 15, 30, 45, 60
20, 40, 60
LCM = 60
Method (B):
15
3
20
5
2
2
5
LCM = 2 x 2 x 3 x 5 = 60
1. What is the length of the side of the smallest possible square formed by
placing rectangles side by side that are 3 inches by 4 inches?
2. Jane and Mark are cycling to the shore but Jane rides at a rate of
15 mph, and Mark rides at a rate of 12 mph. What is the minimum
distance they will ride if they each travel the same distance in a whole
number of hours?
3. Rhonda earned $75 per week for 4 weeks. If Jane only earns $50 per
week how many weeks will she have to work before she will have the
same amount of money?
4. Paul can mow 12 lawns per week for 3 weeks. Robbie mows 6 lawns in 1
week. How many weeks will Robbie have to work before he’s mowed
the same number of lawns as Paul?
5. What is the length of the side of the smallest possible square formed by
placing rectangles side by side that are 5 cm by 6 cm?
6. Tara read 8 books/month for 3 months. How many books will John
have to read for each of 4 months if they are to read the same number
of books?
7. Harold earned $150/week for 8 weeks. Sarah earned $90/week for 10
weeks. What is the least common multiple of both their salaries?
8. Bob types 40 words per min. for 5 minutes. Julie types 50 words per min.
How long before she will have typed the same number of words?
© Milliken Publishing Company
11
MP4055
I. Reduce:
Reduce each fraction
to lowest terms.
80
=
90
15
=
20
6 ÷ 2
=
8 ÷ 2
II. Change mixed number to improper
fraction:
21
=
28
3
9
=
36
17
=
51
32
=
40
48
=
60
38
=
57
12
=
30
3
4
5
7
=
21+ 5
7
=
26
7
III. Find missing number to form equivalent
fraction:
3
3 × 8 24
=
=
:
4 32 4 × 8 32
Change each mixed number to an improper fraction.
1
1 =
2
2
1=
3
3
1=
4
4
1=
5
5
1=
6
3
1 =
5
2
4=
7
3
5=
9
4
3=
4
5
2=
3
2
10 5 =
20
5=
8
30
5=
7
40
1=
2
50
3
19 4 =
7
2=
3
6
1 =
2
8
1=
9
1
11 =
3
3=
5
Find the missing number to form equivalent fractions.
1
=
2
3
6
2 10
=
3
5
=
7
21
7 28
=
9
3
9
=
10
30
6 18
=
11
19
© Milliken Publishing Company
12
=
15
20
=
24
32
=
38
48
9
5
4
=
16
18
=
30
50
=
120
160
MP4055
Write > or < to make a true
statement.
18
3
4
<
2
1
20
5
5 . 3
. .
<
4
6
6
2
3
3
5
4
9
7
8
2
9
3
11
5
6
4
5
2
3
11
2
9
5
7
6
8
7
12
8
11
6
13
4
5
4
17
3
16
7
20
5
16
2
3
18
23
3
8
2
11
27
30
7
10
85
100
16
21
32
50
15
20
21
22
13
14
1
,
3
,
8
:
3
24
<
12
24
<
16
24
Write in order from least to greatest.
5
7
,
,
24 18
8
12
<
<
5
,
8
7
,
20
6
24
<
<
7
,
8
12
,
30
11
15
<
<
3
4
<
<
3
2
1 , 1 ,
5
7
1
© Milliken Publishing Company
13
MP4055
Add the following fractions and mixed
numbers and reduce answer
Write each sum in simplest form.
2
8
+
5
8
+
1
=
8
7
2
+ 5
3
11
11
+ 6 =
4
5
+6
3
21
21
3
+1
=
21
3
+
3
=
4
5
6
=
7
9
1
+
3
5
8
4
+
5
2
5
3
+2
4
7
18
5
+3
12
5
6
2
+ 11
9
3
5
4
+1
9
5
+6
2
3
3
5
1
2
6
3
2
3
5
+5
1
4
+4
9
12
10
12
19
12
=4
7
12
4
3
7
10
4
+5
7
5
4
© Milliken Publishing Company
+
3
2
3
5
+
6
1
3
2
9
14
12
3
8
17
4
5
+ 21
1
4
MP4055
Subtraction of fractional and mixed
numbers
Write each difference in simplest
form.
3
11
–
12
9
=
12
17
–
20
5
=
20
12
7
8
–
5
–1
= 3
3
3
= 1
4
3
=
8
3
7
11
12
2
3
–
7
14
15
12
19
24
–1
4
5
–2
1
4
– 5
2
3
16
5
9
27
7
8
– 8
7
8
– 14
84
28
–4
– 63
5
© Milliken Publishing Company
9
12
= 1
20
12
9
12
4
5
–
5
6
2
3
12
= 2
9
10
3
18
8
11
1
12
19
21
–
2
3
31
3
8
9
10
42
– 16 5
11
–19
15
MP4055
Multiplication of fractions
Write each product in simplest form.
2
5
10
×
×
3
5
9
2
9
×
×
3
10
6
4
10 2
1× 4 × 2
8
×
×
=
=
3
51 6 3
3 ×1× 3
9
=
=
2
1
3
× 3
1
=
4
5
=
3
4
=
1
1
×4
3
8
=
=
16 × 1
2
1
1
× 3
2
2
=
5
2
1
× 3 × 4
=
3
2
2
5
4
=
1
8
×
2
9
1
× 4
2
1
× 10 ×
5
3
7
3
1
×
15 5
4
=
35
4
= × 8
3
4
=
3
3
4
Multiplication of mixed numbers
7
4
2
×
×
12
5
14
2 × 3
21
5
1
× 12 × 1
=
6
10
4
4
× 24 × 1
=
6
12
=
© Milliken Publishing Company
16
MP4055
Write each quotient in simplest form.
Division of fractions
2
3
3
1
÷
=
4
2
4
÷
5
=
2
3
×
5
4
=
10
12
=
5
6
Division of mixed numbers
1
2
÷
=
3
5
2
1
2
÷ 5 =
5
2
×
1
5
=
1
2
6
3
÷
=
5
7
2 ÷
5
÷ 2 =
6
3
=
4
2
1
÷ 3 =
2
5
1
1
÷
=
2
4
3
1
3
÷ 2
=
7
5
4
2
2
÷ 1 =
3
3
3
2
÷ 10 =
5
6 ÷ 8 =
Find the quotients and add to find their sum.
1
3
3
1
÷
=
÷  +

2
4
4
8
1
1
5
1
÷
+
÷
=
6
8
3
4
3

 2 ÷  +

4
2
2
 1
 1
2 ÷  + 3
÷  =
 2
 4
4
3
(3
÷ 9) =
© Milliken Publishing Company
17
MP4055
Evaluate the following expression given
a = 2, b = 3, c = 4.
c
–b+5a
ab+
a
Evaluate each expression given
a = 2, b = 3, c = 4, d = 12.
a+c=
2 (3) +
ab=
4
2
– 3 + 5 (2)
6 + 2 – 3 + 10
4a + 1 =
15
6a ÷ d =
b–a=
d ÷b=
5b – 3 =
abc=
Complete each table.
a
b
1
3a+b
4b–a
c
d
2
1
2
5
2
6
4
2
4
8
2
9
10
9
3
a
b
c
d
e
f
1
5
2
24
7
6
3
8
4
16
3
4
2
9
1
30 12
5
5
8
3
8
2
2a+bc–ac
© Milliken Publishing Company
18
2
cd–2
2c÷d
d÷f+2e
MP4055
Solve the equation for the variable
Solve the following equations.
When finished, add all answers.
The sum should total 310!
x + 85 = 91
y – 14 = 60
– 85 – 85
+ 14 + 14
x=6
y = 74
x + 19 = 26
y – 24 = 30
a–4=9
m + 17 = 26
d – 7 = 30
c – 4 = 12
b + 9 = 81
n + 72 = 78
t + 2 = 20
v – 13 = 6
w – 13 = 4
z + 5 = 20
e+7=9
r – 12 = 6
p–3=4
g+6=6
© Milliken Publishing Company
19
MP4055
Solve the equation for the variable
Solve the following
equations. When finished,
add all the answers.
The sum should
total 1000!
3 × = 21
3
3
× = 7
y
= 3
7
 y = 7 3
7
( )
 7
y = 21
2 × = 42
y
= 6
5
a
= 4
5
7b = 21
R
= 6
24
5 t = 185
2m = 100
n
= 6
82
c
= 18
7
9d = 180
w
= 5
3
4 v = 28
8f = 16
12g = 36
j
= 5
4
k
= 2
5
© Milliken Publishing Company
20
MP4055
WORD PROBLEMS
1. Jake hit 3 more home runs than Rhajon. Jake hit a total of 11 home
runs. How many homers did Rhajon hit?
2. Sallie is 4 inches shorter than Marie. Sallie is 61 inches tall. How tall is
Marie?
3. Josie ate 360 fewer calories than Mark. Josie ate a total of 1250 calories.
How many did Mark eat?
4. Miguel drove 75 miles further than James. Miguel drove 350 miles. How
far did James drive?
5. Rochelle types 23 words per minute faster than Wanda. If Rochelle types
88 words per minute, how fast does Wanda type?
6. Amy’s salary is $2450 less than Joan’s salary. If Amy’s salary is $15,500,
how much is Joan’s salary?
7. Josè scored 13 points higher on the final exam than Frank. Frank earned
a score of 77. What was Josè’s score?
8. Barbara ran the marathon in 3h. 27 min. which was 14 minutes slower
than Julie. What was Julie’s time?
9. There are 36 cookies in a package of Mrs. Buttermaker Cookies. There
are 14 more cookies in a package of Blue Ribbon Cookies. How many
cookies are in the Blue Ribbon package?
10. Last fall Sal read 13 books which was 6 less than the number of books
he read last summer. How many books did he read over the summer?
© Milliken Publishing Company
21
MP4055
WORD PROBLEMS
1. Becky earns $14.25 per hour which is 3 times more than Joe’s hourly
wage. How much does Joe make per hour?
2. Marty drove 1/3 of the distance between St. Louis and Kansas City.
He drove 84 miles. How far is St. Louis from Kansas City?
3. Jennifer’s weight is 3/4 the weight of Sam. Jennifer weighs 120 lbs. How
much does Sam weigh?
4. Fred scored twice the number of points in the basketball game as
John. Fred scored 26 points. How many did John score?
5. Krista tripled her yearly salary. Her new salary is $24,600. What was her
old salary?
6. The cupcakes were divided evenly among the 27 students. Each student
got two cupcakes. How many cupcakes were there originally?
7. Loretta cycled twice as far as Julio. Julio cycled 23 miles. How far did
Loretta cycle?
8. The cost of the party was divided evenly among 18 people. The total
cost was $99.00. How much did each person chip in?
9. Michael drove 3 times as fast as Marla. Michael drove 57 mph. How fast
did Marla drive?
10. Hughie’s yearly salary was divided evenly over 12 months. Each month
he received $800. How much was his yearly salary?
© Milliken Publishing Company
22
MP4055
Solving two-step equations
Solve each equation and check
your solution.
×
– 1 = 4
3
+ 1 = +1
1.“Undo” addition/subtraction
2× + 3 = 7
– 3 = –3
2× = 4
Check:
15
– 1 = 4
3
2.“Undo”multiplication/division
×
3  = 3( 5 )
 3
5 – 1 = 4
× = 15
4 = 4✔
2× = 4
2
2
∴
×=2
4 × + 5 = 21
×
+ 5 = 21
4
3× – 2 = 7
×
+ 2 = 7
3
9 × + 1 = 19
×
– 4 = 2
5
8 × – 2 = 22
×
+ 4 = 8
7
2 × – 42 = 44
×
+ 3 = 3
3
5× – 5 = 5
×
– 2 = 4
11
© Milliken Publishing Company
23
MP4055
Substitute data into formula: e.g.
FORMULAS
area of circle = π x (radius)2
a = π r2
Complete each chart by
substituting data into each
formula.
If r = 3 cm
a = 3.14 (3)2 = 3.14(9)
= 28.26 cm2
2
a=πr
Radius (r)
2
c=2πr
4
Radius (r)
1.5
2
3.5
5
Circumference (c)
Area (a)
a=lxw
p = 2(l + w)
Length (l)
8
9
6
Length (l)
7
12
14.5
Width (w)
5
7
5
Width (w)
5
3
6
300
480
85
6
16
5
Perimeter (p)
Area (a)
d=Rxt
m=
Rate (R)
55
40
62
Distance (d)
Time (t)
3
8
7
Gallons (g)
Distance (d)
© Milliken Publishing Company
d
g
Mileage (m)
24
MP4055
Find the sum.
Addition of integers
3 + – 2 + 4 + –7
2 + –3 + 5 =
–7
+4
–5 + 6 + –7 =
–2
+3
10 + –7 + 6 =
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
–3 + 1 + –5 =
13 + –18 + –5 =
–13 + 16 + –5 =
40 + –32 + 6 =
–40 + 80 + –30 =
–24 + 20 + 6 =
–13 + 15 + –30 =
–6 + 8 + 9 =
26 + –12 + –4 =
32 + –48 + 14 =
–25 + 50 + 75 =
–42 + 38 + –4 =
52 + –34 + 9 =
–29 + –13 + –6 =
56 + 8 + –10 =
20 + 42 + 13 =
–124 + 68 + –56 =
246 + –138 + –56 =
346 + –129 + 48 =
2 + –3 + –4 + 5 =
–6 + –8 + 2 + 4 =
3 + –2 + 5 + 0 =
–7 + 9 + –1 + –2 =
5 + –4 + 6 + –6 =
–3 + 4 + 3 + –4 =
6 + –5 + –4 + –3 =
–9 + 7 + 2 + 3 =
–4 + 5 + –6 + 8 =
4 + –5 + –5 + 4 =
6 + –3 + –8 + 1 =
–7 + 8 + 2 + 5 =
7 + 6 + –2 + –3 =
–5 + –5 + 6 + –7 =
3 + 4 + –3 + –5 =
450 + –230 =
–560 + 450 =
–840 + 760 =
–345 + 500 =
–246 + –424 =
832 + –832 =
–4000 + 842 =
–2000 + –455 =
6000 + –4500 =
–2830 + 1460 =
2465 + –3460 =
8888 + –6468 =
–4256 + 5642 =
–3000 + –467 =
4642 + –5000 =
© Milliken Publishing Company
25
MP4055
Subtraction of integers
(Add the opposite)
Find the difference.
8–5=
24 – 16 = 24 + –16 = 8
–9 – 4 =
24 – –16 = 24 + 16 = 40
10 – –6 =
–24 – 16 = –24 + –16 = –40
–24 – –16 = –24 + 16 = –8
–7– –2 =
12 – –3 =
–20 – 20 =
101 – –303 =
32 – 40 =
32 – –5 =
–13 – –14 =
0 – –4 =
100 – –46 =
37 – 100 =
250 – –75 =
–62 – –41 =
–200 – 48 =
72 – –27 =
–40 – –60 =
82 – –5 =
8 – –4 =
–20 – –20 =
76 – –82 =
–50 – 55 =
2 – 49 =
50 – –50 =
13 – –14 =
3 – –58 =
–202 – 404 =
–242 – –50 =
14 – 464 =
–28 – –36 =
126 – 300 =
7 – 58 =
76 – 84 =
146 – 530 =
99 – 88 =
–43 – –43 =
© Milliken Publishing Company
26
MP4055
Find the product.
Multiplication of integers
–3 x –2 x 4 = 24
–24 x –2 =
“Even” number of signs produces a
positive product.
–4 x 3 =
–7 x –7 =
–3 x –2 x –4 = –24
“Odd” number of signs produces a
negative product.
–41 x 2 =
–2 x –19 =
–3 x 4 x –5 =
2 x –2 x 4 =
–3 x –2 x –1 =
4 x 2 x –1 =
3x 2 x 3=
–2 x 4 x –4 =
5 x 6 x –2 =
–1 x –2 x –3 =
4x5x6=
6 x –2 x 4 =
–3 x 4 x –2 =
5 x 10 x –10 =
2 x –30 x –1 =
–5 x –5 x 5 =
4 x –3 x –3 =
–2 x –3 x –11 =
4x 5 x 2=
–6 x 5 x 2 =
–3 x –4 x 8 =
–2 x –1 x –1 =
4 x –3 x 6 =
–2 x 3 x –4 x –1 =
5 x –2 x 1 x –1 =
–2 x 5 x –2 x –4 =
–1 x –1 x –1 x –2 =
3 x 3 x –1 x –1 x 1 =
–2 x 3 x 4 x –1 =
–2 x –2 x –2 x –2 x –2 =
–1 x –1 x 2 x 3 x –1 =
–3 x 2 x 2 x 2 x –3 =
–2 x –5 x 2 x 3 x 5 =
1x2x 3x 4 x 5=
7 x –6 x 5 x –4 x 3 =
2 x –3 x –2 x –3 x 2 =
–3 x 2 x 3 x –2 x 3 =
2 x 1 x 3 x –2 x –3 x –1 =
1 x –4 x –1 x 1 x –4 x –4 =
© Milliken Publishing Company
27
MP4055
Division of integers
Find the quotient.
24 ÷ –6 = –4
8÷4=
Division involving one negative number
will produce a negative quotient.
–4 ÷ 2 =
–24 ÷ –6 = 4
–1 ÷ –1 =
Division involving two negative numbers
will produce a positive quotient.
6 ÷ –3 =
–9 ÷ –3 =
–10 ÷ –2 =
–6 ÷ –2 =
–8 ÷ 2 =
7 ÷ –1 =
0÷4=
–11 ÷ 1 =
–14 ÷ 7 =
–13 ÷ –1 =
14 ÷ –2 =
15 ÷ –5 =
–16 ÷ –2 =
–15 ÷ 3 =
16 ÷ 8 =
17 ÷ –17 =
18 ÷ –9 =
16 ÷ –4 =
–18 ÷ –2 =
20 ÷ –5 =
–21 ÷ –7 =
–18 ÷ 3 =
–20 ÷ –4 =
22 ÷ –11 =
–20 ÷ –2 =
24 ÷ –8 =
–28 ÷ 7 =
–35 ÷ 7 =
–42 ÷ –21 =
56 ÷ –7 =
–54 ÷ –9 =
63 ÷ –7 =
–65 ÷ –13 =
66 ÷ –11 =
–68 ÷ 17 =
72 ÷ –9 =
75 ÷ 25 =
–58 ÷ 2 =
62 ÷ –31 =
–64 ÷ –16 =
81 ÷ –9 =
–85 ÷ –17 =
100 ÷ –20 =
–100 ÷ 25 =
–100 ÷ –50 =
100 ÷ –10 =
150 ÷ –50 =
–300 ÷ –75 =
225 ÷ –15 =
–275 ÷ 25 =
400 ÷ –20 =
–600 ÷ –40 =
121 ÷ 11 =
–625 ÷ 5 =
–320 ÷ –160 =
–450 ÷ 9 =
490 ÷ 70 =
–630 ÷ 90 =
1000 ÷ –50 =
–2400 ÷ –60 =
–3000 ÷ 600 =
2500 ÷ 50 =
–1500 ÷ –150 =
–2000 ÷ –1000 =
4500 ÷ –900 =
8500 ÷ 1700 =
–12 ÷ –2 =
–3 ÷ 1 =
0 ÷ –2 =
5 ÷ –5 =
© Milliken Publishing Company
28
MP4055
Order of operations
Evaluate.
–2 + 3 x –5
–2 + –15 = –17
–3 + 2 x –5 =
–20 ÷ 4 + –13
–5 + –13 = –18
12 + –3 x –6 =
–18 x 2 + 3 =
16 ÷ –4 + 3 =
28 – 3 x –2 =
–64 + –8 ÷ –4 =
–30 x –2 – 4 =
63 ÷ –9 – 5 =
13 – 4 x –5 =
27 + –8 x 6 =
18 – 3 ÷ –3 =
–65 ÷ 13 – 5 =
14 x –5 – 8 =
© Milliken Publishing Company
29
MP4055
More order of operations
Evaluate.
–2 + 3[–4 x 5 – 6]
4 + 2[–5 x 2 + 1] =
–2 + 3[–20 – 6]
–2 + 3[–26]
–2 + –78 = –80
3 + 5[2 + 3 x –2] =
–1 + 2[5 x 2 – 3] =
8 – 2[6 ÷ 3 – 2] =
–10 – 4[2 + 3 ÷ –3] =
20 + 2[3 ÷ 3 – 1] =
2[5 x 2 – –6] + 1 =
5[8 ÷ 4 – 1] – 2 =
6 – 3[2 + 3 x –4] =
–17 + –2[4 x 5 – 2] =
12 – 3[4 + 8 ÷ –4] =
5[20 ÷ –5 – 1] – 1 =
–2 + –3[6 + 8 x 2] =
© Milliken Publishing Company
30
MP4055
Given a = 2, b = –3, evaluate the expression
If x = –4, y = 2, z = –6
evaluate each of the
following expressions.
5b + a
5 (–3) + 2
–15 + 2
–13
x
+y=
2x + 3y =
y–z=
xy
–2y + 5x =
4z + x =
z
+ 2× =
y
+2=
6y – 4z =
2x + 3y – z =
2y – 4x + 72 =
×y +
3z + 4x – 5z =
10 z
+ 3× – y =
y
xy
© Milliken Publishing Company
31
2y
+ z=
×
+ yz + 2x =
MP4055
INTEGER WORD PROBLEMS
Write and solve a linear equation for each problem.
1. If x is increased by 14, the sum is 12. What is x?
2. If y is decreased by 17, the difference is 3. What is y?
3. If w is multiplied by 7, the product is –63. What is w?
4. If t is divided by –3, the quotient is 42. What is t?
5. –16 is 5 more than q. What is q?
6. 24 less than c is 13. What is c?
7. –3 times m is –105. What is m?
8. A number divided by 9 results in a quotient of –117. What is the
number?
9. When a certain number is increased by 57, the result is 38. What is
the number?
10. A certain number is 14 less than –9. What is the number?
© Milliken Publishing Company
32
MP4055
WORD PROBLEMS USING RATIONAL NUMBERS
1. William gained 4 lbs., lost 3 lbs., gained 6 lbs. and finally lost 11 lbs.
What was his net weight change?
2. Jackie’s average of 84 increased by 2 points, then fell by 4 points,
then increased by 9 points, and finally fell by 1 point. What was her
final average?
3. The average temperature on Monday was 5 degrees below 0. It rose
7 degrees on Tuesay, fell 8 degrees on Wednesday and rose 4 degrees
on Thursday. What was Thurday’s average temperature?
4. In the first quarter of Saturday’s football game, Wayne gained 7 yards,
lost 3 yards, lost another 5 yards, then gained 15 yards. What was his net
gain or loss in the first quarter?
5. The stock market rose 1-1/4 points on Monday, fell 3/4 of a point on
Tuesday, rose 1/2 point on Wednesday, rose 7/8 of a point of Thursday,
and fell 1-1/2 points on Friday. What was the net change for the week?
6. The submarine dove 75 feet, rose 25 feet, then dove 140 feet and finally
ascended 85 feet. What was the net change in depth?
7. Cindy lost 1-3/4 lbs., gained 2-2/3 lbs., gained 1-1/2 lbs., then finally
lost 3-1/2 lbs. What was her net change in weight?
8. Maureen climbed to an altitude of 1500 feet above sea level. She then
descended 750 feet, climbed 850 feet, and then descended 620 feet
where she finally decided to stop. At what altitude did she rest?
9. The temperature at the ski resort was 22° C when the ski lifts opened.
During the course of the day it fell 3.8° C, rose 2.9°C, rose another
5.3° C, and finally fell 4.7°C. What was the temperature at the close
of the day?
© Milliken Publishing Company
33
MP4055
Combining rational numbers
Simplify each expression.
7
3
+
=
8
4
6
7
+
=
–
8
8
1
–
8
–
–. 3 – –. 5 =
–. 3 + . 5 =
.2
–5
2
–
=
3
6
3. 4 + – 5.6 =
–1 – . 46 =
–3. 4 – 5 =
–2
4
+
=
9
3
1
–1 5 + – 2 3 =
4
8
2
7
1
+ –3 =
12
3
– 2. 45 – 3.1 =
4.2 + – 3.14 =
2.5 – – 3. 4 =
3
1
1
+ –1 =
4
2
7
3
+ – =
8
5
– 2.9 + – 5 =
© Milliken Publishing Company
34
2
1
+ –2
=
5
4
–1 2 + 1 =
3
6
1. 4 + – 5 =
MP4055
Absolute value
Determine absolute value.
–3 =
16 =
1.
–2
2.
6 + – 10
3.
–2 × –3
= 2
=
=
–4
6
= 4
=6
– 21 =
–2 + 0 =
3 + –1 =
–5 + – 3 =
8 + –2 =
8 – –3 =
3× –5 =
11 + – 7 =
8 – –9 =
4 + –8 =
–6 + –2 =
8 ÷ –2 =
24 ÷ 6 =
– 20 + 6 =
– 18 ÷ – 3 =
6 + –8 =
– 2 + –1 =
8 + –2
0 + –7
4 – 1
17 + – 2
2 + –3 + –4 =
5 + –8 + 7 =
– 6 + 8 + – 10 =
21 + – 8 + – 2 =
– 13 + – 4 + 9 =
17 + – 2 + 6 =
24 ÷ – 6 + 3 =
2 × –4 –4 =
3 + 8 × –3 =
–6 + 8 + –3 =
8 + 6 + –1 =
8 – 3 – –4 =
© Milliken Publishing Company
35
MP4055
Write > or < in each
Comparing rational numbers
.
–46
1
–3
2
–2
2
–18
>
–50
Ordering rational numbers
–4
23
–5
–5
< –1
–1
<
2
2
<
23
1.2
1
2
–3.2
2
3
7
–10
–21
2
–3.2
–.32
–3
–2.3
–2.31
–3
–3
<
3
4
–2
1
2
–3
1
4
1
2
2. 45
3
6
–
–2.5
2
3
–3
2
4
9
.4
–.6
1
4
–1
2
5
Put numbers in order from least to greatest.
4
5
3
–3.2
–3
–1. 4
1.04
–.14
3
5
–2. 4
–2
1
3
2
–4
1
2
–.2
________ < _______ < _______ <
______
1
4
________ < _______ < _______ <
______
1
2
2
________ < _______ < _______ <
______
–. 45
2
5
–.3
________ < _______ < _______ <
______
4.2
–4.3
1
3
________ < _______ < _______ <
______
© Milliken Publishing Company
1
4
36
MP4055
Solving one-step equations
Solve for x.
Add 8 to each side
of equal sign:
× + –8 = 2
Divide each side of
equal sign by –2:
–2 × = 16
+
8 + 8
–2 × = 16
× +
0 = 10
–2
× = 10
–2
× = –8
× + –9 = 7
× – 2 = – 18
×
–3 = 6
5 × = – 35
× + 12 = 7
× – –3 = 4
–7 × = – 63
×
= –4
7
× – 15 = – 3
× + – 6 = 14
12 × = – 48
×
–
–3 = 4
–3 × = – 51
× – 13 = – 27
×
= – 18
6
× + – 6 = – 50
© Milliken Publishing Company
37
MP4055
Solving two-step equations
Solve for x.
×
+ 2 = –4
5
2 × + – 3 = – 15
Add 3 to
each side
of equal
sign.
→
Divide
each side
of equal
sign by 2.
→
+ 3 = + 3
2 × + 0 = – 12
2 × = – 12
=
2
× = –6
2
–3 × + 5 = 65
4 × – 6 = – 26
×
–2 – 4 = 6
×
+ –4 = –7
3
–7 × – 3 = 18
6 × + 16 = – 2
–12 × + – 4 = 8
×
–
–9 – 7 = 6
–8 × + 6 = – 50
© Milliken Publishing Company
38
MP4055
Negative exponents
Write each decimal as a power
of 10.
.1 =
A. . 01 =
________
.00001 =
________
1000 =
________
.001 =
________
100 =
________
.000001 =
________
B.
1
1
−2
=
= 10
2
100
10
1
= 5 −4
54
C. 3−2 =
1
32
Write each power with a positive exponent.
2
–4
10 –
=
2
=
9 –8 =
5 –2 =
3– =
1
7–4 =
4 –5 =
6 –2 =
8–9 =
6 –6 =
2–5 =
4 –2 =
Write each power with a negative exponent.
1
=
92
1
=
73
1
=
42
1
=
31
1
=
54
1
=
26
1
=
64
1
=
83
1
=
10 2
1
=
79
1
=
34
1
=
25
© Milliken Publishing Company
39
MP4055
Write in scientific notation.
Using scientific notation
.0003 = 3 x 10–4
2000 = 2 x 103
.004 =
Using a decimal
.0012 =
3.2 x 10–4 = .00032
.0204 =
4600 =
.048
.15 =
.00005 =
3200 =
.0003 =
270 =
.018 =
.123 =
Write the equivalent decimal.
2.84 x 10–2 =
3.2 x 10–3 =
4.8 x 102 =
1.2 x 103 =
5 x 10–4 =
1.04 x 10–1 =
5.6 x 10–3 =
2 x 102 =
4.1 x 10–2 =
6 x 10–3 =
5 x 103 =
1.8 x 10–1 =
4.56 x 10–2 =
7.1 x 10–1 =
2.9 x 102 =
© Milliken Publishing Company
40
MP4055
Write > or < in each
3
5
or ?
4
8
Which is bigger:
.
Using cross-products
24
3
4
>
20
5
8
2
3
3
4
5
6
7
8
2
15
1
8
4
9
3
7
9
10
8
9
3
8
4
5
5
6
2
3
1
2
3
5
6
7
4
9
5
11
6
7
2
9
3
11
3
4
7
15
2
3
8
13
6
17
3
4
5
13
6
11
2
3
5
6
9
8
7
6
11
8
10
7
... 3 > 5
4
8
Circle which two ratios are equal in each group.
2
3
3
4
4
6
5
6
20
24
18
22
1
4
3
7
2
8
3
8
15
40
12
36
2
5
3
7
6
15
2
7
3
9
1
3
5
11
18
24
12
16
2
3
4
9
20
30
3
8
21
56
4
9
5
8
4
8
10
16
3
4
14
28
3
6
7
8
14
16
5
9
© Milliken Publishing Company
41
MP4055
Find the “unit price”
Determine the unit price of each
item. Circle the “better buy.”
Item: Milk
Price: $2.56 per gallon
Quantity: 1 gallon or 128 oz.
Unit Price per ounce =
Item
Price
Milk (A)
$1.28
64 oz.
Milk (B)
$ .48
16 oz.
Item
Price Quantity Unit Price
Item
Price Quantity Unit Price
Apples (A)
$3.20
1 lb.
Soap (A)
$1.92
64 oz.
Apples (B)
$1.76
1/2 lb.
Soap (B)
$1.68
48 oz.
Item
Price Quantity Unit Price
Item
Price Quantity Unit Price
Pizza (A)
$1.20
3 slices.
Bologna (A) $1.20
1/4 lb.
Pizza (B)
$1.50
5 slices.
Bologna (B) $2.40
1/2 lb.
Item
Price Quantity Unit Price
Item
Price Quantity Unit Price
Jelly (A)
$ .45
6 oz.
Pencils (A)
$1.50
1 doz.
Jelly (B)
$ .56
8 oz.
Pencils (B)
$1.20
8
Item
Price Quantity Unit Price
Item
Price Quantity Unit Price
Ice crm.(A)
Quantity Unit Price
$2. 56
= $. 02 per ounce
128 oz.
$.90
2 scps.
Gum (A)
$.75
5 sticks
Ice crm.(B) $1.05
3 scps.
Gum (B)
$1.04
8 sticks
© Milliken Publishing Company
42
MP4055
Solve for the variable.
Solving proportions
w
3
6
3
=
8
p
=
8
14
14 w = 24
w =1
5
7
24
×
=
32
4
–3
m
7
=
50
10
4
t
=
5
9
4
3
=
9
n
3
8
=
c
15
5
v
=
6
4
8
7
=
9
d
e
4
=
5
20
6
f
=
8
24
8
3
=
g
12
© Milliken Publishing Company
2
=
15
y
w
11
=
18
12
43
MP4055
How far is it from New York to St. Louis?
Determine the actual distance
between each pair of cities.
Use the scale.
Use the scale: 1" = 500 mi.
NY to St. Louis = 2"
×
2
=
500
1
× = 1000 miles
• Seattle
Boston
Chicago
• San
Francisco
•
Denver
St. Louis
Dallas
•
•
•
• New York
•
Washington,
D.C.
•
• Miami
1.
2.
3.
4.
5.
6.
7.
8.
New York to Miami
Boston to Washington, D.C.
St. Louis to San Francisco
Dallas to Seattle
Chicago to Denver
Boston to Dallas
New York to San Francisco
Miami to St. Louis
© Milliken Publishing Company
9.
10.
11.
12.
13.
14.
15.
44
Seattle to Chicago
Denver to Washington, D.C.
St. Louis to Denver
San Francisco to Washington, D.C.
Boston to Chicago
Miami to Seattle
New York to Dallas
MP4055
Complete the chart, filling in all equivalent values.
Example:
Reduced
Fraction
3
4
Reduced
Fraction
×
100
75
100
×
100
Decimal
Percent
.75
75%
Decimal
Percent
1
2
20
100
.65
30%
15
100
.05
6%
3
8
25
100
60%
7
10
© Milliken Publishing Company
45
MP4055
Write an equation for each question, substituting x for the word
“what.” Solve for x.
8 is 25% of what?
What is 20% of 30?
8 = .25 (x)
x = .20 (30)
8
. 25
=
x = 6
. 25
(x)
. 25
32 = (×x)
Write an equation for each question and solve.
What is 35% of 80?
12 is what % of 60?
15 is 40% of what?
What is 47% of 60?
8 is what % of 32?
24 is 30% of what?
What is 95% of 240?
48 is what % of 64?
34 is 17% of what?
3 is what % of 2?
© Milliken Publishing Company
46
MP4055
WORD PROBLEMS
1. A VCR costs $325. The sales tax is 8% of the price. What is the total price
of the VCR including tax?
2. A coat was originally priced at $280. The department store had a 25%
off sale. What was the sale price of the coat?
3. The bicycle which originally sold for $400 was on sale for $320. What
percent was the bicycle marked down?
4. Last year, Ms Stewart earned $64,000. This year she got a raise and now
makes $67,200. What percent increase did Ms Stewart receive?
5. Sam weighs 160 lbs. but wants to weigh 152 lbs. What percent of his
weight does he want to lose?
6. Terry used to be 125 cm. tall. She is now 7% taller. How tall is she now?
7. Franklin has an average of 75 and wants to raise it to 90. What %
increase would this be?
8. The sale price of a radio is $81. It was marked down 10%. What was the
original price?
9. Tommy bought a sweater at a moonlight madness sale for $20. It used
to be $100. What % savings did he receive?
10. Jeri got a promotion and a 14% increase. Her old salary was $24,000.
What is her new salary?
© Milliken Publishing Company
47
MP4055
What % did the price of the sweater
increase?
Determine increase and %
or decrease and %.
Old Price = $40
New Price = $50
Increase = $10
% Increase = Increase
Old Price
Old Price
New Price
$50
$100
$64
$75
$250
$400
$2500
$5000
$10,000
$70
$125
$96
$90
$300
$440
$3250
$5050
$20,000
Old Price
New Price
$50
$30
$100
$75
$64
$32
$12
$9
$450
$405
$500
$490
$2800
$2772
$4600
$3680
$20,000
$19,400
© Milliken Publishing Company
48
=
10
40
= 25%
Increase
% Increase
Decrease
% Decrease
MP4055
Determine savings and sale price.
25% OFF SALE!
Original Price of Sweater = $40
$40 x .25 = $10 = Discount
$40 – $10 = $30 = Sale Price
Original Price
% Discount
$50
$75
$64
$87
$124
$156
$79
$2000
$850
20%
40%
25%
10%
30%
8%
50%
5%
20%
$250
15%
$450
45%
$600
33%
$4600
18%
$27
20%
$30
25%
$950
12%
$10,000
40%
$25,000
15%
$64,000
8%
$100,000
19%
© Milliken Publishing Company
Savings
49
Sale Price
MP4055
Percents using ratios
(Hint: Always write the % as a
What is 30% of 50? (write a proportion)
fraction with denominator “100.”
×
e.g. 35% = 35
100
50.
=
30
100
100 × = 30 ( 50 )
100 × = 1500
Solve using a proportion
What is 20% of 50?
15 is what % of 60?
17 is 50% of what?
What is 5% of 650?
37 is what % of 370?
48 is 40% of what?
What is 7% of 92?
600 is what % of 500?
24 is 30% of what?
What is 150% of 84?
© Milliken Publishing Company
50
MP4055
Given 3, 8, 7, 1, 0, 3, 6
Find the Mean, Median, Mode
for each set of data.
Mean =
3 + 8 + 7 + 1+ 0 + 3 + 6
= 4
7
Median = 0, 1, 3, 3, 6, 7, 8 = 3
Mode = 3
Test Scores
78, 92, 90, 78, 96
Hourly Wages
Heights
$6.50, $20.00, $32.00
72", 60", 58", 62", 60", 72"
$29.00, $6.50, $10.00,
60", 76"
$15.00
Mean
Mean
Mean
Median
Median
Median
Mode
Mode
Mode
Weights
Homeruns
Driving Speeds
194, 206, 128, 102, 128,
7, 10, 7, 24, 3, 7, 6, 24,
55, 45, 55, 60, 60, 70, 55,
240, 100, 128, 128, 206
7, 11, 26
55, 62, 68, 75, 60
Mean
Mean
Mean
Median
Median
Median
Mode
Mode
Mode
Ages
Hours of Sleep
Phone Calls
14, 15, 13, 13, 15, 12, 17,
2, 8, 10, 8, 7, 10, 6, 4, 7,
3, 10, 1, 0, 2, 5, 5, 3, 2, 7,
15, 15, 12, 15, 12, 14
7, 10, 7, 7, 9, 3
8, 3, 4, 3, 4
Mean
Mean
Mean
Median
Median
Median
Mode
Mode
Mode
© Milliken Publishing Company
51
MP4055
Using the 3 graphs answer the
following exercises.
1. How many schools are in
Summit?
Number of Schools
48
50
40
2. Compare Jamestown and Fox.
How many more schools are in
Jamestown?
28
30
18
20
3. Powell Schools represent what %
of the total number of schools?
S
U
M
M
I
T
10
0
P
O
W
E
L
L
4. How many districts have less than
20 schools?
Fox
6. How many more students does
Summerville have than Jamestown?
Summerville
J
A
M
E
S
T
O
W
N
14
F
O
X
Jamestown
Powell
Summit
7. How many districts have more
than 3000 students?
Each
8. Summit has what % of the total
student population.
10. During what 5 year period was its
fastest rate of growth?
32
Number of Students
5. How many students attend Fox
District?
9. What was the population of
Hope HS in 1965? in 1970? in
1980?
S
U
M
M
E
R
V
I
L
L
E
represents 1000
Population Growth of Hope H. S.
2500
2000
1500
1000
500
1960 1965 1970 1975 1980 1985 1990
© Milliken Publishing Company
52
MP4055
4 +
Simplify each radical.
49
4 = +2
+
NUMBERS WITH PERFECT SQUARE ROOTS
9 = 2 + 3 = 5
3 4 = 3( 2 ) = 6
100
4 = 3 – 2 = 1
9 –
81
36
4 –2
–
25
16
4
64
=
4
2
= 2
Simplify each radical. Then add or subtract.
121 – 100
225 +
16 +
81 – 9
64 + 625
169 – 144
100 + 25
900 –
196 + 256
4
400
49
Simplify each radical. Then multiply by the coefficient.
5 9
7 4
11 16
4 4
6 49
10 225
8 81
100 64
Simplify each radical. Then multiply or divide.
4• 9
16 • 1
25 • 25
49 • 4
64
100
121
225
16
25
49
81
© Milliken Publishing Company
53
MP4055
REDUCING RADICALS
98 =
=
Reduce to simplest form.
49 • 2
49
•
2
= 7 2
12
8
18
6
20
27
45
32
50
28
200
242
162
75
1000
72
44
80
3 48 = 3 16 • 3
= 3 16
= 3 (4) •
= 12 •
2 24
6 125
3 98
9 18
5 40
15 27
7 450
2 90
8 54
4 1350
2 363
5 147
7 7
7 12
18 20
© Milliken Publishing Company
•
54
3
3
3
MP4055
Combining square roots
Simplify the sums and differences.
Only radicals that are “exactly” alike
can be added:
24 +
28 +
54
4•7 +
63
9•7
2 7 + 3 7
5 7
300 –
27
75 +
27
98 –
32
12 –
3
200 +
128
6 +
24
63 –
7
8 +
18
72 –
50
150 –
24
121 +
100
242 +
8
243 +
192
175 –
28
3 +
27
48 –
27
© Milliken Publishing Company
55
MP4055
More combining square roots
Simplify products and quotients.
All square root radicals can be
multiplied or divided.
3
2• 8
•
27 =
3 • 27
=
24
24
=
6
4 = 2
6
=
81 = 9
5 • 125
27
3
6• 6
8
8
3• 2
15
5
5• 7
16
8
7 • 14
32
8
2 • 32
100
4
11 • 3
55
5
13 • 4
75
25
© Milliken Publishing Company
56
MP4055
PYTHAGOREAN THEOREM
c
Find the missing side of each
triangle using the Pythagorean
Theorem.
3
12
a2 + b2 = c2
4
a2 + b2 = c2
32 + 42 = c2
9 + 16 = c2
25 = c2
5=c
a
c
c
b
5
b
6
8
c
3
5
26
13
c
5
b
24
a
3
3
3
5
c
15
10
8
c
a
8
25
a
15
6
a
20
a
12
12
© Milliken Publishing Company
57
MP4055
UNITS OF MEASURE
Equivalent measures
Write the equivalent measure.
12 inch = 1 foot
8 oz. = 1 cup
24 inch = ______ ft.
3 feet = 1 yard
2 c. = 1 pint
36 inches = 1 yard
2 p.t = 1 quart
5280 feet = 1 mile
4 qt. = 1 gal.
16 oz. = 1 lb.
12 feet = ______ yd.
8 qts. =
______ gal.
1000 mm = 1 m
32 oz. =
______ cups
100 cm = 1 m
6 c. =
______ pints
10mm = 1 cm
Add or subtract. (Reduce measure)
Example:
4
15
5 ft.
3 inch
– 2 ft.
7 inch
2 ft.
8 inch
7 ft.
+ 4 ft.
6 inch
9 inch
7 qt.
+2 qt.
1 pt.
1 pt.
3 gal.
–1 gal.
1 qt.
3 qt.
18 yds.
+ 37 yds.
2 ft.
1 ft.
17 m
–4 m
56 cm
85 cm
5 qt.
– 2 qt.
11 oz.
3 oz.
15 mi.
– 8 mi.
370 ft.
2000 ft.
85 cm
+ 17 cm
43 mm
65 mm
18 lbs.
+ 5 lbs.
7 oz.
9 oz.
5 qt.
– 3 qt.
1 cup
7c
–5 c
6 oz.
7 oz.
© Milliken Publishing Company
58
MP4055
Answer Key
page 1
page 6
200
150
330
100
800
2134
15,924
1111
337
5556
1630
1516
792
3195
3087
14,763
128,507
3346
341
47,524
11,978
121,435
3663
1559
2222
4.5
1.25
.74
.4
8.88
1.96
.493
20,000
8892
1041
1111
11,862
page 2
20,172
546,750
2,814,000
106,785
2,356,249
4,577,903
974,848
1,142,301
3,077,745
4,068,240
33
49
31
321
25
43
20
8
25
7
46
24
19
1
27
6
12
4
26
37
13
10
page 4
49
4
53
113
42
53
18
42
126
1.107
.2772
.0912
00492
7.29
.602
42
2.5
8
300
20
4.2
25
50
40
.3
420
.25
.9
.14
.02
42
250
.006
200
2
<
<
>
<
>
<
<
>
<
>
>
>
>
>
>
<
>
7
1
9
0
17
2
22
1
.2
.2
2.4
.5
2.46
.05
1.18
2.01
.046
.123
.234
.060
page 7
page 3
2
12
19
9
45
11
7
.27
1.222
28
54
page 8
33
54
page 5
15
10
13.4
17.5
20
9
4.035
48.72
11.95
11.95
16.88
11.8
29.32
11.73
2.705
6.01
3.65
5.78
4.35
1.763
.99
9.8
29.98
6.7
22.33
22.07
23
(2.4)3
10 4
93
75
84
66
25
49
1
81
27
64
8
512
125
256
16
1
81
625
100 10, 000
10
1
1000
16
3125
(3.2)
2
4
243
64
1
27
87
79
42
611
51
12
3
9
7
(2.5)
© Milliken Publishing Company
59
107
6
MP4055
page 9
page 12
240
3710
50.6
8371
700.8
9230
62,040
1,490,000
5,000,000
74,200
84.05
230,000
3755
810,000
900,000,000
page 16
3
4
8
4
3
4
1
4
1
3
4
5
4
5
2
3
2
5
3
2
7
3
13
4
21
5
31
6
8
5
18
7
32
9
19
4
17
3
52
5
165
8
215
7
81
2
253
5
79
4
23
3
13
2
73
9
34
3
3
15
27
15
36
33
4
4
24
8
3
3
page 14
4
27
24
20
25
6
10
1
15
121
page 13
>
<
>
>
>
14
28
8 43
22
9
11
12 54
2131
2
3
151
1 91
117
40
4 203
8 29
36
16 181
2 425
1123
61
12 70
13125
37
15 45
5117
40
5
14
2 23
5
12
5
6
22
117
25
2 54
3
4
17
50
4
3
3
98
page 15
<
>
<
>
>
<
4 89
page 17
10 74
12
60
6
6
30
6
3600
4 min.
6 21
1
6
page 11
5
18
1
3
14 119
page 10
2
<
<
<
7
24
<
<
>
>
8
12
6
24
<
7
20
<
5
8
12
30
<
11
15
<
7
8
7
172 < 153 < 1134
© Milliken Publishing Company
60
MP4055
page 18
page 24
6
6
9
1
1
4
12
24
12.56
26
165
7.065
22
434
12.56
35
50
21.98
36
30
0,1
22, 3
14, 8
25, 6
10
26
11
19
18
10
30
8
page 19
54
16
19
18
13
72
17
7
9
6
15
0
4
–6
9
–7
–10
10
11
100
–48
–112
0
–1
–6
–2
8
220
155
–3158
–1370
1386
–2
2
10
–8
54
52
–8
1
3
–4
–11
–110
–670
–2455
–955
–3467
14
–28
–2
27
75
265
6
0
3
8
–1
–80
0
1,500
2,420
–358
page 26
page 20
21
144
126
2
30
37
20
3
page 21
20
50
15
20
page 22
8
65”
1610
275
65
$17,950
3h. 13 min.
50
19
$4.75
252
160 lbs.
13
$8,200
54
46 min.
$5.50
19 mph
$9,600
3
492
7
10
3
–13
16
–5
15
–40
–8
1
146
325
–248
20
12
158
–47
27
–606
–450
–174
–8
11
404
37
4
–63
–21
99
87
0
–105
100
61
–192
8
–51
–384
0
page 23
4
15
3
0
31.4
87
17
page 25
5, 7
17, 3
10, 14
37, 31
7
37
18
2
page 27
50.24
32
320
64
2
28
2
3
30
43
66
© Milliken Publishing Company
48
–12
49
–82
38
60
–8
–60
–48
60
–66
96
–24
–80
9
–32
72
120
–72
–36
–16
18
–6
24
125
40
–2
10
2
24
–6
300
2.520
108
64
–6
32
120
–500
36
–60
–72
3
–11
–3
–1
–4
–2
–5
–9
–8
4
–4
4
15
–50
40
2
–3
–4
–2
8
–2
3
10
2
5
3
–9
2
–15
11
7
–5
–5
0
–7
13
–5
–4
–6
–3
–8
–6
–29
5
–10
–11
–125
–7
50
5
–1
13
20
36
–26
8
33
–53
–68
8
–6
92
–44
–28
–11
–15
–28
page 28
–2
–2
1
–2
3
5
0
–7
2
9
5
–4
6
–4
–2
–5
–3
–20
2
–20
10
6
page 30
–14
–17
–14
3
6
page 29
–13
30
34
–12
19
page 31
–33
–62
33
–10
61
–1
56
–21
–78
–2
–2
4
–4
–24
36
MP4055
page 32
page 33
–2
20
–9
–126
–21
37
35
–1,053
–19
–23
page 37
Lost 4
90
–2˚
14 yds.
+3/8
–105 ft.
–1 1/12 lb.
+980 ft.
21.7˚
16
–5
12
17
page 41
–16
1
20
–14
–18
9
–4
–108
–5
–3
–9
–20
–3
7
–7
–28
12
–44
page 38
–30
–20
–9
–1
<
<
>
>
<
<
>
<
>
<
>
>
<
<
<
<
<
<
page 39
page 34
page 43
page 35
3
16
21
2
11
4
14
6
5
11
1
1
page 44
2
15
8
6
7
4
8
12
13
8
4
4
2
3
8
21
21
9
6
17
4
3
15
<
<
<
<
<
>
<
>
>
1 – 1250
2 – 375
3 – 1875
4 – 2000
5 – 1000
6 – 1875
7 – 2875
8 – 1187.5
9 – 1937.5
10 – 1687.5
11 –
875
12 – 2687.5
13 – 1000
14 – 3000
15 – 1687.5
page 40
page 45
page 36
>
<
<
>
>
.0284
1200
.0056
.006
.0456
.0032
.0005
200
5.000
.71
480
.104
.041
.18
290
page 42
$.02 per oz.
$.03 per oz.
$.20 per oz.
$.22 per oz.
$.40/slice
$.30/slice
$.075/oz.
$.07/oz.
$.45/scoop
$.35/scoop
© Milliken Publishing Company
$.03 per oz.
$.035 per oz.
$4.80 lb.
$4.80 lb.
$.125 each
$.15 each
$.15/stick
$.13/stick
62
MP4055
page 46
page 49
28
20
37.5
28.2
25
80
228
75
200
150
page 47
1
2
3
4
5
6
7
8
9
10
$351
$210
20%
5%
5%
133.75 cm
20%
$90
80%
$27,360.
40
25
50
20
20
10
30
1
100
20
25
32
3
45
10
28
920
600
40
25
50
25
10
2
1
20
3
$40.00
$45.00
$48.00
$78.30
$86.80
$143.52
$39.50
$1900.00
$680.00
$212.50
$247.50
$402.00
$3772.00
$21.60
$22.50
$836.00
$6000.00
$21,250.00
$58,880.00
$81,000.00
page 50
page 54
10
25%
34
32.5
10%
120
6.44
120%
80
126
page 48
20
25
32
15
50
40
750
50
10,000
page 53
$10.00
$30.00
$16.00
$ 8.70
$37.20
$12.48
$39.50
$100.00
$170.00
$37.50
$202.50
$198.00
$828.00
$5.40
$7.50
$114.00
$4000.00
$3750.00
$5120.00
$19,000.00
page 51
86.8
90
78
$17.00
$15.00
$6.50
65”
61”
60”
156
128
128
12.63
7
7
60
60
55
14
14
15
7
7
7
4
3
3
page 52
1
2
3
4
5
6
7
8
9
10
© Milliken Publishing Company
18
18
20%
2
approximately 1500
approximately 1750
3
approximately 11.76%
approximately 750, 875, 1800
1973 – 1978
63
MP4055
page 55
page 56
page 57
page 58
2
4
2
4
3
(ex)
1 gal. 2 qts.
3 qts. 1 cup
24 lbs.
© Milliken Publishing Company
12’ 3”
56 yds.
6 mi. 3650 ft.
1 qt. 3 cups
64
10 qt.
12m 71cm
112 cm 8mm
1 cup 7 oz.
MP4055
Grades 7-9
a
Motivate your students to
improve their math skills
with these fun, easy-to-use workbooks!
MP4041 Beginning Algebra grades 6-8
MP4050 Algebra grades 7-9
MP4054 Math Workbook grade 7
MP4055 Math Workbook grade 8
MP4056 Geometry grades 6-8
MP4057 Advanced Geometry grades 7-10
MP4065 Math Workbook grade 5
MP4066 Math Workbook grade 6
MP4075 Fractions grades 5-6
MP4076 Decimals grades 5-7
MP4078 Word Problems grades 4-6
2448 =
2•6
Milliken Publishing Company
a Lorenz company
P.O. Box 802
Dayton, OH 45401-0802
www.LorenzEducationalPress.com
ISBN 978-1-4291-1033-4
17 =
Milliken Publishing Company