Description
In This collection, we will go deep into math.
This collection will help all math and high school students.
Study Set Content:
Luttrell 2012
21
Name: ______________________
Date: _____
2e-Adding Fractions of Like Denominator
For each question, translate the equation and then solve by showing your algebraic steps.
1. The perimeter of the room is 248 inches. If two walls of the rectangular room are 80 ½ long,
how long is each of the other walls?
2. A rectangular room has perimeter of 320 ½ inches. One wall is 70¼. Find the dimensions of
the other three walls.
3. A triangle has sides 1⅜, 2⅝, and 2⅞. Find its perimeter.
4. A rectangle has dimensions 6⅔” by 7⅓”. Find its perimeter.
5. A rectangle has perimeter 8⅛ and one known side of 2⅞. What are the dimensions of the
rectangle?
6. A hexagon (6-
sided shape) has sides 2⅞, 1⅜, 4, 5⅛, 4½ and a perimeter of 20. How long is
the missing side?
7. Find the perimeter of an octagon with equal sides of 4⅝. An octagon has 8 sides.
8. A pentagon has
perimeter of 30⅔. If four sides are known to be 4⅓, 5⅔, 7⅔, and 6⅔, how
long is the remaining side?
Luttrell 2012
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Name: ______________________
Date: _____
2f - Like Terms
Constant
: A number that doesn’t change. In the expression 2x + 3, the constant is 3. The 2 is a
coefficient
.
Variable
: A number that may change, usually is represented by a letter. In the expression 2x + 3, the variable is x.
Term
: Any constant or variable that is being added or subtracted. In the expression 2
x
+ 3
y
, the terms are 2
x
and 3
y
.
Expression
: A collection of terms that together represents a number.
Equation
: When two expressions are equal. Usually the goal is to find the value of the variable that makes the
equation true (equal).
Like Terms
: Two or more terms having the same variables with the same exponents. The variables do not need to
be in the same order; 2
wz
and 3
zw
are like terms. Coefficients are ignored since they refer to the amount of this
term you have.
Combining like terms
: When the terms are alike, you add or subtract (depends on signs) the coefficients. Adding
2
wz
and 3
zw
would give you 5
wz
. Subtracting 2
wz
from 3
zw
would result in 1
wz
(better written as
wz
).
1. Write a like term for each of the following:
a. -16
y
b. -5
c. 4
xy
2
z
d. 5
h
2. Determine which of the following sets are made up of like terms:
a. {4
x
, 3
x
2
, 3
x
3
}
b. {
xyz
, -3
xyz
, 5
yz
}
c. {2
xy
, -3
xy
, -8
xy
}
For the following exercises, combine like terms.
3. 5
x
2
- 6
x
2
+ 4
xy
+ 3
y
2
- 2
y
2
4. (8
y
2
+ 6
y
- ½) + (-
y
2
- 2
y
+ ¾)
5. (3
x
2
+ 4
x
+ 4) + (
x
2
- 2
x
- 2)
6. Add
x
2
- ¼ and -
x
2
- 5
x
+
⅞.
7. (5
x
3
- 10
x
2
+ 3
x
) - (-3
x
3
+ 5
x
2
- 8
x
0
)
8. (5
x
-3) - (- 4
x
2
+ 3
x
- 1)
9. Subtract -3
x
3
- 2
x
2
- 6
x
0
from -2
x
3
- 3
x
2
- 6
x
.
10.
(2 - x) - (4 + 5x)
11. Subtract 2
x
3
- 3
x
2
+ 4
x
from -2
x
3
- 3
x
2
+ 5
x
+4
x
0
.
12.
(3 + 3x) + (1 - 2x)
Luttrell 2012
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Name: ______________________
Date: _____
2g-Solving Basic Algebraic Equations (one step)
Use your calculator to check your answer. Leave answer as simplified proper fractions.
1. 3x = -21
2. -7y = 28
3. -196 = -28x
4. -15a = - 45
5. -x = 17
6. -22 = 2x
7. -12b = -288
8. 12x = -60
9. A÷5 = -6
10. Y ÷ (-5) = -14
11. X ÷ 4 = -24
12. X ÷ (-3) = 4
13. A ÷ 14 = 7
14. 3a = -15
15. A ÷ 4 = -81
16. Y ÷(-1) = 0
17. X ÷ (-3) = 6
18. -12 = 4n
19. - 4w = -12
20. Y ÷ 6 = 2
21. 4x = 0
22. x + 6 = 9
23. x – 9 = 1
24. 7 = x + 3
Luttrell 2012
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Name: ______________________
Date: _____
2h-Solving Basic Algebraic Equations (two steps)
Use your calculator to check your answer. Leave answer as simplified proper fractions.
Solve the following equations:
1. 2x + 1 = 5
2. -3x + 1 = 10
3. 4x - 2 = 10
4. 5x + 3 = -12
5. 8 - x = 13
6. 4x - 3 = 5
7. -3x + 17 = 14
8. - x + 6 = 6
9. -2x + 3 = -5
10. - 4 + 2x = 6
11. -7 + 5x = 13
12. 6 - 5x = 1
13. - 4 - 6x = -4
14. 10x - 45 = 45
15. -9 + 7x = 12
Luttrell 2012
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Name: ______________________
Date: _____
2i-More Solving Basic Algebraic Equations
Use your calculator to check your answer. Leave answer as simplified proper fractions.
1. 3x + 2 = 14
2.
x
−
+ =
3
4
10
3. x + 3 = 3 ¾
4. ½ x - 8 = 16 5. 5x - 10 = 5
6.
x
2
3
7
+ =
7.
x
−
− =
7
5
15
8. ¼ x = 32
9.
x
6
8
10
− =
10.
12
5
8
= +
x
11. 5x - 3 = 17
12. 4x + 7 = -1
13.
2
3
8
6
x
− =
14.
−
+ = −
3
4
4
8
x
15.
−
+ =
x
3
1
4
16. 5x + 3 = -12
*17. 2x +8 = 4x + 2
*18. 3x – 1 = x + 5
Luttrell 2012
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Name: ______________________
Date: _____
2j-Solving for a Variable (multiple steps)
When solving for a variable, reverse the order of operations. The objective is to isolate that
variable by getting it to one side of the equation and all the constants to the other side. If faced
with an equation with a nested variable (see example C), eliminate the outer portion before
messing with the interior of an expression. To eliminate terms or coefficients, you will need to
apply the inverse operations.
Examples:
A. 3
x
= 24
B. 3
x
+ 1 = 25
C. (3
x
-1) ÷ 5 = 3
(3
x
)÷3 = 24÷3
3
x
+1 -1 = 25 - 1
(3
x
-1)÷5×5 = 3×5
x
= 8
3
x
= 24 3
x
- 1 = 15
3
3
24
3
x
=
3
x
-1 + 1 = 15 + 1
x
= 8 3
x
= 16
3
3
16
3
x
=
x
= 5⅓
Solve for the variable in the following equations:
1. 3
x
- 4 = 11
2. 5
x
+ 7 = - 3
3. -3
x
+ 2 = 17
4.
2
3
3
5
x
+
=
5.
2
5
2
3
x
+
=
6. 2(2
x
- 1) = 8
*7.
2
5
2
3
3
x
+
− =
*8.
3
2
3
7
6
x
+
+ =
*9. 3(
x
+5) - 2 = 7
Luttrell 2012
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Name: ______________________
Date: _____
2k-Solving for a Variable (Multiple Like Terms)
When faced with an equation that has multiple terms of the same variable you need to solve, get
those terms together on one side of the equation. It doesn’t matter what side of the equation you
move the terms to, as shown below. You may have to simplify before you can move terms
around. There are many alternate methods of solving for a variable; valid methods require the
use of the Field Axioms and PEMDAS.
Example: 2(3
w
+ 2) - 12 = 3
w
- 11
6
w
+ 4 - 12 = 3
w
- 11
6
w
- 8 = 3
w
- 11
6
w
- 3
w
- 8 = 3
w
- 3
w
- 11
or
6
w
- 6
w
- 8 = 3
w
- 6
w
- 11
3
w
- 8 = - 11 -8 = -3
w
- 11
3
w
- 8 + 8 = 8 - 11
11 - 8 = -3
w
-11 + 11
3
w
= -3
3 = -3
w
3
w
÷3 = -3÷3
3÷(-3) = -3
w
÷(-3)
w
= -1
-1 =
w
Solve for the variable:
1. 8
x
- 2(
x
- 8) + 4
x
= - 4
2. 4
x
- 2(
x
+3) - 4
x
= -3
3. - 3(
x
+ 5) = 10 - 2
x
4. 12
z
- 3(
z
- 7) = -(5
z
+ 7)
5. -6(
w
- 3) = 3
w
- 9
6. 16
x
- 4(
x
- 8) + 8
x
= -8
*7. 5
y
- [7 - (2
y
- 1)] = 3(
y
- 5) + 4(
y
+ 3)
8.
x
x
x
4
2
3
13
6
+ + =
Luttrell 2012
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Name: ______________________
Date: _____
2L-More Practice Solving Equations
You may use your calculator only to check your answers. Leave fractions proper.
Solve for the variable:
1. 3x - 2 = 5x
2. 4x + 6 = 3x
3. 5x - 14 = -2x
4. 6x - 2x - 3 = 9
5. -7x + 4x + 5 = 20
6. -x + 7 = 3x -1
7. 2x + 6 = 7x - 14
8. 7x - 12x + 4 = 19
9. 8 - x = 9 - 2x
10. 7 - 2x = - 3 + 3x
11. 54 -
( ⅔)x = 38
12. -7 - x = 8 + 4x
13. 4x = 3x - 2(5-x)
14. 2x - 3 = 4(x+3) - 5x
15.
x
x
−
+
+
=
3
4
2
5
2
7
Luttrell 2012
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Name: ______________________
Date: _____
2m-Basic Algebraic Equations with Perimeter
For each question, translate the equation and then solve by showing your algebraic steps.
1. The perimeter of the room is 248 inches. If two walls of the rectangular room are 80 long,
how long is each of the other walls?
2. A rectangular room has perimeter of 30 inches. One wall is 10. Find the dimensions of the
other three walls.
3. A triangle has sides 1, 2, and 2. Find its perimeter.
4. A rectangle has dimensions 6” by 7”. Find its perimeter.
5. A rectangle has perimeter 6 and one known side of 2. What are the dimensions of the
rectangle?
6. A hexagon (6-sided shape) has sides 2, 1, 4, 5, 4 and a perimeter of 20. How long is the
missing side?
7. Find the perimeter of an octagon with equal sides of 4. An octagon has 8 sides.
8. A pentagon has perimeter of 26. If four sides are known to be 4, 5, 7, and 6, how long is the
remaining side?
9. A rectangle has dimensions x - 3 and 2x + 6 with a known perimeter of 24, what is x?
10. A triangle has sides 2x, 3x +1, and x - 5 with a known perimeter of 26, what is x?
Luttrell 2012
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Name: ______________________
Date: _____
2n-Basic Algebraic Equations with Area
For each question, translate the equation and then solve by showing your algebraic steps.
Leave answers as proper fractions.
1. A rectangular room is 8 feet by 10 feet. Draw the layout and label. Find the area of the room.
2. A rectangular kitchen is 12 feet by 9 feet. Draw its layout and find the area of the room.
3. A rectangle is 9 ½ by 10 ¼. Find its area.
4. A triangle has a base of 4 ½ and a height of 5¾. Find its area.
5. A rectangle is 3⅝ by 4⅛. Find its area.
6. A square has a side of 3⅓. Find its area.
7. A bedroom is to be carpeted. Its dimensions are 9⅓’ by 11⅞’. How much square feet does it
have? How much square yardage does it have? (3 ft = 1 yd, 9 ft² = 1 yd²)
Luttrell 2012
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Name: ______________________
Date: _____
2o-Basic Algebraic Equations with Fractions
For each question, translate the equation and then solve by showing your algebraic steps.
Leave answers as proper fractions.
1. 2x - 3 = 5
2. 3x - 1 = 11
3. 4x - 1 = 15
4. 3x - ½ = 5 ½
5. 4x - 1¼ = 3¾
6. 6x -
1⅛ = 4⅜
7. 3x - 2 = 5
8. 4x + 5 = 16
9. 5x + 30 = 3
10. -2x = 9
11. -3x + 1 = 6
12. -6x + 4 = -9
13.
3
2
4
5
x
+ = −
14.
5
3
4
6
x
− =
15.
4
5
3
4
1
8
1
10
x
−
=
Luttrell 2012
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Name: ______________________
Date: _____
2p-Basic Algebraic Equations with Fractions
Solve for x by showing your algebraic steps. Leave answers as proper fractions.
1. The a
rea of a rectangle is length times width. If the area of a rectangle is 9⅜ cm² and its
width is 4¼ cm, what is its length?
2. If a rectangle’s area is 100¾ cm² and its length is 9½ cm, what is its width?
3. Area of a triangle is half its height times base. If the area of a triangle is 10 cm² and has base
of 3 ½ cm, what is its height?
4. If the area of a triangle is 4 ½ and its height is 3¼, what is its base?
5.
3
2
27
44
x
=
6.
4
5
6
25
x
=
7.
4
9
30
21
x
=
8.
3
4
3
4
11
12
x
− =
9.
8
9
1
2
7
8
3
10
x
+
=
10.
4
1
11
1
3
3
4
1
8
x
+
=
Luttrell 2012
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Name: ______________________
Date: _____
2q-Basic Algebraic Equations with Fractions
Use your calculator to check your answer. Leave answer as simplified proper fractions.
1. 2x - 4 = 9
2. 3x + 5 = 7
3. 7x - 1 = 8
4.
2
3
4
10
x
− =
5.
3
4
5
14
x
+ =
6.
x
2
3
4
1
8
3
5
−
=
7.
3
4
6
10
x
− =
8.
5
3
15
x
=
9.
3
4
15
16
x
=
10. If a square has a side of ½ inch, what is its area?
11. If a rectangle has dimensions of ¾” and ⅝”, what is its area?
12. If a rectangle has area of 32 ⅔” and a length of 14", what is its width?
13. If a rectangle has area of 38" and a length of 4¾”, what is its width?
14. If a square has area of
25
64
squared feet, what is its length of side?
Luttrell 2012
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Name: ______________________
Date: _____
2r- More Fractions with Perimeter and Area
Use your calculator to check your answer. Leave answer as simplified proper fractions.
1. Find the perimeter of the following triangles with sides of:
a. 3¾, 4½, 5⅝
b. 7⅔, 8⅓, 13⅓
c. 4⅓, 3½, 5⅛
2. Find the perimeter of the following rectangles with dimensions of:
a. 4⅜” by
5
3
5
”
b
. 4" by 4⅜”
c. 3½” by 3½”
3. Find the area of the following rectangles, with dimensions of:
a. 4⅜” by
5
3
5
”
b
. 4" by 4⅜”
c. 3½” by 3½”
4. Find the area of the following triangles, with dimensions of:
a. Height 3½”, base 8"
b. Height 2⅛”, base
4
4
17
“
c. Height 4, base 5 ½
5. If the perimeter of a square is 16, what is its area?
6. If a triangle has area of 35 and length of 16, what is its base?
7. If a triangle has area of 32 and height of 5, what is its base?
*8. If the perimeter of a rectangle is 14 and the length is 5 more than the width, what is its area?
*9. If perimeter of a rectangle is 14 and the length is 4 more than the width, what is its area?
Luttrell 2012
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Name: ___________________________
Date: __________
Chapter 2 Test
SHOW WORK
. A calculator is NOT allowed on this test. You must work alone.
Questions regarding interpreting the directions are allowed. Simplify your fractions!
1. Put in order from smallest to greatest.
a.
1
2
1
3
1
5
1
4
1
1
, , , ,
B. - ½, ¼, -
⅝, ⅔
2. A. Write as an improper fraction:
6
3
4
.
B. Write as a mixed number:
23
8
.
3. Add the fractions, leaving answer as a proper fraction:
a.
3
8
7
8
+
B.
4
5
2
3
3
+
4. Subtract the fractions, leaving answer as a proper fraction:
a.
17
5
8
9
4
9
−
B.
6
3
1
2
7
8
−
5. Multiply and simplify to proper fractions:
a.
5
6
9
10
×
B. 5
2
3
9
17
×
6. Divide and simplify to proper fractions:
a.
3
4
7
8
÷
B.
8
4
1
2
÷
5
5
5
5
5
5
Luttrell 2012
36
Chapter 2 Test, continued
7. Match the terms
_____ a. Vinculum
A. Top part of a fraction
_____ b. Denominator
B. Bar separating parts of fraction
_____ c. Numerator
C. Bottom part of a fraction
_____ d. Decimal
D. Dot separating whole from parts
_____ e. Improper fraction E.
41
5
_____ f. Proper fraction F.
3
2
5
_____ g. 0.06
G. 6 thousandths
_____ h. 600
H. 6 hundredths
_____ I. 0.006
I. 6 hundreds
_____ j. 0.6
J. 6 tens
K. 6 tenths
8. Find the perimeter of the rectangle whose dimensions are 1¾ cm by 3 ⅓ cm.
9. Find the area of the rectangle whose dimensions are 5 ½” by 10".
Bonus: (2 pts) Let N = 20×30×50×70×110×130. What is the smallest positive
prime number which is NOT a factor of N?
(3 pts) What is the value of
x
which satisfies the equation
5
20
7
28
9
36
11
44
80
1
+ + + + =
x
?
0
5
5
10
Luttrell 2012
37
Chapter 3 Decimals
Prior Skills:
•
Fractions
•
Area
•
Perimeter
•
For sheet 3k, volume and surface area
•
For sheet 3m and 3o, Pythagorean Theorem
•
For chapter 3 test, order of operations and definition of addend
Luttrell 2012
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Name: ______________________
Date: _____
3a-Decimal Notation
Use your calculator only to check your answer.
Write the following number in standard form:
1. (3×100) + (5×10) + (1×1) + (8×0.001)
2. (5×1000) + (3×10) + (4×0.1) + (5×0.01)
3. (4×1000) + (8×100) + (4×0.1) + (3×0.001) + (3×0.0001) + (5×0.00001)
4. (8×1) + (7×0.1) + (4×0.001) + (2×0.000001)
Write the following in expanded form, use the example as a guide:
27.6581 = (2×10) + (7×1) + (6×0.1) + (5×0.01) + (8×0.001) + (1×0.0001)
=
(
)
(
)
(
)
(
)
(
)
(
)
2 10
7 1
6
5
8
1
1
10
1
100
1
1000
1
10000
×
+ × + ×
+ ×
+ ×
+ ×
5. 4.8712
6. 3.140092
7. 62.34
8. 144.987
Write the following lengths of a rectangle as fractions:
9. 0.47 m
10. 0.3609 m
11. 5.63 ft
12. Which area is largest? 3.404, 3.44, 3.40004, 3.4004
13. Which perimeter is smallest? 2.3, 2.31, 2.311, 2.3111, 2.31111
Luttrell 2012
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Name: ______________________
Date: _____
3b-Switching Between Fractions and Decimals
Use your calculator only to check your answer.
The perimeter of the room is given in fraction form. Convert the fractions into decimal
1. 40½
2. 31¾
3. 61⅝
4. 56⅔
5.
4
5
6.
7
4
7.
19
9
8.
212
100
9.
82
5
10.
6
12
15
11.
5
8
12.
3
14
20
13. Which questions had repeating decimals? When does the fraction cause repeating decimals?
14. Rational numbers are those numbers that can be written as fractions. Thus the numbers in
problems #1- #12 are all rational numbers. Which of the following decimals are not rational
numbers?
A. 0.121212...
F. 2.718281828459045...
B. 3.444554445544455...
G. 1.4142643...
C. 0.5
H. 1.4141414...
D. 0.567
I. 6.55789789789...
E. 3.141592...
Luttrell 2012
40
Name: ______________________
Date: _____
3c- Operations with Decimals
Only use your calculator to check your answer!
Order the decimals from least to greatest.
1. 9.33, 9.4, 9.44, 9.45, 9.446
2. 2.11, 2.111, 2.121, 2.112
3. 3.4, 3.43, 3.424, 3.4509, 3.43509
4. 1.2, 1.302, 1.3002, 1.30002
Simplify the expressions.
5. 8.275 – 5.857
6. 18.93 + 149.42
7. 87.944 – 6.58
8. 14.923 + 1.8
9. 13.245 + 1.4467
10. 12.3 + 1.43 + 1.5607
11. 2.3 – 3.12
12. 0.45 + 45
13. 0.45 – 45
14. 4.01 + 0.034
15. -0.475 – 3.78
16. -7.2 + 10.56
17. 1.8 + 2.401 + 1.05
18. 3.45 + 2.356
