CareerTrain - HVAC

Urban and Civil

Nov 29, 2013 (4 years and 5 months ago)

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1

Career
Train

Contextualized Learning Packet

HVAC

2

Career
Train

Contextualized Learning Packet

Applied Mathematics

HVAC

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
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 th
e 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

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.

A forced convection heating system is installed in a house. The length 6
-
inch circular duct
that a
r
e needed are: kitchen, 6 feet
;
dining room, 12 feet; living room
,
3 feet; maste
r
bedroom
,
5 feet
;
second bedroom
,
7 feet; third bedroom
,
14 feet
.

How many feet of 6
-
inch round duct are needed?

2.

An air
-
conditioning shop orders the following amounts of refrigerant: 125 pounds of R
-
134a
,
150 pounds of R
-
125, 70 pounds of R
-
124
,
90 pounds of R
-
32
,
and 140 pounds of R
-
152a.

What is the tota
l
weight of refrigerant ordered?

3.

Last year, the Keep Kool Company's five repair trucks covered 7
,
252 miles; 8,917 miles; 4,266 mi
l
es
;
7
,
793 miles; and 9,214 miles.

What is the total mileage the Keep Koo
l
Comp
any should
r
eport for last year?

4.

A 162
-
foot roll of #10 wire is used when installing a residential air
-

conditioning unit. Lengths of 17
feet and 38 feet are cut from the roll.

How many feet are left?

5.

A full oil tank holds 280 gallons of #2 fuel oil. Du
ring 4 months, the amounts of fuel oil used are: 19
gallons, 18 gallons, 53 gallons, and 123 gallons.

How much fuel oil is left in the tank?

6.

A large house has 5,676 sq. ft. of floor space and is divided into 3 heating/ cooling zones.
The first zone, the
upstairs, has 1,625 sq. ft. The main living area is the second zone covering
2,130 sq. ft of space. The third zone is the basement where the rec room, laundry, half bath,
and home shop are located.

How much floor space is included in the third zone?

7.

Ther
e are 144 electrical connectors in a box.

How many connectors are there in 8 boxes?

8.

A technician charges \$16 per hour for labor.

How much should be charged for an 18
-
hour job?

9.

There are 26 cylinders of refrigerant R
-
125 in a stockroom. Each cylinder contains 137
pounds of refrigerant.

How many pounds of R
-
125 are in the stockroom?

10.

A crate contains 6 compressors. The 6 compressors weigh 332 pounds.

What is the weight of one co
mpressor?

11.

New tires are purchased for service trucks. Each service truck is given 4 tires and a spare.

If there are 75 new tires, how many trucks will receive new tires?

6

12.

A 1,800
-
sq ft. attic is to be insulated. One roll of 6
-
inch thick insulation cove
rs 60 sq. ft. of
the attic.

How many rolls are needed to insulate the entire attic?

13.

At high speed, a blower delivers 3010 cu ft. /min... This volume is divided equally among 14
ducts.

Find in cubic feet the amount of air that flows through each duct ever
y minute.

14.

A hot water baseboard radiator has 12 fins per inch.

How many fins are in 108 inches of the radiator?

15.

An office building with three air
-
conditioning units has an air
-
conditioning load of 121,000
Btu/hr. The first unit handles 44,000 Btu/hr. an
d the second handles 42,000 Btu/hr.

How many Btu/hr. does the third unit handle?

16.

A defective section of soft copper tubing must be replaced. A 21
5
/
16
-
inch section is cut out.
The tube to replace the defective section will overlap ¼ inch at each end.

How many inches long is the piece of new tubing?

17.

A technician had to work on a single job on three different days. Monday 6 ¾ hours were
spent on the job; Tuesday3 ½ hours were spent on the job; and Thursday, 6 hours were spent
on the job.

What was the t
otal time spent on the job?

18.

A technician needs to drill a hole through a wall for a ½
-
inch fuel line. There should be a
1
/
32
-
inch clearance on each side of the tube.

What should be the diameter of the drill?

19.

A piece of polyvinylchloride (PVC) tubing,
4
3
/
8

feet long, is needed for a drain of an air
-
conditioning unit. The piece is cut from an 8
-
foot
-
long coil of tubing.

How much tubing is left?

20.

In one day, a technician works 9 ½ hours and finishes two jobs. It takes 4 ¾ hours to finish
the first job.

How long does it take to finish the second job?

21.

Five air
-
conditioning units are checked and recharged with refrigerant R
-
134a. A full
cylinder of R
-
134a contains 25 pounds of refrigerant. These amounts are taken from a full
cylinder: 3 ½ pounds, 2
2
/
3

po
unds, 4
2
/
3
pounds, 2
4
/
5

pounds and 5
1
/
16

pounds.

How much R
-
134a is left?

7

22.

A house is being built. The contractor states that the cost to install electric baseboard heat
will be \$2,580. The contractor also states that a forced air, heat pump system wi
ll be 2 ¼
times as much.

How much will the heat pump system cost?

23.

A technician is recovering refrigerant R
-
22 to be properly disposed. Each unit emptied
produced 25
5
/
6

ounces of R
-
22.

If 9 units are emptied, how many ounces of R
-
22 did the technician r
ecover?

24.

At 63°F, refrigerant R
-
22 has a weight of 2
1
/
5

pounds for each cu ft. of the R
-
22 in gaseous
form. A technician determines that the tubing from the evaporator back to the condenser has
a volume of
1
/
8

cu ft.

If the technician recovers the R
-
22 i
n this tubing, how many pounds will be recovered?

25.

A certain house has floor joists every 1
1
/
3

feet. The heating duct for the house runs under the
floor hanging from straps that are attached to some of the joists. The distance between the
support straps i
s 8 feet.

How many joists have straps attached to them?

26.

An installer can work 6 ¾ hours each day at a worksite. The installer will need 27 hours to do
a complete installation of a heating and cooling system.

How many days will the installer be at the j
ob site?

27.

A technician recovered 22 ½ pounds of refrigerant R
-
12 for disposal by emptying
5
/
8

of a
pound of R
-
12 from each of a number of portable dehumidifier units.

How many dehumidifier units did the technician empty?

28.

A technician uses #12 wire for a
repair job. Before starting the job, the roll has 78 ½ feet of
wire on it. The technician uses
1
/
3

of the roll.

How many feet of wire does the technician use?

29.

The tubes going to and from an air
-
conditioning condensing unit must pass through a wall.
The tu
bes have diameters of 1
7
/
8

inches and ½ inch.

What is the smallest diameter hole that can be used?

30.

A circular duct has an outside diameter of 7 ½ inches. Insulation that is 1
3
/
8

inches thick is
wrapped around the duct.

What is the diameter of the insulated duct?

31.

In a refrigeration cycle, the refrigerant gains heat in the evaporator and in the suction line. In
a certain refrigeration system, R
-
134a gains 67.8 British thermal units per pound (Btu/lb.) in
the evaporator. I
t then gains 3.5 Btu/lb. in the suction line.

Find in Btu/lb. the total heat gained by the R
-
134a.

8

32.

A partially filled cylinder of refrigerant R
-
124 weighs 47.3 pounds. Another

14.5 pounds of
R
-
124 are put into the cylinder.

How much does the cylinder
now weigh?

33.

Upon being started, a motor draws additional current until the motor is running. A large fan
motor draws4.45 amperes of current when running. When starting, the fan motor draws an
additional 9.27 amperes of starting current.

What is the total
current drawn by the fan motor while starting?

34.

Some refrigerants are a mixture of three different refrigerants, R
-
32, R
-
125, and R
-
134a. In 2
pound of such a mixture, R
-
32 and R
-
125 together weighs 0.706 pound.

How much R
-
134a is in this mixture?

35.

A part
ially filled cylinder of R
-
407c weighs 57.3 pounds. When the cylinder is empty, it
weighs 12.6 pounds.

What is the weight of the R
-
407c in the partially filled cylinder?

36.

Absolute pressure = Gauge pressure + Atmospheric pressure

A pressure gauge is being

checked for accuracy. The gauge is connected to a tank that has an
absolute pressure of 742.11 psi. Atmospheric pressure is 15.7 psi.

What should the gauge read?

37.

A mullion heater prevents condensation on the refrigerator cabinet between the two doors of

the cabinet. One mullion heater has a value of 12.5 watts. One watt produces 2.415 Btu of
heat.

How much heat does the mullion heater

produce?

38.

When 1 pound of refrigerant R
-
134a vaporizes, 90.2 Btu of heat are removed from the
surroundings.

How many B
tu of heat are removed when 8.4 pounds of R
-
134a vaporize?

39.

The manufacturer's manual for a fan motor states that the motor draws a starting current that
is 5.2 times larger than it’s running current.

If its

running current is 2.147 amperes, find the expected current reading on an ammeter
when starting the fan.

40.

A technician is troubleshooting a problem in an electrical circuit. Six identical air
conditioners are running on one circuit.

With all of them runn
ing, 12.845 amperes of current flow through the circuit, what
should each air conditioner have as current running through its unit?

9

41.

Air
-
conditioning units come in sizes of whole and half tons. A 1
-
ton air
-
conditioning unit
will cool a typical 1,100 sq.
ft. house in the southern part of the United States.

What size unit would be needed to cool a 3,973 sq. ft. house in
t
he southern part o
f t
he
United States? (Round up to
t
he next
h
igher whole or hal
f
ton)

42.

An oil burner ran a total of 4.5 hours in one day and used 7.425 gallons of fuel.

How many gallons would be used if it ran only 1 hour?

43.

The slight sideward motion of a shaft is called end play. The end play in the shaft of a rotor
for an electric motor s
hould not be more than
1
/
32

inch. One motor has an end play of 0.0305
inch.

Is this end play more than
1
/
32

inch?

44.

A heating supply dealer has taken ¼ off the price of an acetylene torch.

If the price was \$233.57, what is the savings for buying the torch
today?

45.

A humidifier is designed to put 8.3 gallons of water into air flowing through an air duct every
24 hours when running continuously.

How much water is put into the air when the system runs only
3
/
5

of the time?

46.

A store replaced 60 of its 150
-
watt
incandescent light bulbs with new 45
-
watt compact
fluorescent light bulbs. The light output is the same; the difference in wattage is the
difference in heat output by the bulbs.

If 1 watt is equal to 3.41 Btu’s for each hour the light is on, how much less

heat must be
removed by the air conditioner each hour, due to changing the light bulbs?

47.

If a room that was used as a part of a house becomes office space and is air
-
conditioned
rather than just heated like the house, the number of air changes per hour is

increased. For the
room in question, the number of air changes per hour becomes 2.3 times larger.

If the old number of changes was 3.4 changes per hour and the room is 952.7 cu ft., what
flow should the new ventilating system be able to handle in 1 hour?

48.

Two technicians from the All Cool Company worked on a repair job. Janet

earns \$13.25
an hour and worked 10.5 hours. Bill, who earns \$11.70 an hour, worked 9.75 hours.

How much did the All Cool Company have to payout for labor?

49.

The weight of 1 cu ft. of

#2 fuel oil is about 53.125 pounds. The weight of 1 cu ft. of water is

Find the ratio of the weight of the fuel to the weight of the water.

10

50.

A compressor takes in refrigerant at a pressure of 80.34 psia
. The discharge pressure of the
refrigerant is 401.7 psia.

What is the compression ratio of the compressor? (Round your answer to the nearer
tenth)

51.

In one minute, 90 cu ft. of air flow through a duct into a room. The room contains 1050 cu ft.
of space.

W
hat is the ratio of the flow of air into the room to the volume of the room?

52.

The weight of 10 gallons of #2 fuel oil is 71 pounds.

What is the weight of 325 gallons?

53.

A 5
-
foot section of 14" x 8" rectangular metal ducting weighs 18 pounds.

What would be

the weight of a 28
-
foot section of 14" x 8" rectangular duct?

54.

Two triangles are similar. Triangle 1 has side a = 16 and side b = 8. Triangle 2 has side B =
30.

How long is side A of triangle 2?

55.

An air
-
conditioning installer works part time and has a

taxable income of \$6,520.00. The
state income tax is 8% of the taxable income.

How much money does the installer pay in state taxes?

56.

During a 40
-
hour work week, a technician spends 17% of the time driving to and from
various jobs.

How many hours are sp
ent driving?

57.

A repair company borrows money to purchase new trucks. The interest paid on the loan is
\$1,440.

This is 7% of the loan. How much money is borrowed
?

58.

A shop needs 480 pounds of refrigerant R
-
134a. A supplier charges \$0.93 per pound. If the
ref
rigerant is ordered in 125
-
pound cylinders, a 14% discount is given. If ordered in 30
-
pound cylinders, a 9% discount is given.

a.

What is the cost of 750 pounds of R
-
134a if it is ordered in 125
-
pound cylinders?
(Round the answer to the nearer whole cent)

b.

Wh
at is the cost of 750 pounds of R
-
134a if it is ordered in 30
-
pound cylinders?
(Round the answer to the nearer whole cent)

59.

When purchasing a heat pump unit, an installer is given discounts of 12% and 3%. The unit
is priced at \$3,400.

What is the final price of the pump?

11

60.

A bill for duct insulation and furnace filters is \$237.15 with the notation 2% 1 O/Net 30.

How much is saved by paying the bill within 10 days?

61.

The temperature difference between the floor and the ceiling of a room
is 7° F.

Express this difference in degrees Celsius. (Round the answer to the nearest tenth)

62.

On a cold day, the temperature difference between the inside and the outside of a certain
house is 23°C.

Express this value on the Fahrenheit scale.

63.

A hydronic
heating system is designed to use 180°F water leaving the furnace and returns the
water to the furnace at 168°F.

a.

Find in degrees Celsius the temperature of the water leaving the furnace.

b.

Find in degrees Celsius the temperature of the water returning to t
he furnace.

64.

Identify the type of angle and its measurements.

65.

Identify the type of angle and its measurements.

12

66.

Identify the type of angle and its measurements.

67.

A
window is 2 feet 8 inches ac
r
oss
.

W
h
a
t
i
s the l
a
r
ges
t
wi
d
th a
ir
co
n
d
i
tione
r
i
n
i
n
ches
th
a
t
w
i
ll
fi
t in tha
t
window?

68.

A strap to support a round duct is 1.28 meters long.

Find the length of the strap in centimeters.

69.

A domestic heat pump system has the condenser coils and evaporator coils separated by
1
0.48 meters.

What would be the length of the hose, expressed in centimeters, connecting these two
coils?

70.

Express 1 foot 9 inches as centimeters.

71.

Express 14 centimeters as inches.

72.

Express 8 meters as feet and inches.

73.

A refrigerator door is sealed with a
magnetic gasket. The rectangular door is 34 inches wide
and 39 1/2inches long.

Find in feet and inches the total length of the gasket.

74.

Pieces of copper tubing are used to install a hot water heating system.

How many pieces, each 2 feet 4 inches long, ca
n be cut from a 20
-
foot length of tubing?

75.

To repair a certain refrigerator, these lengths of wire are needed: 6 feet 4 inches, 2 feet 3
inches, 1 foot 10 inches, and 4 feet 9 inches. The lengths are cut from a 25
-
foot coil.

Find in feet and

inches the amount of coil left after the lengths are cut.

13

76.

What is the area of the opening in a duct that has a diameter of 8 inches (Round the answer to
the nearer thousandth square inch)

77.

The filter for a room air
-
conditioning unit has an area of 1,600

sq. cm.

How many square inches are there in the filter?

78.

An 8
-
inch by 12
-
inch rectangular duct splits into two branch ducts. The area of the two
branches is equal to the area of the 8
-
inch by 12
-
inch duct. One of the branches is a square
duct measuring 6

inches on each side.

What is the area of the opening in the second branch?

79.

The opening in an air duct is 81 sq. in.

What is the area to the nearer hundredth square centimeter?

80.

The filter for a room air
-
conditioning unit has an area of 1,800 sq. cm.

H
ow many square inches are there in the filter?

81.

The installation instructions for an imported condensing unit for a domestic heat pump
system state that it should sit on a slab at least 1.3 sq. m in area.

What is the minimum size of the slab in square fee
t? (Round to the nearer tenth square
foot)

82.

The inside dimensions of a refrigerated tractor trailer are 91 inches across, 100
7
/
8

inches
high, and 44 feet ½ inch long.

Find the volume in cubic feet that must be cooled by the refrigeration unit. (Round the

answer to the nearer tenth cubic foot)

83.

An imported freezer lists its interior dimensions as 152 centimeters long, 65 centimeters
wide, and 80 centimeters deep.

What is the volume of the freezer in cubic feet? (Round to the nearer hundredth cubic
foot)

84.

A room measures 12 ½ feet wide and 15 ½ feet long. The walls are 9 feet

high. The
volume of air in the room changes six times each hour.

How many cubic feet of air enters the room each minute?

85.

A cylinder containing propane has an inside diameter of 3.5
inches and is

10 inches
long.

How many cubic inches of propane can the container hold?

86.

Find the total volume of air in 46 feet of 6
-
inch round duct.

Find the volume in cubic feet.

14

87.

An expansion tank for a domestic hot water system measures 7 ¾ inches
in diameter and 21
¼ inches long.

What is the maximum volume the tank can hold?

88.

A furnace for an electric heating system is rated at
121

amperes and 27,600 watts.

What is the voltage of this system?

89.

A
t
echnician
u
ses a 75
-
wa
tt
bu
l
b i
n
a
p
o
rt
a
bl
e
li
g
ht
.
T
h
is
i
s plugged
i
n
t
o a
r
egu
l
a
r
1
20
-
v
o
lt
h
o
u
seho
l
d
ou
t
l
et
.

Wh
a
t
c
urr
e
nt
does
t
he bu
l
b
h
ave
f
lo
w
i
n
g
th
roug
h it
w
h
en
it i
s o
n
?

90.

A
½

-
h
o
r
se
p
o
w
e
r (
3
7
2
.
85
-
w
a
tt)
co
mpr
essor
m
oto
r h
as 3
.
3
4
ampe
r
es o
f
cu
rr
e
nt fl
ow
i
ng
thr
o
u
gh
it w
he
n
r
u
nn
i
ng
.

Wh
a
t
vo
lt
age
i
s supp
l
y
i
ng the cu
r
ren
t t
o
th
is mo
t
o
r?

91.

An air compressor begins
it
s cyc
l
e
with 0
.
8 cu
in of a
i
r a
t
atmospheric pressure (1
4
.7 ps
i
or 0
psig
) in i
ts
cy
l
ind
er. The air
l
eaving
t
he cylinder has an absolute p
r
essure o
f 4
2
p
s
i
a
.
Th
e
t
e
m
pe
r
at
ur
e re
m
ai
n
s the same
.

What is the ne
w
volume of t
h
e air lea
ving th
e co
mp
resso
r
?

92.

An oxygen cylinder for an oxyacetylene setup registers a pressure of 1,724 kPa in the afternoon when
the technician is finished using it
.
The temperature of the cylinder in the afternoon is 30
°
C. I
n the
morn
i
ng the temperature
i
s 20
°
C and the cylinder registers a pressure of 1
,
668 kPa
.

Has the cylinder developed a leak?

93.

A
large e
l
ectric generato
r
is cooled by a gas that then passes
t
hrough a heat exchanger a
n
d is
cooled
it
self
.
O
n
e c
u
bic mete
r
of
gas enters t
h
e heat exchanger w
it
h a
t
empe
r
a
tu
re of 7
7
°
C
.
W
h
en it
l
eaves the heat exchanger, it occupies 0
.
95 cu m
.

Wha
t i
s
t
he
t
emperature of
t
he gas as
i
t leaves the heat exchange
r
?

Express the answer to the nearer British thermal unit per hour.

94.

A warehouse measures 40 feet by 50 feet and has 20
-
foot
-
high walls. The warehouse was
built on a concrete slab and has 6 inches of insulation in the wood frame walls and
9 ½
inches in the ceiling. There are no windows in the building
,
and the door is made
just like the
walls.

What is the heat load for this warehouse in an area where there is a 75
°
F design
temperature difference?

95.

The dimensions of a rectangular duct with a lap seam are:

h
=
35 cm;
w
=
20 cm;
I
=
75 cm; M
=
0.8 cm

a.

W
h
a
t
is
t
he
l
e
n
g
th
o
f t
he

s
tr
e
t
ch ou
t
i
n c
en
t
imete
r
s?

b.

What
i
s
th
e
w
id
t
h o
f th
e stre
t
c
h
o
ut
i
n
ce
nt
imete
r
s?

15

96.

A r
ec
t
a
n
g
u
la
r duct i
s 2
f
ee
t wi
de
,
30
i
nches h
i
g
h
and 3 fee
t
long
.
The
l
ap sea
m i
s ¼
inc
h.
T
he
ove
rl
ap
i
s
¾ inch.

a.

W
ha
t i
s
th
e
l
e
n
g
th
o
f th
e st
r
e
t
c
h
ou
t i
n
i
nches?

b.

W
hat i
s

t
he
width
o
f
the s
tr
e
t
c
h
o
ut in in
ches?

97.

A 9
1
/
8
-
i
nc
h
s
qu
are
du
c
t h
as a
l
a
p
sea
m
o
f
¾

i
nches.

The duc
t
has a length o
f
4 fee
t
.

a.

Find the
l
eng
t
h o
f th
e stretc
h
o
ut
.

b.

Find the width of the stretch out

98.

A 26
-
centimeter diameter duct is 50 centimeters long and has a butt seam.

a.

Find to the nearer hundredth centimeter the value of
L
.
S.

b.

Fin
d in centimeters the value of
WS.

99.

A circular duct has a radius of 15.3 centimeters. It is 1.1 meters long and has a
welded duct
.

a.

Find to the nearer hundredth centimeter the length of the stretch
-
out
.

b.

Find the width of the stretch
-
out
.

100.

A circular duct is to measure 22.5 centimeters in diameter and 75 centimeters long
.
It has
a butt seam.

a.

Find to the nearer hundredth
centimeter the length of the stretch
-
out
.

b.

Find the width of the stretch
-
out
.

101.

The
l
eng
t
h of an
ar
c o
f
a c
i
rcle
i
s
11.77
5 fee
t
.
Th
e diamete
r
of the c
i
rcle is
9 f
ee
t
.

H
o
w m
an
y d
eg
r
ees a
r
e i
n th
e
c
en
tral an
gle o
f th
e a
r
c? (Round the a
n
s
w
e
r t
o
t
he
nea
r
e
r
de
g
r
ee)

102.

The c
yl
i
nd
er o
f
a
r
o
t
a
ry compr
esso
r
is
12 c
e
nt
i
m
e
t
ers i
n d
i
a
mete
r
.

The an
g
le be
t
wee
n t
he i
n
ta
k
e a
nd
e
xh
a
u
s
t p
o
rt
s o
f
t
h
e co
m
presso
r
i
s 4
0
°
.

Wha
t
is the
d
is
t
ance be
t
ween
th
e
p
o
rts m
easu
r
e
d
a
l
o
ng th
e a
r
c? (R
o
u
n
d th
e
a
n
swe
r
to
th
e nea
r
e
r hund
re
dt
h
c
e
ntim
e
t
er)

103.

An oil g
u
n is
f
astened
t
o a fu
rn
ace with s
ix sc
rews. T
h
e screws are equa
l
ly spaced and
fo
rm
a 6
-
i
nch diam
e
t
e
r
circ
l
e
.

Wh
a
t
is
th
e arc leng
t
h
t
o the nearer hu
n
d
r
ed
t
h
i
nch be
t
wee
n t
h
e cent
e
r
s
of th
e
s
c
r
e
w
s
?

104.

How much heat does the full 8 ounces remove from the refrigerator area as it boils back
to a vapor?

105.

How much heat is added by the pump to 1 pound of the refrigerant and must be removed
without doing any cooling?

16

106.

I
f
a s
y
s
t
e
m
is o
verch
a
r
ge
d,
t
h
e
d
isc
h
arge

pressure w
i
ll rea
d
higher than it s
h
ou
ld. If th
e
sy
stem
is un
d
e
r
c
h
ar
g
e
d,
t
h
e
d
isc
h
ar
g
e p
r
essu
r
e wi
ll
r t
han
it
s
h
o
u
l
d.
T
he
c
o
m
press
or
suc
tion p
ressu
r
e
i
s
6
2 psig a
n
d
t
he disc
h
a
r
ge press
u
r
e indi
ca
t
e
s 290 p
sig
when t
h
e o
ut
side
t
e
mp
eratu
r
e is 85°F
.

Sho
u
ld the refrige
r
a
nt b
e
a
d
d
e
d
to the s
y
ste
m
o
r t
a
k
en out?

107.

Find a. Round to the nearest hundredth of a degree.

108.

Find a. Round to the nearest hundredth of a degree.

109.

Find C. Round to the nearest hundredth.

17

110.

Find B. Round
to the nearest hundredth.

111.

Find C. Round to the nearest hundredth.

112.

Find a. Round to the nearest hundredth of a degree.

113.

Find B. Round to the nearest hundredth.

114.

Find the area of a circle with a diameter of 10”.

18

115.

Find

the area of a circle with a radius of 4”.

116.

How many square feet are in a floor measuring 10’ by 18’?

117.

How many inches squared in a right triangle with sides of 5” x 4” x 3”?

118.

Find the volume in inches squared of a cylinder 30” tall with a diameter of 12
”?

119.

What is the volume of a room 10’ wide 18’ long with a 9’ ceiling?

120.

If a refrigerant cylinder weights 35# 6oz and 5# 10oz of refrigerant is removed, what does the
cylinder weight?

121.

If 14.7 psia is the pressure at sea level what is the pressure at the

bottom of a 50’ high cylinder
sitting on the seashore full of h20 if 27.7” wc = 1 psia?

Use the following information for the next four questions:

It takes 144 btus to change 1 lb of ice at 32
o
F to 32
o
F water

It takes 1 btu to raise 1lb of water 1
o
F

It takes 970 btus to change 1lb of water at 212
o
F to 1lb of vapor at 212
o
F

It takes .5 btus to raise 1lb of ice 1
o
F.

122.

How many btus will it take to change 5# of ice at 32
o
F to water at 32
o
F?

123.

How many btus will it take to change 2# of water at 32
o
F to va
por at 212
o
F?

124.

How many btus will it take to change 4# of ice at 24
o
F to vapor at 212
o
F?

125.

How many btus would have to be removed to cool 30# of water from 75
o
F to 50
o
F?

126.

What is atmospheric pressure per feet squared at sea level?

19

127.

A Freon blend
consists of 3 Freon, if 23% is Freon A, 13% is Freon B, What percentage is Freon
C?

128.

If R
-
22 sells for \$210.00 for a 30 lb. cylinder, what is the cost per pound?

129.

If a duct work truck line is 38’ long and must be supported on both sides every 4’ how many
su
pports will be required?

130.

If a furnace cost a contractor \$750,000 and he pays 7% sales tax on it what is his total cost?

131.

If a furnace cost a contractor \$825.00 and sells it for \$1501.00 including 7% tax, what did he
mark up the cost of the furnace?

132.

If

an
AC

unit costs a contractor \$1200.00 and he marks up the cost of the ac 80% and charges the
customer 7% tax, what does he sell it for?

133.

If a resistor is rated at 1500 ohms and its tolerance is + or
-
5%, what is the maximum and
minimum acceptable range
of resistance?

134.

If an installation takes 13 hours at a labor rate of \$22 per hour what is the labor cost?

135.

The tubes going to and from an air conditioning condensing unit must pass through a wall. The
tubes have diameters of 1
7
/
8

inches and ¾ inch. Wha
t is the smallest diameter hole that can be
used?

136.

An 1800 sq. ft. attic is to be insulated. One roll of 6” thick insulation covers 40 sq. ft. of the attic.

How many rolls are needed to insulate the entire attic?

137.

One man hour is one working for 1 hour. A housing development has 12 buildings. Each
building has 45 condominiums in it. Each condo needs a heating/air conditioning system
installed. Each system installation will take 23 man hours to complete.

The con
tractor must plan for how many man hours to complete the job?

20

138.

When on cubic feet of gas is burned 1060 btus is produced. If a building uses 750,000 btus of
heat a day how many cubic feet of gas is used?

139.

A blower delivers 2600 cubic feet per minute. This
volume is divided equally between 12 ducts.

How much is being delivered through each duct?

140.

In one day, technician works 9 ½ hours and finishes two jobs. It takes 3 ¾ hours to finish the first
job.

How long does it take to finish the second job?

141.

A house
is being built. The contractor states that the cost in install electric baseboard heat will be
\$2480. The contractor also states that a forced air, heat pump system will be 2 ¼ times as much.

How much will the heat pump system cost?

142.

An installer can work

6 ¾ hours each day at a worksite. The installer will need 30
3
/
8

hours to do
a complete installation of a heating and cooling system.

How many days will the installer be at the job site?

143.

A technician recovered 22 ½ pounds of refrigerant R
-
12 for disposal by emptying
5
/
6

of a pound
of R
-
12 from each of a number of portable dehumidifier units.

How many dehumidifier units did the technician empty?

144.

In a refrigeration cycle, the refrigerant gains heat in the evaporator and in the suction line. In a
certain refrigeration system, R
-
134a gains 80 btus per pound in the evaporator. It then gains 2.5
btus per pound in the suction line.

Find in btu/lb the t
otal heat gain by the R
-
134a?

145.

The north wall of a house measures 30ft long 10ft high. It contains 3 windows measuring 3ft x 4ft
and two doors 36” wide by 84” high. The wall has a btu loss of 5btus per square foot per hour.
The windows have a btu loss of 25
btus per hour. The doors have a btu loss of 10btus per hour.

How much btu loss is there in one hour?

146.

Two wells are dug for a ground source heat pump system. Each well is 6” in diameter and 116ft
deep.

How many cubic feet of dirt will be displaced for the

two wells?

147.

An 8 x 14 rectangular duct splits into two branch ducts. The area of the two branches is equal to
the 8 x 14 duct. One branch duct measures 6” x 6”.

What is the area of the opening of the other duct?

Using Ohms Law

Volts = amps x ohms

Ohms=v
olts/amps

Amps=volts/ohms

21

148.

If a motor is 120 volts and draws 4 amps, what are the ohms?

149.

If a motor is 240 volts and draws 4 amps, what are the ohms?

150.

If a relay draws 1.2 amps and has 20 ohms resistance, what is the voltage?

151.

If a light bulb draws 1.2

amps and 100 ohms, what is the voltage?

152.

If a motor is 240 volts with a resistance of 20 ohms, what is the amperage?

153.

If starting amperage is 17 amps and run amps are
1
/
3

start amperage what are run amps?

154.

If a motor is 120 volts with a resistance of 1
5 ohms, what is the amperage?

155.

If running amperage of a motor is 4.3 amps and start amps are 3.6 time higher, what is the start
amperage?

156.

If a section of duct is 10’ long and 8” x 16” in height and width, how many inches of sheet metal
are in the duct?

157.

How much sheet metal in ft
2

are needed for 50’ of 8” round duct?

Use the following information for the next two questions:

(A)

Service Call charge \$50

(B)

Labor \$80 hour minimum of 1 hour, \$20 per every 15 min or portion thereof after 1
st

hour

(C)

Material marked up 75%

(D)

7% tax on material

158.

A service call lasts 1 hour and 45 minutes. A motor that cost the company \$135 installed.

What is the customer’s total bill?

159.

A service call last 1 hour and 20 minutes. A thermostat costing \$88 is installed
.

How much is the customer billed?

22

160.

An installation is going to take 78 man hours. If two men are assigned to the job and they work 8
hours per day, how many day will it take to do the job if they take a half hour to get to the job and
a half hour back to
the shop and they get two fifteen minutes paid breaks?

161.

If 187 service calls are done and there are callbacks on 11 jobs, what percent of the jobs require
callbacks?

162.

If a wall is 18’ high how far should the base of the ladder be from the wall if it should
equal 25%
of the wall height?

163.

A room measure 20’ x 15’ with a 9’ ceiling, two registers blow air into the room at a rate of
200cfm each.

How long will it take to change out all of the air in the room?

23

1

47 FT

2

575 LBS

3

3,7442 MILES

4

107 FT

5

67 GAL

6

1,921 SQFT

7

1,152

8

\$288.00

9

3562 LBS

10

55
1
/
3

LBS

11

15

12

30 ROLLS

13

215 CUFT/MIN

14

1296 FINS

15

35000 BTU/HR

16

21
13
/
16

17

16 ¼

18

NINE/16 IN

19

91 5/8 IN

20

4 ¾ HRS

21

18AND167/240

22

\$5,805.00

23

232 ½ OZ

24

11/44 LB

25

6 JOISTS

26

4 DAYS

27

36

28

26
1
/
6

FT.

29

2
3
/
8

IN

30

9
5
/
8

31

71.3 BTU/LB.

32

6,181 LBS.

33

13.72 AMPS

34

1.294 LBS.

35

44.7 LBS.

Q #

Q #

36

726.41 PSI

37

30.18

38

757.68 BTU

39

11.16 AMPS

40

1.83 AMPS

41

4 TON

42

1.65GAL

43

NO

44

\$58.39

45

\$49.80

46

21483 BTU

47

7450.114
CUFT/HR

48

\$253.21

49

17/20

50

5 TO1

51

3 OVER 35

52

2307.5 LBS

53

100.8 LBS

54

60

55

\$521.60

56

6.8 HRS

57

\$20, 571.43

58

A=\$675.00
B=\$634.73

59

\$2,902.24

60

\$4.74

61

14.4 DEG C

62

73.4 DEG C

63

356 DEG C

334.4 DEG C

64

OBTUSE 112
DEG

24

Q #

65

ACUTE 49 DEG

66

RIGHT 90 DEG

67

23 IN

68

128 CENTE

69

1048 CENTE

70

53.34 CENTE

71

5.51 IN

72

26.24FT

314.88IN

73

12FT 3IN

74

8

75

9FT 8IN

76

50.266 IN

77

12 IN

78

60 SQ IN.

79

522.61 SQ FT

80

278.98 SQ IN.

81

13.98 SQ FT

82

2807.5 CU FT

83

27.91 CU FT

84

174.37 CU
FT/MIN

85

96.21 CU IN

86

9.03 CU FT

87

1002.42 CU IN

88

228.09 VOLTS

89

0.62 AMPS

90

111.63 VOLTS

91

0.28 CU IN

92

1667.1 KPA:
NO

93

59.5 DEG CEL

94

15240 BTU/HR

95

110.8CM

75CM

96

108 ¼ IN

24 3/4IN

97

37 ¼ IN

38IN

98

81.64CM

50CM

99

96.08CM

110CM

100

141.3CM

75CM

101

150 DEG

102

4.19 CM

103

3.14 IN

104

33.5 BTU

105

15
BTU

106

TAKE SOME
OUT

107

32.58 DEG

108

52.25 DEG

109

47.32 IN

110

20.53 IN

111

21.21 FT

112

26.57 DEG

113

6.18 FT

114

78.5 SQ IN

115

50.24 SQ IN

116

180 SQ FT

117

6 SQ IN

118

339.12 CU IN

119

1620 CU FT

120

29LB 12OZ

121

16.5 PSIA

122

720

BTU

123

2300 BTU

124

4616 BTU

125

750 BTU

126

2116.8 SQ FT

127

64%

128

\$7.00

129

19 SUPPORTS

130

\$802.50

131

\$569.93

132

\$2,311.20

133

1575

1425

134

\$264.00

135

2 ANDN
5
/
8

IN

136

45ROLL

137

HRS

138

707.55 CU FT

139

216.6 CFM

140

5 AND

¾
HOURS

25

141

\$5,580.00

142

4.5 days

143

27

144

82.5 btu/lb

145

2430 btu

146

45.53 Cu. ft.

147

76 Sq. in.

148

30 ohms

149

60 ohms

150

24 volts

151

120 volts

152

12 amps

153

5.66 amps

154

8 amps

155

15.48 amps

156

5760 Sq. in.

157

104.66 Sq.

ft.

158

\$462.79

159

\$318.53

160

6 days

161

0.06% rounded

162

4.5 ft.

163

13.5 minutes

26

Mathematical Points to Remember

and

Problem Solving Tips

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 c
an be very helpful.

27

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 elaps
ed 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

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 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
.

Multiplyi
ng 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.

28

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 letter x or n.

Solving equations for a missing value requires you to understand opposite operati
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 propo
rtions. 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, follow
ed by multiplication and division, addition and subtraction
29

come last. An easy way to remember the order of operation is: PEMDAS or Please Excuse My
Dear Aunt Sally

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.

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 en
closed within this edge
is measured by area. Area is a two
-
dimensional measurement that measures the number of square
units of a surface.

30

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 o
f 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.

31

The Customary/Standard System

The customary or standard system of measurement is the sys
tem 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:

=

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
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 ra
te, 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
32

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

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 ≈ means “appro
ximately 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

Example:

-
4 + 7 equals +3

3.

WHEN YOU ARE ADDING OR SUBTRACTING NUMBERS:

IF YOU HAVE MORE NEGATIVES THAN POSITIVES NUMBERS YOUR

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

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 2
nd

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 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.

42

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

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
43