CareerTrain
Contextualized Learning Packet
HVAC
1
CareerTrain
Contextualized Learning Packet
Applied Mathematics
HVAC
2
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

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
Translate easily from a word
problem to a math equation
All needed information is
presented in logical order
No extra information
Skills
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
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

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




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)
Calculate percent discounts or
3
markups
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
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
4
1. A forced convection heating system is installed in a house. The length 6-inch circular duct
that are needed are: kitchen, 6 feet; dining room, 12 feet; living room, 3 feet; master
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 total weight of refrigerant ordered?
3. Last year, the Keep Kool Company's five repair trucks covered 7,252 miles; 8,917 miles; 4,266 miles;
7,793 miles; and 9,214 miles.
What is the total mileage the Keep Kool Company should report 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. During 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. There 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 compressor?
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?
5
12. A 1,800-sq ft. attic is to be insulated. One roll of 6-inch thick insulation covers 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 every 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. and 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 total 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 airconditioning 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 pounds, 4 2/3pounds, 2 4/5 pounds and 5 1/16 pounds.
How much R-134a is left?
6
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 will 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 recover?
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 in 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 is 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 job 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 tubes 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. It then gains 3.5 Btu/lb. in the suction line.
Find in Btu/lb. the total heat gained by the R-134a.
7
32. A partially filled cylinder of refrigerant R-124 weighs 47.3 pounds. Another
R-124 are put into the cylinder.
14.5 pounds of
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 partially 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 Btu 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 running, 12.845 amperes of current flow through the circuit, what
should each air conditioner have as current running through its unit?
8
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 the southern part of the
United States? (Round up to the next higher whole or half 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 should 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
about 62.5 pounds.
Find the ratio of the weight of the fuel to the weight of the water.
9
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.
What 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.
30.
Triangle 2 has side B =
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 spent 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
refrigerant is ordered in 125-pound cylinders, a 14% discount is given. If ordered in 30pound 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. What 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?
10
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 the furnace.
64. Identify the type of angle and its measurements.
65. Identify the type of angle and its measurements.
11
66. Identify the type of angle and its measurements.
67. A window is 2 feet 8 inches across.
What is the largest width air conditioner in inches that will fit in that 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
10.48 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, can 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.
12
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.
How 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 feet? (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
volume of air in the room changes six times each hour.
high. The
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
long.
10 inches
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.
13
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 technician uses a 75-watt bulb in a portable light. This is plugged into a regular 120-volt
household outlet.
What current does the bulb have flowing through it when it is on?
90. A ½ -horsepower (372.85-watt) compressor motor has 3.34 amperes of current flowing
through it when running.
What voltage is supplying the current to this motor?
91. An air compressor begins its cycle with 0.8 cu in of air at atmospheric pressure (14.7 psi or 0
psig) in its cylinder. The air leaving the cylinder has an absolute pressure of 42 psia. The
temperature remains the same.
What is the new volume of the air leaving the compressor?
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. In the
morning the temperature is 20°C and the cylinder registers a pressure of 1,668 kPa.
Has the cylinder developed a leak?
93. A large electric generator is cooled by a gas that then passes through a heat exchanger and is
cooled itself. One cubic meter of gas enters the heat exchanger with a temperature of 77°C.
When it leaves the heat exchanger, it occupies 0.95 cu m.
What is the temperature of the gas as it leaves the heat exchanger?
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. What is the length of the stretch out in centimeters?
b. What is the width of the stretch out in centimeters?
14
96. A rectangular duct is 2 feet wide, 30 inches high and 3 feet long. The lap seam is ¼ inch. The
overlap is ¾ inch.
a. What is the length of the stretch out in inches?
b. What is the width of the stretch out in inches?
97. A 9 1/8-inch square duct has a lap seam of ¾ inches.
The duct has a length of 4 feet.
a. Find the length of the stretch out.
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. Find 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 length of an arc of a circle is 11.775 feet. The diameter of the circle is 9 feet.
How many degrees are in the central angle of the arc? (Round the answer to the
nearer degree)
102.
The cylinder of a rotary compressor is 12 centimeters in diameter.
The angle between the intake and exhaust ports of the compressor is 40°.
What is the distance between the ports measured along the arc? (Round the
answer to the nearer hundredth centimeter)
103.
An oil gun is fastened to a furnace with six screws. The screws are equally spaced and
form a 6-inch diameter circle.
What is the arc length to the nearer hundredth inch between the centers of the
screws?
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?
15
106.
If a system is overcharged, the discharge pressure will read higher than it should. If the
system is undercharged, the discharge pressure will read lower than it should. The
compressor suction pressure is 62 psig and the discharge pressure indicates 290 psig
when the outside temperature is 85°F.
Should the refrigerant be added to the system or taken 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.
16
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”.
17
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 32oF to 32oF water
It takes 1 btu to raise 1lb of water 1oF
It takes 970 btus to change 1lb of water at 212oF to 1lb of vapor at 212oF
It takes .5 btus to raise 1lb of ice 1oF.
122.
How many btus will it take to change 5# of ice at 32oF to water at 32oF?
123.
How many btus will it take to change 2# of water at 32oF to vapor at 212oF?
124.
How many btus will it take to change 4# of ice at 24oF to vapor at 212oF?
125.
How many btus would have to be removed to cool 30# of water from 75oF to 50oF?
126.
What is atmospheric pressure per feet squared at sea level?
18
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
supports 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. What 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 contractor must plan for how many man hours to complete the job?
19
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 total 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 25btus 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=volts/amps
Amps=volts/ohms
20
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 15 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 ft2 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
hour
(C) Material marked up 75%
(D) 7% tax on material
158.
after 1st
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?
21
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?
22
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
ANSWER
47 FT
575 LBS
3,7442 MILES
107 FT
67 GAL
1,921 SQFT
1,152
$288.00
3562 LBS
55 1/3 LBS
15
30 ROLLS
215 CUFT/MIN
1296 FINS
35000 BTU/HR
21 13/16
16 ¼
NINE/16 IN
91 5/8 IN
4 ¾ HRS
18AND167/240
$5,805.00
232 ½ OZ
11/44 LB
6 JOISTS
4 DAYS
36
26 1/6 FT.
2 3/8 IN
9 5/8
71.3 BTU/LB.
6,181 LBS.
13.72 AMPS
1.294 LBS.
44.7 LBS.
Q#
ANSWER
36
37
38
39
40
41
42
43
44
45
46
47
726.41 PSI
30.18
757.68 BTU
11.16 AMPS
1.83 AMPS
4 TON
1.65GAL
NO
$58.39
$49.80
21483 BTU
7450.114
CUFT/HR
$253.21
17/20
5 TO1
3 OVER 35
2307.5 LBS
100.8 LBS
60
$521.60
6.8 HRS
$20, 571.43
A=$675.00
B=$634.73
$2,902.24
$4.74
14.4 DEG C
73.4 DEG C
356 DEG C
334.4 DEG C
OBTUSE 112
DEG
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
23
Q#
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
93
94
95
96
97
98
99
ANSWER
ACUTE 49 DEG
RIGHT 90 DEG
23 IN
128 CENTE
1048 CENTE
53.34 CENTE
5.51 IN
26.24FT
314.88IN
12FT 3IN
8
9FT 8IN
50.266 IN
12 IN
60 SQ IN.
522.61 SQ FT
278.98 SQ IN.
13.98 SQ FT
2807.5 CU FT
27.91 CU FT
174.37 CU
FT/MIN
96.21 CU IN
9.03 CU FT
1002.42 CU IN
228.09 VOLTS
0.62 AMPS
111.63 VOLTS
0.28 CU IN
1667.1 KPA:
NO
59.5 DEG CEL
15240 BTU/HR
110.8CM
75CM
108 ¼ IN
24 3/4IN
37 ¼ IN
38IN
81.64CM
50CM
96.08CM
110CM
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141.3CM
75CM
150 DEG
4.19 CM
3.14 IN
33.5 BTU
15 BTU
TAKE SOME
OUT
32.58 DEG
52.25 DEG
47.32 IN
20.53 IN
21.21 FT
26.57 DEG
6.18 FT
78.5 SQ IN
50.24 SQ IN
180 SQ FT
6 SQ IN
339.12 CU IN
1620 CU FT
29LB 12OZ
16.5 PSIA
720 BTU
2300 BTU
4616 BTU
750 BTU
2116.8 SQ FT
64%
$7.00
19 SUPPORTS
$802.50
$569.93
$2,311.20
1575
1425
$264.00
2 ANDN5/8 IN
45ROLL
HRS
707.55 CU FT
216.6 CFM
5 AND ¾
HOURS
24
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
$5,580.00
4.5 days
27
82.5 btu/lb
2430 btu
45.53 Cu. ft.
76 Sq. in.
30 ohms
60 ohms
24 volts
120 volts
12 amps
5.66 amps
8 amps
15.48 amps
5760 Sq. in.
104.66 Sq. ft.
$462.79
$318.53
6 days
0.06% rounded
4.5 ft.
13.5 minutes
25
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
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
31
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%
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
32
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
33
34
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
35
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
36
37
38
39
PERCENT PROBLEMS
The Percent (%)
The Whole (OF)
The Part
(IS)
40
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.
41
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
42
43