CareerTrain
Contextualized Learning Packet
CIM
(Computer Integrated Manufacturing)
CareerTrain
Contextualized Learning Packet
Applied Mathematics
CIM
(Computer Integrated Manufacturing)
What the WorkKeys Applied Mathematics Test Measures
There are five levels of difficulty. Level 3 is the least complex, and Level 7 is the most complex. The levels
build on each other, each incorporating the skills assessed at the previous levels.
Level
3
Characteristics of Items



Translate easily from a word
problem to a math equation
All needed information is
presented in logical order
No extra information
Skills




Level
4
Characteristics of Items



Information may be presented out
of order
May include extra, unnecessary
information
May include a simple chart,
diagram, or graph
Skills







Level
5
Characteristics of Items

Problems require several steps of
logic and calculation (e.g., problem
may involve completing an order
form by totaling the order and then
computing tax)
Solve problems that require a single
type of mathematics operation
(addition, subtraction, multiplication,
and division) using whole numbers
Add or subtract negative numbers
Change numbers from one form to
another using whole numbers,
fractions, decimals, or percentages
Convert simple money and time
units (e.g., hours to minutes)
Solve problems that require one or
two operations
Multiply negative numbers
Calculate averages, simple ratios,
simple proportions, or rates using
whole numbers and decimals
Add commonly known fractions,
decimals, or percentages (e.g., 1/2,
.75, 25%)
Add up to three fractions that share
a common denominator
Multiply a mixed number by a whole
number or decimal
Put the information in the right order
before performing calculations
Skills






Decide what information,
calculations, or unit conversions to
use to solve the problem
Look up a formula and perform
single-step conversions within or
between systems of measurement
Calculate using mixed units (e.g.,
3.5 hours and 4 hours 30 minutes)
Divide negative numbers
Find the best deal using one- and
two-step calculations and then
compare results
Calculate perimeters and areas of
basic shapes (rectangles and
circles)

Level
6
Characteristics of Items


May require considerable
translation from verbal form to
mathematical expression
Generally require considerable
setup and involve multiple-step
calculations
Skills









Level
7
Characteristics of Items



Content or format may be unusual
Information may be incomplete or
implicit
Problems often involve multiple
steps of logic and calculation
Calculate percent discounts or
markups
Use fractions, negative numbers,
ratios, percentages, or mixed
numbers
Rearrange a formula before solving
a problem
Use two formulas to change from
one unit to another within the same
system of measurement
Use two formulas to change from
one unit in one system of
measurement to a unit in another
system of measurement
Find mistakes in questions that
belong at Levels 3, 4, and 5
Find the best deal and use the
result for another calculation
Find areas of basic shapes when it
may be necessary to rearrange the
formula, convert units of
measurement in the calculations, or
use the result in further calculations
Find the volume of rectangular
solids
Calculate multiple rates
Skills







Solve problems that include
nonlinear functions and/or that
involve more than one unknown
Find mistakes in Level 6 questions
Convert between systems of
measurement that involve fractions,
mixed numbers, decimals, and/or
percentages
Calculate multiple areas and
volumes of spheres, cylinders, or
cones
Set up and manipulate complex
ratios or proportions
Find the best deal when there are
several choices
Apply basic statistical concepts
1.
The stockroom has eight boxes of No. 10 hex head cap screws.
How many screws of this type are in stock if the boxes contain 246, 275, 84, 128,
325, 98, 260, and 120 screws, respectively?
2.
In calculating her weekly expenses, a machinist found that he had spent the following amounts:
materials, $11,860; labor, $3854; salaried help, $942; overhead expense, $832.
What was his total expense for the week?
3.
The head machinist at Tiger Tool Co. is responsible for totaling time cards to determine job
costs. He found that five different jobs this week took 78, 428, 143, 96, and 284 minutes
each.
What was the total time in minutes for the five jobs?
4.
A machinist needs the following lengths of l in. diameter rod: 8 in., 14 in., 6 in., 17 in., and 42 in.
How long a rod is required to supply all five pieces? (Ignore cutting waste.)
5.
A machinist needs 25 lengths of steel each 11 in. long.
What is the total length of steel that he needs? No allowance is required for cutting.
6.
The Ace Machine Company advertises that one of its machinists can produce 2 parts per
hour.
How many such parts can 26 machinists produce if they work 45 hours each?
7.
If a machine produces 15 screws per minute, how many screws will it produce in 24 hours?
8.
A machinist has a piece of bar stock 243 in. long.
If she must cut 12 equal pieces, how long will each piece be? (Assume no waste to get a
first approximation.)
9.
A machine shop bought 14 steel rods of 7/8 in.-diameter steel, 23 rods of ½ in. diameter, 9
rods of ¼ in. diameter, and 19 rods of l-in. diameter.
How many rods were purchased?
10. If it takes 45 minutes to cut the teeth on a gear blank.
How many hours will be needed for a job that requires cutting 34such gear blanks?
11. The Ace Machine Tool Co. received an order for 15,500 flanges.
If three dozen flanges are packed in a box, how many boxes are needed to ship the order?
12. A machinist is deciding between two job offers. One is with Company A, paying him $24 per hour
plus health benefits. The other is with Company B, paying him $28 per hour with no health benefits.
Health insurance would cost him $500 per month. Assume he works an 8-hour day and an average of
22 work days per month, then answer the following questions:
a. How much would he earn per month with Company A?
b. How much would he earn per month with Company B?
c. What are his net earnings with Company B after paying for his health benefits?
d. Which company provides the best overall compensation?
13. What is the shortest bar that can be used for making 5 chisels each 6 1/8 in. in length?
14. How long will it take to machine 44pins if each pin requires 6 3/4 minutes?
Allow 1 minute per pin for placing stock in the lathe.
15. How many pieces 6 1/2 in. long can be cut from 35 metal rods each 40 in. long?
Disregard waste.
16. The architectural drawing for a room measures 3 5/8 in.by 4 1/4in.
If 1/4 in. is equal to 1 ft. on the drawing, what are the actual dimensions of the room?
17. The feed on a boring mill is set for 1/64 in. How many revolutions are needed to advance the tool 3 3/8
in.?
18. If the pitch of a thread is 1/18 in., how many threads are needed for the threaded section of a pipe to
be 2 1/2in. long?
19. What is the combined thickness of these five shims: 0.008, 0.125, 0.150, 0.185, and 0.007 in.?
20. A certain machine part is 2.327 in. thick.
What is its thickness after 0.078 in. is ground off?
21. The diameter of a steel shaft is reduced 0.006 in. The original diameter of the shaft was
0.850 in.
Calculate the reduced diameter of the shaft.
22. A shop tech earns a base pay of $19.28 per hour, plus "time-and-a-half' for overtime (time
exceeding 40 hours).
If he works 43.5 hours during a particular week, what is his gross pay?
23. For the following four machine parts, find W, the number of pounds per part; C, the cost of
the metal per part; and T, the total cost.
Metal
Parts
Number
of Inches
Needed
Number
of Pounds
per Inch
Cost
per
Pound
A
44.5
0.38
$0.98
B
122.0
0.19
$0.89
C
108.0
0.08
$1.05
D
9.5
0.32
$2.15
Pounds
(W)
Cost
per Part
(C)
T=
24. How much does 15.7 sq. ft. of No. 16 gauge steel weigh if 1 sq. ft. weighs 2.65 lb.?
25. A machinist estimates the following times for fabricating a certain part: 0.6 hour for setup,
2.4 hours of turning, 5.2 hours of milling, 1.4 hours of grinding, and 1.4 hours of drilling.
What is the total time needed to make the part?
26. A shop technician earns $18.26 per hour plus time-and-a- half for overtime (time exceeding 40
hours).
If he worked 42.5 hours during a particular week, what would be his gross pay?
27. A mower motor rated at 2.0 hp is found to deliver only 1.6 hp when connected to a transmission
system.
What is the efficiency of the transmission?
28. A machinist can produce 14 parts in 40 min.
How many parts can the machinist produce in 4 hours?
29. A machinist creates 2lb of steel chips in fabricating 16 rods.
How many pounds of steel chips will be created in producing 120 rods?
30. If 28 tapered pins can be machined from a steel rod 12 ft. long, how many tapered pins can be made
from a steel rod 9 ft. long?
31. A 9-in. pulley on a drill press rotates at 960 rpm. It is belted to a 5-in. pulley on an electric
motor.
Find the speed of the motor shaft.
32. A cylindrical oil tank 8 ft. deep holds 420 gallons when filled to capacity.
How many gallons remain in the tank when the depth of oil is 51 ft.?
33. It is known that a cable with a cross-sectional area of 0.60 sq. in. has a capacity to hold 2500
lb.
If the capacity of the cable is proportional to its cross-sectional area, what size cable is
needed to hold 4500 lb.?
34. A pair of belted pulleys has diameters of 20 in. and 12 in., respectively.
If the larger pulley turns at 2000 rpm, how fast will the smaller pulley turn?
35. A casting weighed 1461b out of the mold. It weighed 134 lb. after finishing.
What percent of the weight was lost in finishing?
36. Specifications call for a hole in a machined part to be 2.315 in. in diameter.
If the hole is measured to be 2.318 in., what is the machinist's percent error?
37. Six steel parts weigh 1.8 lb.
How many of these parts are in a box weighing 142lb if the box itself weighs 7 lb.?
38. In a closed container, the pressure is inversely proportional to the volume when the temperature is
held constant.
Find the pressure of a gas compressed to 0.386 cu ft. if the pressure is 12.86 psi at 2.52 cu ft.
39. If 250 ft. of wire weighs 22 lb., what will be the weight of 150 ft. of the same wire?
40. A 156-in. twist drill with a periphery speed of 50.0 ft. /min has a cutting speed of 611 rpm
(revolutions per minute).
Convert this speed to rps (revolutions per second) and round to one decimal place
41. A piece 7 1/16 in. long is cut from a steel bar 28 5/8 in. long.
How much is left
Find the lengths marked on the following rules:
42
43
44
45
46
47
48. The total length of three pipes is 86 inches. The middle-sized pipe is 6 inches longer than the
smallest. The largest is twice as long as the smallest.
How long is each pipe?
49. The Formula L=2d+3.26 (r+4) can be used under certain conditions to approximate the length L of
belt need to connect two pulleys of radii rand R if their centers are a distance d apart.
How far apart can two pulleys be if their radii are 8 inches and 6 inches and total
length of the belt connecting them is 82 inches? (Round to the nearest inch)
50. Suppose that on the average, 3 % of the parts produced by a particular machine have proven to be
defective. Then the formula N-0.03N=P will give the number of parts N that must be produced in
order to manufacture a total of P non defective ones.
How many parts should be produced by this machine in order to end up with 7500
non defective ones?
51. One-tenth of the parts tooled by a machine are rejects.
How many parts must be tooled to ensure 4500 acceptable ones?
52. Identify the type of angle and give its measurement.
53. Identify the type of angle and give its measurement.
54. Identify the type of angle and give its measurement.
55. Find the Area. Round to the nearest tenth.
56. Find the area. Round to the nearest tenth. Use n = 3.1416
57. Find the area of the shaded region. Round to the nearest tenth. Use n-3.1416
58. Find the area of the ring. Round to the nearest tenth.
59. Find the area.
60. Find the area of the shaded region. Round to the nearest tenth.
61. Find the area of the entire figure.
62. Find the missing dimension.
63. Find the missing demension. Round to the nearest tenth.
64. Find the missing dimension.
65. Find the missing dimension.
66. Find the missing dimension. Round to the nearest tenth.
67. Find the
lateral surface area and the volume.
68. Find the lateral surface area and the volume.
69. Find the lateral surface area and the volume.
70. Find the
lateral surface area and the volume. Round to the nearest
hundredth.
71. Find the lateral surface area and the volume. Round to the nearest thousandth.
72. Calculate the gallons of water needed to fill this swimming pool. Round the the nearest gallon.
73. Find the lateral surface area and the volume. Round to the nearest hundreth.
74. Find the lateral surface area and volume. Round to the the nearest hundreth.
75. Find the lateral surface area and volume. Round to the the nearest hundreth.
76. Write the angle in radians. Round to the nearest hundredth.
45 o
77. Write the angle in radians. Round to the nearest hundredth.
122 ¾ o
78. Write the angle in radians. Round to the nearest hundredth.
20.28 o
79. Write the angle in degrees. Round to the nearest hundredth.
3.14 radians
80. Write the angle in degrees. Round to the nearest hundredth.
1.34 radians
81. Write the angle in degrees. Round to the nearest hundredth.
0.08 radians
82. Find the value of C and a. Round to the nearest tenth.
83. Find the value of A. Round to the nearest tenth.
84. Find the values of A, B, and a. Round to the nearest tenth.
85. Find the value of C. Round to the nearest tenth.
86. Find the values of A, C and b. Round to the nearest tenth.
87. Find the value of A. Round to the nearest tenth.
88. Find the values of A, B, and a. Round to the nearest tenth.
89. Find the values of B and C. Round to the nearest tenth.
90. Find the trig value. Round to 3 decimal places.
sin 25o 40’
91. Find the trig value. Round to 3 decimal places.
65o 35’
92. Find the trig value. Round to 3 decimal places.
sin 85o
93. Find the acute angle x. Round to the nearest minute.
tan x = 0.454
94. Find the acute angle x. Round to the nearest minute.
cos x= 0.683
95. Find the acute angle x. Round to the nearest minute.
cos x = 0.821
96. Find the acute angle x. Round to the nearest minute.
sin x = 0.962
97. Find the acute angle x. Round to the nearest minute.
tan x = 0.332
98. Find a. Round to the nearest hundredth of a degree.
99. Find a. Round to the nearest hundredth of a degree.
100.
Find C. Round to the nearest hundredth.
101.
Find B. Round to the nearest hundredth.
102.
Find C. Round to the nearest hundredth.
103.
Find a. Round to the nearest hundredth degree.
104.
Find B. Round to the nearest hundredth.
105.
Find a. Round to the nearest hundredth degree.
106.
Find A. Round to the nearest thousandth.
107.
Find B. Round to the nearest tenth.
ANSWER KEY
Q#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
ANSWER
1536 SCREWS
$17,488
1129 MIN.
87 IN.
275 IN
2430 PARTS
21600 SCREWS
18 IN.
65 RODS
25.5 HRS.
431 BOXES
A=4224
B=4928
C=4428
D=COMPANY B
36 ¾ IN.
341
210
216
104
45
0.475
2.249
0.844
$857.96
$52.83
41.6 LBS.
11 HRS.
$798.87
80%
84 PARTS
15 LBS.
21 PINS
1728 RPM
267.75 GAL.
1.08 SQ. IN.
3320 RPM
8.30%
13%
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
450 PARTS
83.96 PSI
13.2 LBS.
10.2 RPS
21 9/16
A= 5/8
B= 17/8
C=2½
D=31/8
E= 4/16
F=11/16
G=210/16
H=3½
A= 10/32
B= 24/32
C= 11/32
D= 117/32
E= 8/24
F= 34/64
G= 53/64
H= 128/64
A= 2/10
B= 5/10
C= 1 3/10
D= 16/10
E= 25/100
F= 72/100
G=148/100
H=174/100
20,26 AND 40 IN
18 IN.
7,732 PARTS
5,000 PARTS
OBTUSE 112 DEG.
ACUTE 49 DEG.
RIGHT 90 DEG.
19.6 SQ. IN.
219.3 SQ. CM
75.4 SQ. CM.
141.4 SQ. FT.
1728
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
45.5
106
20
13.7
86.6
66
18.3
140 SQ. IN
512 SQ. IN
680 SQ. FT.
72 SQ. FT.
355.572 SQ. CM.
732.939 CU. CM.
19747 GALLONS
424.12 SQ. MTRS.
1,017.88 CU. CM.
431.97 SQ. FT.
1237 CU. FT.
472.37 SQ. FT.
829.38 CU. FT.
0.79 RAD.
2.14RAD.
0.35RAD.
179.91 DEG.
76.78 DEG.
4.58 DEG.
C=7.1 IN.
A=45DEG.
13.6 IN.
A=5IN
B=8.7IN.
A=30 DEG.
24.2 CM.
A=13.9IN
C=16 IN
B=60 DEG.
10.6 IN..
A=7.5IN
B=7.5 IN
A=45 DEG.
B=6IN.
C=8.5IN.
0.433
2.203
0.996
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
24 DEG. 25 MIN.
46DEG 55MIN.
34 DEG 49 MIN
74 DEG 9 MIN
48.1 DEG
32.58 DEG
52.25DEG
47.32IN
20.53IN
21.21FT
26.57DEG
6.18FT
51.99DEG
13.192 MTRS
19.3CM
Mathematical Points to Remember
and
Problem Solving Tips
Addition
Use addition in order to find the total when combining two or more
amounts.
Subtraction
Use subtraction in order to:
 Determine how much remains when taking a particular amount
away from a larger amount
 Determine the difference between two numbers
Multiplication
Use multiplication to find a total when there are a number of equally sized groups.
Division
Use division to:
 Split a larger amount into equal parts
 Share a larger amount equally amount a certain number of people or groups
Calculating Time
When solving problems that involve time, using a visual aid such as an analog
clock can be very helpful.
26
Time
When adding time, be careful to distinguish between A.M. and P.M
times. If you begin at a P.M. time and the elapsed time takes you past
midnight the ending time will likely be in A.M. If you start from an
A.M. time and the elapsed time takes you past noon, the ending time
will likely be in P.M. time. For instance, if you start sleeping at 10
P.M. and you sleep for 8 hours, the time you will wake up is going to
be in the A.M. To calculate, add the hours, and then subtract 12 from
the total – 10 + 8 = 18 hours; 18 hours – 12 hours = 6 hours past midnight or 6 A.M.
Fraction/Decimal/Percent
 Fraction – identifies the number of parts (top number) divided by the
total number of pars in the whole (bottom number)
 Decimal – place values to identify part of 1, written in tenths,
hundredths, thousandths, etc.
 Percent – part of 100.
Remember!
A decimal number reads the same as its fractional equivalent. For example, 0.4 = four tenths =
4
/10; 0.15 = fifteen hundredths = 15/100
When working with fraction and decimal quantities that are greater than 1,
remember that these numbers can be written as the number of wholes plus the
number of parts. For example, 2.5 can be written as 2 + 0.5 (two wholes plus
five-tenths of another whole). The mixed number 2 ½ can be written as 2 + ½
(2 wholes plus half of another whole). When converting these numbers, the
whole number stays the same. Always remember to add the whole number back
to the fraction or decimal after you have completed converting.
Multiplying fractions by fractions
Decimals are named by their ending place value – tenth’s, hundredths, thousandth’s, etc. This
makes it easy to convert to fractions.
27
0.3
“3 tenths”
3
0.76
“76 hundredths”
76
0.923
“923 thousandths”
923
1.7
“1 and 7 tenths”
/10
/100
/1000
1 7/10
When you multiply a fraction by another fraction, the result is the product of the numerators over
the product of the denominators.
4
/5 x 2/3 = 8/15
To multiply a fraction by a decimal, convert the fraction to a decimal:
½ x .25 = .5 x .25 = .125
Basic Algebra
Basic algebra involves solving equations for which there is a missing value. This value is often
represented as a letter; such as the letter x or n.
Solving equations for a missing value requires you to understand opposite operations. Addition
and subtraction are opposite operations as well as multiplication and division. You use opposite
operations so that an equation can remain “balanced” when solving the missing value.
Proportions
Multiple operations are using when solving proportions. After the proportion statement is set up,
multiply in order to find cross products. Then divide each side of the equation by the factor being
multiplied by the unknown variable to solve for the unknown variable.
𝑛
8
=
16
40
40 x n = 16 x 8
40n = 128
n=
128
40
1
=35
Order of Operations
When calculations require you to more than one operation, you must follow the order of
operations. Any operation containing a parenthesis must be calculated first. Exponents come next
in the order of operations, followed by multiplication and division, addition and subtraction
28
come last. An easy way to remember the order of operation is: PEMDAS or Please Excuse My
Dear Aunt Sally – Parenthesis/Exponents/Multiplication/Division/Addition/Subtraction
Exponents
An exponent is an expression that shows a number is multiplied by itself. The base is the number
to be multiplied. The exponent tells how many times the base is multiplied by itself.
23
The base is 2. The exponent is 3.
2x2x2=8
Multiplying Negative Numbers
Multiplying negative numbers is similar to multiplying positive numbers
except for two rules:
 When multiplying a positive number and a negative number, the
answer is always negative
8 x (-6) = -48
 When multiplying two negative numbers, the answer is always
positive.
-2 x (-7) = 14
By knowing the rules of multiplying positive and negative numbers, you can rule out
incorrect answers before performing any calculations.
Perimeter Measures
Perimeter measures the length of the outer edge of a shape. The space enclosed within this edge
is measured by area. Area is a two-dimensional measurement that measures the number of square
units of a surface.
29
Formulas for Perimeter and Area of Rectangles
To understand the formulas for finding perimeters and area, consider the figure on the next
page, which is 3 units wide by 5 units long.
 Perimeter: by counting the number of units on each side of the rectangle, you find that
the perimeter is 16 units.
 Area: Area is a 2 dimensional (2D) measurement that measures a surface. By counting
the total number of squares that make up the rectangle, you find that its area is 15 square
units. So the formula is:
area = length x width
Volume is a 3 dimensional (3D) measurement that measures the amount of space taken up by an
object. Like area, you need to know the length and width of an object in order to calculate
volume. In addition to this, you need to know the object’s height. Volume is measured in cubic
units.
Use the formula V = 1 x W x h
Convert Measurements
In the United States, there are two systems of measurements; the
traditional (standard) system and the metric system. Gasoline is usually
sold by the gallon (standard), and large bottles of soda are sold by the
liter (metric).
The Metric System
The metric system of measurement is used by most of the world. Units
of length are measured in centimeters, meters, and kilometers. Units of
volume (capacity) include liters and milliliters. Units of weight include
milligrams, grams, and kilograms. The metric system follows the base -10 system of numeration.
This system is commonly used in sciences and medicine.
30
The Customary/Standard System
The customary or standard system of measurement is the system most commonly used in
everyday life in the United States. Units of length include inches, feet, and miles. Units of
volume include cups, quarts, and gallons. Units of weight include ounces, pounds and tons.
Unlike the metric system, the standard system of measurement does not follow the base -10
system.
If you are unsure of whether to multiply or divide to convert from one unit of measurement to
another, you can set up the problem as a proportion. Here is an example:
1 liter
=
x liters
0.264 gallons 21 gallons
By finding the cross products, you see that:
0.264x = 21
The final step needed to solve is to divide both sides of the equation by 0.264, which gives you
the answer of x = 79.5 liters.
What’s the best deal? Use Ratios and Proportions to find the outcome
A rate is a kind of ratio. Rates compare two quantities that have different units of
measure, such as miles and hours.
Unit Rates
Unit rates have 1 as their second term. An example of unit rate is $32 per
hour.
$32
1 hour
Another example of a unit rate is $6 per page
$6
1 page
Proportions
Proportions show equivalent ratios. You may find it helpful to use proportions to solve problems
involving rates. Calculate the total cost based on the hourly rate.
To find the total cost based on an hourly rate, multiply the number of hours worked by the hourly
rate.
$32
$480
=
1 hour 15 hours
Convert Between Systems of Measurement
When solving problems that involve converting from one unit of
measurement to another, you typically should first determine to which unit of
measurement you should be converting.
For example:
You are the service manager for a corporation and are responsible for a fleet
of vehicles. You need to determine which brand of engine oil to use with
31
your fleet. There are two brands that you are deciding between. So, you decided to run a test
between the two brands. On average, a vehicle burned 5 milliliters of the more expensive
synthetic blend. The average consumption of regular engine oil was 64 milliliters. Each vehicle
holds 5.8 quarts of engine oil. What percentage of the regular oil was lost during the test?
A. 0.5%
B. 1.2%
C. 3.2%
D. 5.6%
E. 9.1%
Plan for Successful Solving
What am I asked
to do?
What are the
facts?
How do I find
the answer?
Is there any
unnecessary
information?
What prior
knowledge will
help me?
Find the percent
of regular engine
oil that was used
The engine holds
5.8 quarts, 64 ml
of oil was lost
Convert one
measurement to
the same system
as the other.
5 milliliters of
the synthetic oil
was consumed
1 gallon = 4 qts.
1 liter = 0.264
gal.
Calculate the
percentage that
was lost.


4 quarts = 1 liter
1 liter = 1,000
milliliters
Confirm your understanding of the problem and revise your plan as needed.
Based on your plan, determine your solution approach: I am going to convert the quarts
to milliliters and then find the percent of the total that was lost.
5.8 quarts ÷ 4 = 1.45 gallons
Divide to convert
1.45 gallons ÷ 0.264 ≈ 5.492 liters
Divide to convert gallons to liters
5.492 liters x 1,000 = 5,492 milliliters
Multiply to convert liters to milliliters
64 𝑚𝑖𝑙𝑙𝑖𝑙𝑖𝑡𝑒𝑟𝑠
5,492 𝑚𝑖𝑙𝑙𝑖𝑙𝑖𝑡𝑒𝑟𝑠



= 0.012 x 100% = 1.2%
quarts to gallons
Divide the amount of oil that was lost by the initial
total to calculate the percent of lubricant that was
consumed.
Check your answer. You can solve the problem another way by converting the milliliters
to quarts and finding the percent.
Select the correct answer: B. 1.2%
By converting the units of measure to the same system, you can calculate the percent of
oil lost in the test by dividing the amount consumed by the total capacity and multiplying
by 100%
32
The symbol ≈ means “approximately equal to” and is used because the
conversion formula between gallons and liters is not exact. When
calculating conversions between measurements for which the
conversions are not exact, you must take into account the fact that the
numbers are often rounded at some point during the calculation
33
BASIC ALGEBRA RULES
1.
DO BRACKETS FIRST
Example: ( )
2.
[ ]
WHEN YOU ARE ADDING OR SUBTRACTING NUMBERS:
IF YOU HAVE MORE POSITIVES THAN NEGATIVES NUMBERS YOUR
ANSWER WILL BE A PLUS ANSWER.
Example: -4 + 7 equals +3
3.
WHEN YOU ARE ADDING OR SUBTRACTING NUMBERS:
IF YOU HAVE MORE NEGATIVES THAN POSITIVES NUMBERS YOUR
ANSWER WILL BE A MINUS ANSWER
Example: -7 + 4 equals -3
4.
WHEN YOU ARE MULTIPLYING OR DIVIDING NUMBERS
LIKE SIGNS ARE POSITIVE AND UNLIKE SIGNS ARE MINUS
Example: (+ and + or -+- +) equal a plus sign
(- and +) equals minus
5.
WHEN ADDING OR SUBTRACTING EXPONENTS
LIKE EXPONENTS CAN ONLY BE ADDED TOGETHER
Example: x to the second power can be combined
With another x to the second power only
6.
WHEN YOU ARE MULTIPLYING WHOLE NUMBERS
7.
THEY ARE MULTIPLIED, AND EXPONENTS ARE ADDED TOGETHER
Example: 3x to the third power times 2x to the second power
equals 6x to the fifth power
8.
WHEN YOU DIVIDE NUMBERS THEY ARE DIVIDED AS USUAL AND
EXPONENTS ARE SUBTRACTED FROM EACH OTHER
Example: 16m to the third power divided by 4m
equals 4m to the second power
34
35
Formulas 1
Gear Ratio = Number of Teeth on the Driving
Gear
Number of Teeth on the Driven
Gear
Reduce to Lowest Terms
Pulley Ratio = Diameter of Pulley A
Diameter of Pulley B
Reduce to Lowest Terms
Compression Ratio = Expanded Volume
Compressed Volume
Reduce to Lowest Terms
A Proportion is 2 Ratios that are =
Example 1/3 = 4/12
Cross Product Rule
A
/B = C/D or A x D = B x C
Pitch = Rise
Run
Changing a Decimal to a %
Multiply by 100
Changing a Fraction to a %
Divide the Numerator by the
Denominator and Multiply by 100
Changing a % to a Decimal
Divide by 100
P
/B = R/100
When P is unknown
When R is unknown
When B is unknown
Changing a decimal to a fraction
.375 hit 2nd hit prb hit enter
Sales Tax
Sales Tax = Tax Rate
Cost
100
Interest
Annual Interest = Annual Interest
Rate
36
Principal
100
Commission
Commission Sales = Rate
Sales
100
Efficiency
Output = Efficiency
Input
100
Tolerance
Tolerance = % of Tolerance
Measurement
100
% of Change
Amount of Increase = % of
Increase
Original Amount
100
Discounts
Sales Price = List Price –
Discount
37
38
39
40
PERCENT PROBLEMS
The Percent (%)
The Whole (OF)
The Part
(IS)
41
Trig Formulas
1. Change an angle to radians = angle times pie divided by 180
2. Change an angle to degrees = radians times 180 divided by pie
3. 30 deg., 60 deg., 90 deg., triangle; the short end is equal to ½ the hypotenuse or the
hypotenuse = 2 times the short end
4. 45 deg., 45 deg., 90 deg., triangle – the 2 shorter sides are the same length and the
hypotenuse is 1.4114 times the leg
5. Find trig value – put in SIN, COS, or TAN followed by degrees and hit enter
6. Find acute angle X – hit 2nd button, then SIN, COS, or TAN; enter number and hit equals. Hit
RP move arrow to DMS hit enter twice
You would use this when you need an answer in degrees, minutes, and or seconds
7. Find acute angle X – hit 2nd button, then SIN, COS, or TAN; enter
You would use this when you need an answer in degrees.
number and hit equals.
42
Applied Mathematics Formula Sheet
Distance
Rectangle
1 foot = 12 inches
1 yard = 3 feet
1 mile = 5,280 feet
1 mile ≈ 1.61 kilometers
1 inch = 2.54 centimeters
1 foot = 0.3048 meters
1 meter = 1,000 millimeters
1 meter = 100 centimeters
1 kilometer = 1,000 meters
1 kilometer ≈ 0.62 miles
perimeter = 2(length + width)
area = length x width
Area
Triangle
1 square foot = 144 inches
1 square yard = 9 square feet
1 acre = 43,560
sum of angles = 180o
area = ½(base x height)
Volume
1 cup = 8 fluid ounces
1 quart = 4 cups
1 gallon = 4 quarts
1 gallon = 231 cubic inches
1 liter ≈ 0.264 gallons
1 cubic foot = 1,728 cubic inches
1 cubic yard = 27 cubic feet
1 board = 1 inch by 12 inches by 12 inch
Weight
1 ounce ≈ 28.350
1 pound = 16 ounces
1 pound ≈ 453.592 grams
1 milligram = 0.0001 grams
1 kilogram = 1,000 grams
1 kilogram ≈ 2.2 pounds
1 ton = 2,000 pounds
Rectangle Solid (Box)
volume = length x width x height
Cube
volume = (length of side)3
Circle
number of degrees in a circle = 360o
circumference ≈ 3.14 x diameter
area ≈ 3.14 x (radius)2
Cylinder
volume ≈ 3.14 x (radius)2 x height
Cone
2
volume ≈ 3.14 × (radius) × height
3
Sphere (Ball)
volume ≈ 4/3 x 3.14 x (radius)3
Electricity
1 kilowatt-hour = 1,000 watt-hours
Amps = watts ÷ volts
Temperature
o
C = 0.56(oF-32) or 5/9(oF-32)
o
F = 1.8(oC) + 32 or (9/5 x oC) + 32
43
44