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ESM 1 Materials
Topic 1: Order of Operations
Objective #1: Master Order of Operations (understand misleading nature of “PEMDAS”)
Objective #2: Evaluate formulas, expressions and functions
Order of Operations
Step 1: Grouping Symbols: [( )],
, ab. value
Step 2: Exponents
Step 3: Multiplication/division Left → Right
Step 4: Addition/subtraction
Left → Right
“inside-out”
Example 1
15 − 7  −1 − (2 − 5) 2 
Example 2
−40 ÷ 23 (5) + 20
Example 3
−6 3 − 7 − 5 100 − 36
15 − 7  −1 − (−3) 2 
−40 ÷ 8(5) + 20
−6 −4 − 5 64
−5(5) + 20
−6(4) − 5(8)
−25 + 20
−24 − 40
−5
−64
15 − 7 [ −1 − 9]
15 − 7 [ −10]
15 + 70
85
I. Perform the operations - no calculator! Simplify completely.
1) (−16) − (+8) + 9 − (4 − 10)
2) −6 − (−13) + (−11) − (+7)
3) 7 + 8 − [ 4 + 2(−1 − 3) ]
4) 7 + 2 16 + 9
5) −32 ÷ 4 ⋅ 23 − 6
6) −24 − (4 − 7) 2 + 13
7) −9 −2 + 5
8) 8 − 5 − 1 − 7
9)
10) (− 5) 2 + 42 − 62
11)
13) 2(24) ÷ 42 − 4 9
14) 6 − (− 2 )
16) − 5 − 5
17) − 12 + 20 + − 7 − 11
16
19) 3 +
16
5 11
−
6 4
52 − 4 2 − 52 ⋅ 4 2
20)
2 + 4⋅6
− 12 − 6
− 1 + 24 − 7
(5
2
− 33
9
18) 2 − 3[ 2(5 − 8) + 1]
21) 0 ÷ 3
23) −18 ÷ 0
24) 1 187 ÷ 3 89
25) −15  8  ⋅ −35
26) 12 ÷ 9 ⋅ 15
27)
5
10 4
4
3
22) 4 13 ÷ 26
25  21  −24
)
15) 6(− 2 )
3
−4
12)
1 − 32
4 − 2 ⋅10
8 9
1
28)
31)
−2 −6
36 + 64
−5 + 52 − 4(1)(−14)
2(1)
29)
−8 + 82 − 4(−1)(−15)
2(−1)
30)
−(−6) + (−6) 2 − 4(1)(−7)
2(1)
32)
2 − (−2) 2 − 4(2)(−12)
2(2)
33)
− (−1) − (−1) 2 − 4(3)(−4)
2(3)
II. a, b: Evaluate each formula or expression at the given value(s). For c, solve for unknown value.
1) =
a) C = 45
b) C = −10
c) F = 50
F 95 C + 32 ;
2)=
C
( F − 32) ;
a) F = 59
b) F = −31
c) C = −25
3) K = 12 m v 2 ;
a)=
m 12;
=
v 5
b) m = 20; v = −8
c)=
K 90;
=
v 6
an −1
;
a −1
a) a 4;=
=
n 3
b) a =
−2; n =
4
5) x 3 yz 2 ;
a) x =
3; z =
−2; y =
−1
b) x = 3; y = 5; z = − 23
6 y − z2
xy
a) x =
3; z =
−2; y =
−4
b) x =
6; y =
2
− 4; z =
4) S =
6)
5
9
III. Evaluate each function at the given value; write result as an (Ordered, Pair).
1) f ( x) =
2) g ( x=
3) g ( x=
) 53 x − 2 ; x = 6
−9 x + 20 ; x = 4
) 2 x + 3 ; x = − 125
4) f ( x) = x 2 − 4 x − 11 ; x = −3
5) h( x=
) x3 − 2 x 2 ; x = −2
IV. Place < , > or = to make each statement true.
5
2
−5+4
1) − (−8)
2) −
9
9
5
4
17
5) ( −2 )
6) 5
3.4
( −2 )
5
6
7) 5.3
3)
6) h=
( x) x 2 1 − 2 x ; x = −4
7
9
5.049
− −5
4) 0
8)
( 23 )
2
( 23 )
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Topic 2: Place Value, Rounding, Fractions
Objective #1: Identify place-value by name and “exponential equivalent”
Objective #2: Round numbers; express numbers in written form
Objective #3: Perform operations with fractions
I. a) Place Value name; b) Values in decimal form; c) Values in fraction form; d) Values in 10n form
a) _____________ ____________ _________
.
__________ ______________
______________
b)
c)
d)
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II. a) Round 1,752,493 to the nearest:
________________ ten
b) Round 8.60527 to the nearest:
________________ unit
________________
thousand
________________
thousandth
________________
hundred
________________
ten
________________
ten million
________________
hundredth
________________
ten thousand
________________
tenth
________________
million
________________
ten-thousandth
________________
hundred thousand
________________
hundred
III. Express each number in written form.
Examples:
Acceptable
Unacceptable
a) 0.062
sixty-two thousandths
point zero, six, two
b) 2.15
two and fifteen hundredths
two point fifteen
c) 74.0502
seventy-four and five hundred two ten-thousandths - - - Number
1) 0.9008
Written form
2) 12.7
3) 5.91
4) 0.654
5) 453.02
6) 0.00035
7) 0.7426
8) 0.085309
IV. Simplify the following fractions.
1)
35
56
2)
540
360
60
108
6)
30
4) −2 107 ( − 18
)
5) 5 13 ÷ 2 76
6)
9) 6 − 2 95
10) 3 14 + 1 23
11)
15) 2 − 18 + 125
16)
3)
90
75
3)
125
81
4)
72
56
5)
840
6300
V. Perform the operations; simplify.
24
1) 3 81 ⋅ 40
7)
19
8
− 87
13) 1+ 34 − 107
2
2) − 64
25 ÷ 2 15
8) 3 83 + 2 87
14)
5
6
− 92 + 23
75
÷ 90
( − 32 )
2
34
5
2)
54
11
3)
115
12
4)
72
56
+ 107
+ 3 17) (1− 13 ) − 56
VI. Write each improper fraction as a simplified mixed number, e.g.,
1)
8
15
2
28
3
35
72
24
⋅ 60
12)
17
18
11
− 12
18)
13
20
− 85 − 107
= 9 13
5)
95
8
6)
131
24
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Topic 3: Fractions, Decimals, %
Objective #1: Improve computational skills with fractions, decimals and per cents
Objective #2: Develop “mental math” skills and number sense
Objective #3: Convert numbers to Scientific Notation/Standard forms
I. Determine the unknown part of each circle.
1)
2)
3)
4)
5)
6)
0.21
0.279
0.32
0.6
0.17
0.25
7)
0.406
8)
9)
21.3%
33.1%
6%
53%
35.3%
18.0%
18%
19.5%
23.5%
II. Perform the operations; simplify.
1) 3.708 + 0.56
2) −3.708 − 0.56
3) −3.708 + 0.56
4) 3.708 − 0.56
5) 1.2 − 0.12 + 0.012
6) 1.2 − 0.12 − 0.012
7) −1.2 − 0.12 + 0.012
8) −1.2 + 0.12 − 0.012
9) 3.2 + ( − 0.3)
10) 3.2 − ( − 0.3)
11) ( −0.4 ) − 2.351
12) (1.2 ) − (1 − 0.9 )
2
13) 0.7 ( 2.6 ) − ( 4.3 − 5 )
2
2
3
14) (−0.03) 2 + (0.08) 2 15) ( 0.2 ) − 1.2 ( 0.1)
3
2
2
16) 3 ( 0.5 ) + 2 ( 0.5 )
2
3
2
III. “Divide” by moving the decimal point(s), then simplifying fraction. Do not use long division.
1)
2.7
6
2)
1
0.32
3)
0.12
0.8
4)
0.6
0.25
5)
0.54
0.036
6)
0.00048
0.008
7)
61
3.4
4
IV. Express each fraction as a decimal and a per cent.
92
863
1) 1,000
2) 100
3) 2 14
6)
59
40
7)
2
25
8) 3 17
20
3
8
4)
4
5
5)
9)
23
10,000
10)
39
50
V. Express each decimal as a per cent and a simplified fraction/mixed number.
1) 0.6
2) 0.85
3) 3.04
4) 20.26
6) 11.5
7) 0.08
8) 0.802
9) 0.0005
5) 2.9
10) 0.048
VI. Express each per cent as a decimal and as a simplified fraction.
1) 24%
2) 700%
3) 5%
4) 1,234%
6) 1.5%
7) 325%
8) 750%
9) 0.08%
1
3
4
11) 9 2 %
12) 4 %
13) 2 5 %
14) 39
50 %
5) 0.6%
10) 12.9%
15) 8 14 %
VII. Express each per cent as a simplified fraction. (Hint: Avoid long division)
1) 8 13 %
2) 5 95 %
3) 7 12 %
4) 3 117 %
5) 4 83 %
VIII. True or False
2) (23 + 23 ) 2 =
1) (23 ) 2 = 86
26 + 26
5) 9 2 − 4 2 =
52
3) 32 ⋅ 42 =
122
4) ( −2 2 )3 =
−64
Topic 4: Applications to Fractions, Decimals, %
Objective #1: Apply fraction/decimal/per cent skills in application problems
Objective #2: Organize verbal information into a useful chart, diagram, etc.
1. At a school of 38 students, 20 are girls. Ten of the 16 students who play hockey are boys.
a) How many boys attend the school? __________
b) How many girls play hockey? ___________
c) How many students do not play hockey? _____________
d) What simplified fraction of the hockey players are girls? ___________
e) What simplified fraction of the girls plays hockey? ____________
f) What simplified fraction of the boys plays hockey? _____________
g) What % of the girls plays hockey? _____________
2. In a business of 40 employees, 16 are men. Eight of the 18 employees that are accountants are women.
a) How many women work in the office? __________
b) How many male accountants are there? __________
c) What simplified fraction of the employees is male? ___________
d) What simplified fraction of the accountants is female? _____________
e) What % of the employees works in accounting? _____________
f) What % of the men does not work in accounting? ____________
3. A sports club has 40 members, 35% of whom surf. Five-eighths of the club members snow board.
a) How many club members surf? __________
b) How many club members snow board? _________
c) What simplified fraction of the members surfs? ________
d) What simplified fraction of the members does not snow board? ________
e) What % of the members snow boards? ___________
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4. Fifty people were asked to name their favorite ice cream. The results of the survey are below.
Other
a) What % chose “Other”?
Strawberry,
20%
Chocolate,
46%
_________
b) What simplified fraction named strawberry? ________
c) What simplified fraction chose vanilla? _________
d) How many selected:
Vanilla,
24%
vanilla? ___________
chocolate? ___________
5. a) A 32-gram pendant is 12% gold and 5% silver. Find the amount of each precious metal in the pendant.
b) An 80-ounce bottle of sports drink contains 12 ounces of fruit juice. Find the concentration of fruit juice.
c) Calculate a 15% tip for a $72 restaurant bill.
d) During a sale, a pair of $58 shoes is discounted by 20%. Find the sale price of the shoes.
6. Find the perimeter and area of each rectangle. Use correct units.
a)
5 13 in.
b)
1.4 ft.
2 56
0.8
0.43 m
c)
0.2
Topic 5: Expressions, Equations and Applications
Objective #1: Determine whether an expression or equation
Objective #2: Simplify expressions; solve equations
Objective #3: Determine whether a value is a solution to an equation.
I. Classify as either an expression or an equation; simplify/solve, as appropriate.
1. a) 2n − 11 − 3n + 9
b) 2n − 11 =
−3n + 9
2. a) (5a − 1) − (6 − 2a ) = a + 5
b) (5a − 1) − (6 − 2a ) + a + 5
3
3. a) c − 9c + 13 =
5
b) c − 9c − 13 + 53
y − 2 − 65 y + 16
4. a)
1
4
5. a)
3a
a
−5−6−
6
4
y =−2 + 56 y + 16
b)
1
4
b)
a
3a
−5 = 6−
6
4
6. a) 3[ r − (5r − 9)] =−r + 5
b) 3[ r − (5r − 9)] − r + 5
7. a) 0.2 p + 1 − 0.2 − 0.6 p
b) 0.2 p =+
1 0.2 − 0.6 p
II. Simply the expressions; solve the equations.
b
0
1) y − 0.4 y
2)
3) ( 7 r − r + 1) − (3r + 16) =
= 0.3
2.1
4) 4 [w + (5w − 9)]
3
6
5) 3 − n =
11
5
6) −40 =
− n
9
9) 0.12k = 0.009
10)
7 3 1 3
m − m
15
10
7) (5c − 3d )6 + 2(3d + 9c)
8) 3v − 89 = 8v − 89
11) −2 [ 4 − 6(n − 3) ] =
−40
12) 0.3r − 0.19r − 0.24 − 0.85
3
21n 2 ) + 3(−2n + 5n 2 )
(
7
13)
5y
45
= −
36
60
14) y − 0.4 y =
0.42
17)
3c 1 5c 2
+ =
−
8 2 12 3
18)
2 xy 7 xy
3
1
y + 9=
y − 3 19) 7 y −
+
− 15 y
5
5
9
12
21)
−m
= − 0.06
0.3
22)
3
8
y + 76 − 125 y + 65
15) −
3c 1 5c 2
+ − −
8 2 12 3
16)
20) r + 0.2r − 2 =−8
23) 9n − n − 8 = 5 − 13 (9 − 12n)
24) 2.718 − z − 0.27 − 3 z
III. Determine whether the given number is a solution to the equation. Show work!!!
1) Does n = −2 ? YES
NO
2) Does x = −3?
4n 3 + 6 = n − 2n 4
YES
NO
5 − 3x 2 = x3 − 2 x − 1
3) Does y = − 32 ?
YES
NO
y 2 − 12 y = 3
IV. a) The area is 28 meters 2 . b) The angles are complementary.
i) Find y - write equation; solve. i) Find n (in degrees).
ii) Find perimeter.
ii) Find measure of each angle.
c) The angles are complementary.
i) Find x (in degrees).
ii) Find measure of each angle.
3y +1
4
x
2n + 10
3x − 2
42
V. a) Find the supplement of a 32° angle.
VI. The angles are supplementary.
a) Find y (in degrees).
b) Find measure of each angle.
b) Find the complement of a 27.8° angle.
7 y + 30
9y +6
2y
VII. At 2 AM, the temperature registered − 8  F . [NOTE: A difference in temperatures must be positive!]
a) By 6 AM, the temperature had decreased by 6  F . Find the new temperature.
b) By 11 AM, it had increased by 11  F from the 6 AM temperature. Find the new temperature.
c) At 3 PM, the temperature read 19  F . Find the difference between the 3 PM and 2 AM temperatures.
d) Find the difference between the 2 AM and 11 AM temperatures.
VIII. a) Find two points on the number line 4 units from 1.
b) The elevation of the lowest point in Death Valley is -282 feet, while the highest point on
Mt. Whitney is 14,505 feet. Find the difference in their elevations.
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