1
Career
Train
Contextualized Learning Packet
Welding Technology
2
Career
Train
Contextualized Learning Packet
Applied Mathematics
Welding Technology
3
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
Characteristics of Items
Skills
3
Translate easily from a word
problem to a math equation
All needed information is
presented in logical order
No extra information
Solve problems that require a single
type of mathematics operation
(addition, subtraction, multiplication,
and division) usi
ng 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)
Level
Characteristics of Items
Skills
4
Information may be presented out
of order
May include extra, unnecessary
information
May include a simple chart,
diagram, or graph
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 numb
er by a whole
number or decimal
Put the information in the right order
before performing calculations
Level
Characteristics of Items
Skills
5
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)
Decide what information,
calculations, or unit conversions to
use to solve the problem
Look up a for
mula 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 resul
ts
Calculate perimeters and areas of
basic shapes (rectangles and
circles)
Calculate percent discounts or
4
markups
Level
Characteristics of Items
Skills
6
May require considerable
translation from verbal form to
mathematical expression
Generally require considerable
setup and involve multiple

step
calculations
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 vo
lume of rectangular
solids
Calculate multiple rates
Level
Characteristics of Items
Skills
7
Content or format may be
unusual
Information may be
incomplete or implicit
Problems often involve
multiple steps of logic and
calculation
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 ar
eas 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
5
1.
Bob added up the hours on his timecard. He got a total of 37 6/8
hrs.
Reduce his hours to lowest terms.
2.
You need to drill a hole in a piece of metal for an
11
/
16
inch bolt to pass thru
Will a
¾
inch hole be a little too big or little too small?
3.
Arrange these four drill bits in order from smallest (left) to largest (
right)
3
/
8
11
/
32
7
/
16
13
/
32
4.
Which steel bar is longer?
Bar 1 25
1
/
8
inches
or
Bar 2 25
3
/
32
inches
5.
Eleven 2 and1/2 inch pieces are cut from a bar of round stock.
How much material is used?
(Ignore any loss due to cutting
.
)
6.
A rectangular plate which is 36
" w
ide and 49
"
long is to be cut into strips that are 36
"
long and
7
/
8
in. wide
.
Ho
w
many strips can be cut from the p
l
ate?
7.
4
7
/
8
in long pieces are to be cut from a 120" long bar.
How many
complete
pieces can be cut from the bar?
8.
Seven weldments
,
each 1 ¾ in, long
,
are made.
What is the total length of weldments?
6
9.
A support bracket as shown below has equally spaced uprights welded to it. If the
distance between the centers of the first and fourth
uprights is 16
½
"
F
ind the distance between the centers of the first and second uprights. 16 1
1
2"
10.
If one piece of angle iron weighs 24
½
lbs., how much will three pieces weigh?
11.
If three tanks of propane gas lasted 22
3
/
8
days, how long did each tank last?
12.
A 7
¼
in piece and a 9
3
/
8
in are cut from a 49" length of round stock
.
If
1
/
8
in. of waste is
allowed for each cut how much of the round stock remains?
13.
A steel frame is constructed in the welding shop
.
At one corner
,
three pieces of steel
overlap.
If their thicknesses are
3
/
8
”,
3
/
16
”,
5
/
32
”
what will the total thickness be?
14.
Given the following diagram
,
determine the di
s
tance
"
X
"
.
Note: You may assume that the
3
/
8
" dia. hole is centered on the width of
t
he piece
7
15.
What is the total clearance when a
5
/
16
"
bolt is inserted into a
3
/
8
"
hole?
16.
Con
v
ert each of the follo
w
ing dr
i
l
l
bit si
z
e
s
to dec
i
mal form then rearrange the
m in
order
f
rom the
s
m
a
llest
(
le
f
t
)
to the
l
argest
(
r
i
gh
t)
.
3
/
8
5
/
16
7
/
1
6
11
/
3
2
13
/
32
17.
A
¾
i
n.
w
rench is n
e
eded to loosen
a
he
x

head bolt.
What would be the decimal
si
ze
w
rench that could be u
s
ed?
18.
A speci
fi
cation calls for
t
he clearance bet
w
een piston and c
y
linder in an engine to be
3
/
64
in.
What
i
s its decimal equivalent
?
(Nearest ten
thousandth)
19.
Four sheet
s
of sheet metal have the following thicknesses:
3
/
16
in
.
11
/
16
in
.
7
/
32
in
.
1
/
8
in.
Convert these to decimal fo
r
m and then arrange them from thinnest to thickest
.
20.
You are to bore a hole which is 0
.
27 " in d
i
ameter
.
If you
have a drill bit set that is made in increments of 64ths of an inch
,
what size
dri
l
l bit will give a hole
of
at least
0
.
27
"?
21.
Two pieces o
f
metal o
v
e
r
lap as sho
w
n
.
If
th
e
s
e
a
r
e
w
elded together
,
what
is
the total length o
f
the welded piece
?
22.
Three rectangular sections of plate steel are connected by overlaps as shown below.
Using the dimensions given, f
in
d the total length of the welded section.
8
23.
A cross

section of steel channel as shown below has a width of 4.25 inches, with each
flange having a thickness of 0.1875 inches.
Determine the inside width of the channel
.
(
nearest thousandth)
24.
Using the diagram below
,
determine the thickness of the mai
n stem of the I

beam
(dimension "X")
25.
Two holes, 3.6cm and 3
.
2cm in diameter, are drilled into a 20cm long strip of steel bar as
shown below.
Using the given dimensions, find the missing dimension
,
labeled "x” that
separates the edges of the two ho
les.
26.
What is the final cost for this stock order?
Quantity
Description
Unit Price
15
¾" 8' stock
$
3
.
49 per 8' section
10
112" 8' stock
$
2.75 per
8'
section
25
114" 8' stock
$
2.23 per
8'
section
9
27.
Eleven 8 inch long sections of ½” inch diameter round stock is needed.
If 0
.
125" of material is lost from each cut, what will be the total length of material
removed from a 120 inch bar? (Nearest thousandth)
28.
Thirty

five L

shaped pieces are cut from sheet metal which is 0.310" thick. The pieces
are then stacked as shown below.
Wh
at is the height of that stack of metal pieces? (Nearest thousandth)
29.
A strip of metal 20" long has four rectangular slots each 2
.
5" long stamped from it
.
If the spacing between the slots is 1
.
75", how
far
is it from the end of the metal
strip to the edge of the first slot?
30.
A 72" length of angle iron is subdivided into 7 pieces of equal length
.
If
1
/
8
" is allowed for each cut, what is the length of each piece? (Nearest
hundredth)
31.
Calculate
the weight of a product if the parts before welding have the following
weights
:
5 lengths of angle iron, each 3.6Ibs
4 lengths of bar steel, each 1
.
1lbs
2 lengths of round stock
, each 0.85 lbs.
32.
A portable welding unit was purchased at a list price of $795
less a 10% discount plus a
5% sales ta
x
.
What is the selling price
?
20”
10
33.
A series of holes are drilled into a circular flange. The spec calls for diameters of 0.
65
in.
± 0.05"
.
Restate the
t
olerance in terms of percentage. (i
.
e
.
, 0.75 in.±
?%)
34.
The diameter
of certain round stock is 112 in
.
±0.3%.
What maximum diameter could a buyer expect from the stock?
35.
The supply of metal stock in a shop weighs about 3 tons
.
If 0
.
2 ton of this is used
,
what percentage remains? (Nearest percent)
36.
A motor is rated at 85%
efficiency
.
If it has an input power of
1
/
2
HP
,
what would be its output power? (Nearest
hundredth)
37.
A shop does $1
3
00 worth of business one month and $1700 worth of business the next
.
What is the rate of increase? (Nearest whole percent)
38.
A welder
receives a $0
.
24/hour raise.
I
f he used to get $7.50
for
1hr
,
what is the
percent increase
in his salar
y
? (Nearest
tenth of a percent)
39.
A frame is made up of five l

ft sections of solid round stock steel that is 0
.
5" in diameter
each steel bar weighs 0
.
67
lbs. A solid aluminum rod with the same dimensions weighs
0
.
23 lbs.
If the aluminum rods replaced the steel bars mentioned above, what will be the
percent decrease
in weight of the frame
?
Find the measurements for the
questions 40 to 45:
40.
.
41.
11
42.
43.
44.
45.
46.
Find the weight of a 1 ft. long piece of steel round stock if the diameter is 2".
Use the formula:
lbs. per linear ft.
=
3.67
X
D2
47.
Using
t
he
i
nfo
rmation
found
in p
r
oblem #46
,
w
ha
t
'
s
th
e
w
eight o
f
a 65 ft. piece of round
stock?
48.
Find the weight of a
1
ft. piece of aluminum tubing that has dimensions W=.125"
,
and
OD=2
.
375":
U
se the formula
:
lb
s
.
p
er li
n
e
ar
ft.
=
2.69
x
(
OD

W
)
x
W
49.
Ho
w
much would a 23.5 ft. piec
e of the a
bove tubing weigh?
50.
A
rectangular frame measu
r
es
9
6
"
by
48"
.
F
ind the length of the diag
o
nal
,
d
i
mension C
,
using this formula:
C
=
.
J
a
2
+
b
51.
De
t
erm
i
ne the vo
l
um
e
of an o
x
ygen cylinder wi
t
h a height of 24" and a diameter of 6"
,
b
y
using this formu
l
a:
V
=
3.14
x
H
x
D
2 divided by
4
12
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 total length of stock needed to make this table frame?
56.
Find the total area of these plates that are welded together?
13
57.
Find the area and perimeter of this flame

cut steel plate.
s
58.
Compare the areas of these two figures. Since the sides are the same length, are the areas
the same?
59.
Calculate the perimeter and the area of the following.
60.
Calculate the perimeter and the area of the f
ollowing.
14
61.
Calculate the perimeter and the area of the following.
62.
Calculate the perimeter
:
63.
Calculate the perimeter.
64.
Find the missing dimension. (Round to the nearest tenth.)
65.
Find the missing
dimension.
15
66.
Find the missing dimension. (Round to the nearest tenth)
67.
Find the missing dimension.
68.
Find the Area.
16
69.
Find the area. (Round to the nearest tenth)
70.
Find the area. (Round to the nearest whole number)
71.
Find the circumference. (Round to the nearest tenth)
17
72.
Find the area. (Round to the nearest tenth)
73.
Find the area. (Round to the nearest tenth)
74.
Find the area of the ring. (Round to the nearest tent
h)
75.
A welder is required to sheer

cut a piece of sheet steel as shown in the illustration.
After the cut piece is removed, how much sheet, in inches, remains from the original piece?
18
76.
Welded support is illustrated.
A customer
orders 34 supports
:
A.
What, in inches, is the total length of weld needed?
B.
The support plate is 13 inches long and 9 inches wide. How much 9

inch

wide bar
stock, in inches, is used for the completed order?
C.
Each support weighs 14 pounds. What is the weight
in pounds of the total order?
19
M
ake fractions out of the following information
:
(
Reduce, if possible
)
77.
An inch into 8ths
78.
Read the distances from the start of the steel tape measure to the letters. Record the
answers in the proper blanks.
20
79.
Find
the total combined length of these 2 pieces of bar stock
80.
Find the total combined weight of these 3 pieces of steel.
21
81.
To make shims for leveling a shear, three pieces of material are welded together.
What is the total thickness of the welded
material, in inches?
82.
Determine the missing dimension on this welded bracket.
22
83.
A frame

cut wheel is to have the shape shown. Find the missing dimension.
84.
Three of these welded brackets are needed.
What is the total
length, in inches, of the bar stock needed for all of the brackets?
23
85.
This piece of angle is to be used for an anchor bracket.
If the holes are equally spaced, what is the measurement between hole 1 and hole
2?
86.
Nine sections of steel bar, each 12 ¼” long, are welded together. The finished piece is cut
into 4 equal parts.
What is the length of each new piece? Disregard cut waste.
87.
Round off to the nearest whole number.
a.
7.7
_________________
b.
12.1
_________________
c.
9.7
_________________
d.
17.398
_________________
88.
A welder uses 4.18 cubic feet of acetylene gas to cut one flange.
How much acetylene gas is used to cut 19 flanges?
89.
A welder shears key stock into pieces 3.75” long.
How many whole piec
es are sheared from a length of key stock 74.15” long?
24
90.
Express each decimal dimension as a fractional number.
91.
A piece of steel channel and a piece of I beam are needed. Express each dimension as a
decimal number.
25
92.
Find the length of slot 2.
93.
The fillet weld shown has 2’, plus 18” of weld on the other side of the joint.
Express the total amount of weld in feet.
94.
Six welding jobs are completed using 33 pounds, 13 pounds, 48 pounds, 14 pounds, 31
pounds, and 95 pounds of electro
des.
What is the average poundage of electrodes used for each job?
26
95.
This I Beam is 180 cm long and 14.5 cm high.
(
Round ea
ch answer to two decimal
places)
96.
How many square inches are in 2 square foot?
27
97.
These two triangular shapes are cut from sheet metal.
What is the area of each piece in square inches?
1.
Triangle A
________________
ii)
Triangle C ___________
98.
Two pieces of square stock are welded together.
Find, in cubic feet, the total volume of the pieces.
(
Round the answer to three
decimal places
)
28
99.
Circles A and B are cut from 3/8 steel plate.
What is the circumference of both circles in inches
and
i
n
f
eet
?
A.
__________Inches
B. _____________Inches
__________Feet
_____________Feet
100.
How many degrees are in each of these parts of a circle?
i)
1
/
3
Circle
_______________
ii)
5
/
6
Circle
_______________
101.
Find the size of ¼” plate needed to construct this semicircular
ventilation
section.
The average diameter is 19
3
/
16
.
29
102.
A weld shop supplies 104 shaft blanks, each 4” wide and 5” long.
How many can be cut from the piece of plate shown?
103.
The circular bland is used to make sprocket drives.
How many sprocket
drive blanks can be cut from a plate of steel having
the dimensions of 44” x 44”?
30
ANSWER SHEET
Q #
ANSWER
1
37 ¾
2
TO BIG
3
11
/
32,
3
/
8,
13
/
32,
7
/
12
4
BAR 1
5
27.5
6
56 STRIPS
7
24 WHOLE PIECES
8
12 ¼
9
5 ½
10
73 ½ LBS.
11
7.5 DAYS
ROUNDED
12
32
1
/
8
13
23
/
32
14
3
/
16
15
1
/
16
16
0.3125
0.34375
0.375
0.40625
0.4375
17
0.75
18
0.0469 ROUNDED
19
0.125
0.1875
0.21875
0.6875
20
18
/
64
21
13.9
22
44.5 CM.
23
3.875
24
1.5
25
5.2 CM.
26
$135.60
27
89.375
28
10.85
29
2.375
30
10.2
31
23 LBS.
32
$751.27
33
6.7
34
0.5025
35
96.7 ROUNDED
36
0.45 HP.
37
30.80%
38
2.90%
39
65.70%
40
2
5
/
16
41
1
7
/
16
42
4
11
/
16
43
5
3
/
16
44
9
1
/
2
45
9 ¾
46
14.68 LBS.
47
95.42 LBS.
48
1.03 LBS.
49
24.2 LBS.
50
16.73
51
678.24 CU.IN.
52
OBTUSE 112 DEG.
53
ACUTE 49 DEG.
54
RIGHT 90 DEG.
55
249.5
56
28.25 SQ. IN.
57
69 CM.
58
NOT THE SAME
59
P=36IN.
A=54SQ. IN.
60
P=91YDS.
A=390SQ. YD.
61
P=32FT.
A=45SQ. FT.
62
P=16CM.
63
P=44FT.
64
18.3IN..
65
20
IN.
66
13.7 FT.
67
86.6 MTRS.
68
4.5SQ. FT.
69
210.4SQ. FT.
70
65 SQ. FT.
71
100.5 in.
31
72
16.6 sq. mtrs
73
19.6 sq. in.
74
141.4 sq. ft.
75
17
76
272. IN
442 IN
476 LBS
77
1
/
8
IN
3
/
8
IN
½ IN
¾ IN
78
¼ in
5
/
8
IN
1 AND
1
/
8
2 AND
1
/
8
79
8 AND
5
/
8
80
29 AND
3
/
8
81
1 AND
5
/
8
82
6 AND
1
/
8
83
2 AND
9
/
16
84
53 AND
7
/
16
85
3 AND
9
/
74
86
27 AND
9
/
16
87
8
12
10
17
88
79.42CUFT
89
19 PIECES
90
9 AND ½
2 AND ¾
91
2.25 IN
0.1875 IN
6.5625 IN
92
7.0375
93
3 AND ½ FT
94
39 LBS
95
70.87 IN
5.71 IN
96
288 SQ IN
97
32 SQ IN
72 SQ IN
98
2.949 CU FT
99
A=30.35 FT
364.24 IN
B=13.345 FT
160.14 IN
100
120 DEG
300 DEG
101
¾ X
63 AND ¾ IN
102
72 PIECES
103
9 PIECES
32
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.
33
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 wil
l 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
F
raction
–
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 equi
valent. 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 ad
d 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.
34
0.3
“3 tenths”
3
/
10
0.76
“76 hundredths”
76
/
100
0.923
“923 thousandths”
923
/
1000
1.7
“1 and 7 tenths”
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 le
tter 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 “balance
d” 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.
40 x
n =
16 x 8
40
n
= 128
n =
= 3
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, add
ition and subtraction
35
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 i
s multiplied by itself. The base is the number
to be multiplied. The exponent tells how many times the base is multiplied by itself.
2
3
The base is 2. The exponent is 3.
2 x 2 x 2 = 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.
36
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 use
d 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.
37
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, fee
t, 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:
=
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.
Another example of a unit rate is $6 per 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.
=
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 responsib
le for a fleet
of vehicles. You need to determine which brand of engine oil to use with
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
38
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 Suc
cessful 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.
Calculate the
percentage that
was lost.
5 milliliters of
the synthetic oil
was consumed
1 gallon = 4 qts.
4 quarts = 1 liter
1 liter = 0.264
gal.
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
quarts to gallons
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
= 0.012 x 100% = 1.2%
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 per
cent.
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%
The symbol ≈ mean
s “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 a
re often rounded at some point during the calculation
39
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 combin
ed
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 t
he 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
40
41
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 2
nd
hit prb hit enter
Sales Tax
Sales Tax
=
Tax Rate
Cost 100
Interest
Annual Interest
=
Annual Interest
Rate
42
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
43
44
45
46
PERCENT
PROBLEMS
The Percent (%)
The Whole (OF)
The Part (IS)
47
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 2
nd
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 2
nd
button,
then SIN, COS, or TAN; enter number and hit equals.
You would use this when you need an answer in degrees.
48
Applied Mathematics Formula Sheet
Distance
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
Area
1 square foot = 144 inches
1 square yard = 9 square feet
1 acre = 43,560
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
perimeter = 2(length + width)
area = length x width
Rectangle Solid (Box)
volume = length x width x height
Cube
volume = (length of side)
3
Triangle
sum of angles = 180
o
area = ½(base x height)
Circle
number of degrees in a circle = 360
o
circumference ≈ 3.14 x diameter
area ≈ 3.14 x (radius)
2
Cylinder
volume ≈ 3.14 x (radius)
2
x height
Cone
volume
≈
3.14 × (radius)
2
× 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(
o
F

32)
or
5
/
9
(
o
F

32)
o
F = 1.8(
o
C) + 32
or
(
9
/
5
x
o
C) + 32
49
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