CareerTrain

Contextualized Learning Packet

CIM (Computer Integrated Manufacturing)

CareerTrain

Contextualized Learning Packet

Applied Mathematics

CIM (Computer Integrated Manufacturing)

What the WorkKeys Applied Mathematics Test MeasuresThere are five levels of difficulty. Level 3 is the least complex, and Level 7 is the most complex. The levels build on each other, each incorporating the skills assessed at the previous levels.

Level Characteristics of Items Skills

3 Translate easily from a word problem to a math equation

All needed information is presented in logical order

No extra information

Solve problems that require a single type of mathematics operation (addition, subtraction, multiplication, and division) 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)

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 number 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 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 markups

Level Characteristics of Items Skills

6 May require considerable translation from verbal form to mathematical expression

Generally require considerable setup and involve multiple-step calculations

Use fractions, negative numbers, ratios, percentages, or mixed numbers

Rearrange a formula before solving a problem

Use two formulas to change from one unit to another within the same system of measurement

Use two formulas to change from one unit in one system of measurement to a unit in another system of measurement

Find mistakes in questions that belong at Levels 3, 4, and 5

Find the best deal and use the result for another calculation

Find areas of basic shapes when it may be necessary to rearrange the formula, convert units of measurement in the calculations, or use the result in further calculations

Find the 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 areas and volumes of spheres, cylinders, or cones

Set up and manipulate complex ratios or proportions

Find the best deal when there are several choices

Apply basic statistical concepts

1. The stockroom has eight boxes of No. 10 hex head cap screws.

How many screws of this type are in stock if the boxes contain 246, 275, 84, 128, 325, 98, 260, and 120 screws, respectively?

2. In calculating her weekly expenses, a machinist found that he had spent the following amounts: materials, $11,860; labor, $3854; salaried help, $942; overhead expense, $832.

What was his total expense for the week?

3. The head machinist at Tiger Tool Co. is responsible for totaling time cards to determine job costs. He found that five different jobs this week took 78, 428, 143, 96, and 284 minutes each.

What was the total time in minutes for the five jobs?

4. A machinist needs the following lengths of l in. diameter rod: 8 in., 14 in., 6 in., 17 in., and 42 in.

How long a rod is required to supply all five pieces? (Ignore cutting waste.)

5. A machinist needs 25 lengths of steel each 11 in. long.

What is the total length of steel that he needs? No allowance is required for cutting.

6. The Ace Machine Company advertises that one of its machinists can produce 2 parts per hour.

How many such parts can 26 machinists produce if they work 45 hours each?

7. If a machine produces 15 screws per minute, how many screws will it produce in 24 hours?

8. A machinist has a piece of bar stock 243 in. long.

If she must cut 12 equal pieces, how long will each piece be? (Assume no waste to get a first approximation.)

9. A machine shop bought 14 steel rods of 7/8 in.-diameter steel, 23 rods of ½ in. diameter, 9 rods of ¼ in. diameter, and 19 rods of l-in. diameter.

How many rods were purchased?

10. If it takes 45 minutes to cut the teeth on a gear blank.

How many hours will be needed for a job that requires cutting 34such gear blanks?

11. The Ace Machine Tool Co. received an order for 15,500 flanges.

If three dozen flanges are packed in a box, how many boxes are needed to ship the order?

12. A machinist is deciding between two job offers. One is with Company A, paying him $24 per hour plus health benefits. The other is with Company B, paying him $28 per hour with no health benefits. Health insurance would cost him $500 per month. Assume he works an 8-hour day and an average of 22 work days per month, then answer the following questions:

a. How much would he earn per month with Company A? b. How much would he earn per month with Company B? c. What are his net earnings with Company B after paying for his health benefits? d. Which company provides the best overall compensation?

13. What is the shortest bar that can be used for making 5 chisels each 6 1/8 in. in length?

14. How long will it take to machine 44pins if each pin requires 6 3/4 minutes? Allow 1 minute per pin for placing stock in the lathe.

15. How many pieces 6 1/2 in. long can be cut from 35 metal rods each 40 in. long?

Disregard waste.

16. The architectural drawing for a room measures 3 5/8 in.by 4 1/4in.

If 1/4 in. is equal to 1 ft. on the drawing, what are the actual dimensions of the room?

17. The feed on a boring mill is set for 1/64 in. How many revolutions are needed to advance the tool 3 3/8

in.?

18. If the pitch of a thread is 1/18 in., how many threads are needed for the threaded section of a pipe to

be 2 1/2in. long?

19. What is the combined thickness of these five shims: 0.008, 0.125, 0.150, 0.185, and 0.007 in.?

20. A certain machine part is 2.327 in. thick.

What is its thickness after 0.078 in. is ground off?

21. The diameter of a steel shaft is reduced 0.006 in. The original diameter of the shaft was

0.850 in.

Calculate the reduced diameter of the shaft.

22. A shop tech earns a base pay of $19.28 per hour, plus "time-and-a-half' for overtime (time

exceeding 40 hours).

If he works 43.5 hours during a particular week, what is his gross pay?

23. For the following four machine parts, find W, the number of pounds per part; C, the cost of the metal per part; and T, the total cost.

Number Number Cost Cost Metal of Inches of Pounds per Pounds per Part Parts Needed per Inch Pound (W) (C)

A 44.5 0.38 $0.98

B 122.0 0.19 $0.89

C 108.0 0.08 $1.05

D 9.5 0.32 $2.15

T=

24. How much does 15.7 sq. ft. of No. 16 gauge steel weigh if 1 sq. ft. weighs 2.65 lb.?

25. A machinist estimates the following times for fabricating a certain part: 0.6 hour for setup, 2.4 hours of turning, 5.2 hours of milling, 1.4 hours of grinding, and 1.4 hours of drilling.

What is the total time needed to make the part?

26. A shop technician earns $18.26 per hour plus time-and-a- half for overtime (time exceeding 40 hours).

If he worked 42.5 hours during a particular week, what would be his gross pay?

27. A mower motor rated at 2.0 hp is found to deliver only 1.6 hp when connected to a transmission system.

What is the efficiency of the transmission?

28. A machinist can produce 14 parts in 40 min. How many parts can the machinist produce in 4 hours?

29. A machinist creates 2lb of steel chips in fabricating 16 rods. How many pounds of steel chips will be created in producing 120 rods?

30. If 28 tapered pins can be machined from a steel rod 12 ft. long, how many tapered pins can be made from a steel rod 9 ft. long?

31. A 9-in. pulley on a drill press rotates at 960 rpm. It is belted to a 5-in. pulley on an electric motor.

Find the speed of the motor shaft.

32. A cylindrical oil tank 8 ft. deep holds 420 gallons when filled to capacity.

How many gallons remain in the tank when the depth of oil is 51 ft.?

33. It is known that a cable with a cross-sectional area of 0.60 sq. in. has a capacity to hold 2500 lb.

If the capacity of the cable is proportional to its cross-sectional area, what size cable is needed to hold 4500 lb.?

34. A pair of belted pulleys has diameters of 20 in. and 12 in., respectively. If the larger pulley turns at 2000 rpm, how fast will the smaller pulley turn?

35. A casting weighed 1461b out of the mold. It weighed 134 lb. after finishing.

What percent of the weight was lost in finishing?

36. Specifications call for a hole in a machined part to be 2.315 in. in diameter. If the hole is measured to be 2.318 in., what is the machinist's percent error?

37. Six steel parts weigh 1.8 lb. How many of these parts are in a box weighing 142lb if the box itself weighs 7 lb.?

38. In a closed container, the pressure is inversely proportional to the volume when the temperature is held constant.

Find the pressure of a gas compressed to 0.386 cu ft. if the pressure is 12.86 psi at 2.52 cu ft.

39. If 250 ft. of wire weighs 22 lb., what will be the weight of 150 ft. of the same wire?

40. A 156-in. twist drill with a periphery speed of 50.0 ft. /min has a cutting speed of 611 rpm

(revolutions per minute).

Convert this speed to rps (revolutions per second) and round to one decimal place

41. A piece 7 1/16 in. long is cut from a steel bar 28 5/8 in. long.

How much is left

42

43

44

45

46

47

Find the lengths marked on the following rules:

48. The total length of three pipes is 86 inches. The middle-sized pipe is 6 inches longer than the smallest. The largest is twice as long as the smallest.

How long is each pipe?

49. The Formula L=2d+3.26 (r+4) can be used under certain conditions to approximate the length L of belt need to connect two pulleys of radii rand R if their centers are a distance d apart.

How far apart can two pulleys be if their radii are 8 inches and 6 inches and total length of the belt connecting them is 82 inches? (Round to the nearest inch)

50. Suppose that on the average, 3 % of the parts produced by a particular machine have proven to be defective. Then the formula N-0.03N=P will give the number of parts N that must be produced in order to manufacture a total of P non defective ones.

How many parts should be produced by this machine in order to end up with 7500 non defective ones?

51. One-tenth of the parts tooled by a machine are rejects.

How many parts must be tooled to ensure 4500 acceptable ones?

52. Identify the type of angle and give its measurement.

53. Identify the type of angle and give its measurement.

54. Identify the type of angle and give its measurement.

55. Find the Area. Round to the nearest tenth.

56. Find the area. Round to the nearest tenth. Use n = 3.1416

57. Find the area of the shaded region. Round to the nearest tenth. Use n-3.1416

58. Find the area of the ring. Round to the nearest tenth.

59. Find the area.

60. Find the area of the shaded region. Round to the nearest tenth.

61. Find the area of the entire figure.

62. Find the missing dimension.

63. Find the missing demension. Round to the nearest tenth.

64. Find the missing dimension.

65. Find the missing dimension.

66. Find the missing dimension. Round to the nearest tenth.

67. Find the lateral surface area and the volume.

68. Find the lateral surface area and the volume.

69. Find the lateral surface area and the volume.

70. Find the lateral surface area and the volume. Round to the nearest hundredth.

71. Find the lateral surface area and the volume. Round to the nearest thousandth.

72. Calculate the gallons of water needed to fill this swimming pool. Round the the nearest gallon.

73. Find the lateral surface area and the volume. Round to the nearest hundreth.

74. Find the lateral surface area and volume. Round to the the nearest hundreth.

75. Find the lateral surface area and volume. Round to the the nearest hundreth.

76. Write the angle in radians. Round to the nearest hundredth.

45 o

77. Write the angle in radians. Round to the nearest hundredth.

122 ¾ o

78. Write the angle in radians. Round to the nearest hundredth.

20.28 o

79. Write the angle in degrees. Round to the nearest hundredth.

3.14 radians

80. Write the angle in degrees. Round to the nearest hundredth.

1.34 radians

81. Write the angle in degrees. Round to the nearest hundredth.

0.08 radians

82. Find the value of C and a. Round to the nearest tenth.

83. Find the value of A. Round to the nearest tenth.

84. Find the values of A, B, and a. Round to the nearest tenth.

85. Find the value of C. Round to the nearest tenth.

86. Find the values of A, C and b. Round to the nearest tenth.

87. Find the value of A. Round to the nearest tenth.

88. Find the values of A, B, and a. Round to the nearest tenth.

89. Find the values of B and C. Round to the nearest tenth.

90. Find the trig value. Round to 3 decimal places.

sin 25o 40’91. Find the trig value. Round to 3 decimal places.

65o 35’92. Find the trig value. Round to 3 decimal places.

sin 85o

93. Find the acute angle x. Round to the nearest minute.

tan x = 0.45494. Find the acute angle x. Round to the nearest minute.

cos x= 0.68395. Find the acute angle x. Round to the nearest minute.

cos x = 0.82196. Find the acute angle x. Round to the nearest minute.

sin x = 0.96297. Find the acute angle x. Round to the nearest minute.

tan x = 0.332

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

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

100. Find C. Round to the nearest hundredth.

101. Find B. Round to the nearest hundredth.

102. Find C. Round to the nearest hundredth.

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

104. Find B. Round to the nearest hundredth.

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

106. Find A. Round to the nearest thousandth.

107. Find B. Round to the nearest tenth.

ANSWER KEYQ # ANSWER1 1536 SCREWS2 $17,4883 1129 MIN.4 87 IN.5 275 IN6 2430 PARTS7 21600 SCREWS8 18 IN.9 65 RODS10 25.5 HRS.11 431 BOXES12 A=4224

B=4928C=4428D=COMPANY B

13 36 ¾ IN.14 34115 21016 21617 10418 4519 0.47520 2.24921 0.84422 $857.96 23 $52.83 24 41.6 LBS.25 11 HRS.26 $798.87 27 80%28 84 PARTS29 15 LBS.30 21 PINS31 1728 RPM32 267.75 GAL.33 1.08 SQ. IN.34 3320 RPM35 8.30%36 13%

37 450 PARTS38 83.96 PSI39 13.2 LBS.40 10.2 RPS41 21 9/1642 A= 5/8

B= 17/8

C=2½ D=31/8

43 E= 4/16

F=11/16

G=210/16

H=3½ 44 A= 10/32

B= 24/32

C= 11/32

D= 117/32

45 E= 8/24

F= 34/64

G= 53/64

H= 128/64

46 A= 2/10

B= 5/10

C= 1 3/10

D= 16/10

47 E= 25/100

F= 72/100

G=148/100

H=174/100

48 20,26 AND 40 IN49 18 IN.50 7,732 PARTS51 5,000 PARTS52 OBTUSE 112 DEG.53 ACUTE 49 DEG.54 RIGHT 90 DEG.55 19.6 SQ. IN.56 219.3 SQ. CM57 75.4 SQ. CM.58 141.4 SQ. FT.59 1728

60 45.5

61 10662 2063 13.764 86.665 6666 18.367 140 SQ. IN68 512 SQ. IN69 680 SQ. FT.70 72 SQ. FT.71 355.572 SQ. CM.

732.939 CU. CM.72 19747 GALLONS73 424.12 SQ. MTRS.

1,017.88 CU. CM.74 431.97 SQ. FT.

1237 CU. FT.75 472.37 SQ. FT.

829.38 CU. FT.76 0.79 RAD.77 2.14RAD.78 0.35RAD.79 179.91 DEG.80 76.78 DEG.81 4.58 DEG.82 C=7.1 IN.

A=45DEG.83 13.6 IN.84 A=5IN

B=8.7IN.A=30 DEG.

85 24.2 CM.86 A=13.9IN

C=16 INB=60 DEG.

87 10.6 IN..88 A=7.5IN

B=7.5 INA=45 DEG.

89 B=6IN.C=8.5IN.

90 0.43391 2.203

92 0.99693 24 DEG. 25 MIN.94 46DEG 55MIN.95 34 DEG 49 MIN96 74 DEG 9 MIN97 48.1 DEG98 32.58 DEG99 52.25DEG100 47.32IN101 20.53IN102 21.21FT103 26.57DEG104 6.18FT105 51.99DEG106 13.192 MTRS107 19.3CM

Mathematical Points to Remember and

Problem Solving TipsAdditionUse addition in order to find the total when combining two or more amounts.SubtractionUse subtraction in order to:

Determine how much remains when taking a particular amount away from a larger amount

Determine the difference between two numbers

MultiplicationUse multiplication to find a total when there are a number of equally sized groups.DivisionUse 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” 3/10

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 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.

n8=16

40

40 x n = 16 x 840n = 128

n = 12840 = 3

15

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.

2 x 2 x 2 = 8

Multiplying Negative Numbers

Multiplying negative numbers is similar to multiplying positive numbers except for two rules:

When multiplying a positive number and a negative number, the answer is always negative

8 x (-6) = -48

When multiplying two negative numbers, the answer is always positive.

-2 x (-7) = 14

By knowing the rules of multiplying positive and negative numbers, you can rule out incorrect answers before performing any calculations.

Perimeter Measures

Perimeter measures the length of the outer edge of a shape. The space enclosed within this edge is measured by area. Area is a two-dimensional measurement that measures the number of square units of a surface.

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 liter0.264 gallons =

x liters21 gallons

By finding the cross products, you see that:

0.264x = 21The 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

1hour

Another example of a unit rate is $6 per page$ 6

1 pageProportions

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.

$ 321hour =

$ 48015 hours

Convert Between Systems of Measurement

When solving problems that involve converting from one unit of measurement to another, you typically should first determine to which unit of measurement you should be converting.

For example:

You are the service manager for a corporation and are responsible for a fleet of vehicles. You need to determine which brand of engine oil to use with

31

your fleet. There are two brands that you are deciding between. So, you decided to run a test between the two brands. On average, a vehicle burned 5 milliliters of the more expensive synthetic blend. The average consumption of regular engine oil was 64 milliliters. Each vehicle holds 5.8 quarts of engine oil. What percentage of the regular oil was lost during the test?

A. 0.5% B. 1.2% C. 3.2% D. 5.6% E. 9.1%

Plan for Successful Solving

What am I asked to do?

What are the facts?

How do I find the answer?

Is there any unnecessary information?

What prior knowledge will help me?

Find the percent of regular engine oil that was used

The engine holds 5.8 quarts, 64 ml of oil was lost

Convert one measurement to the same system as the other.

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 milliliters64 milliliters

5,492milliliters = 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 percent.

Select the correct answer: B. 1.2% By converting the units of measure to the same system, you can calculate the percent of

oil lost in the test by dividing the amount consumed by the total capacity and multiplying by 100%

32

The symbol ≈ means “approximately equal to” and is used because the conversion formula between gallons and liters is not exact. When calculating conversions between measurements for which the conversions are not exact, you must take into account the fact that the numbers are often rounded at some point during the calculation

33

BASIC ALGEBRA RULES

1. DO BRACKETS FIRST

Example: ( ) [ ]

2. WHEN YOU ARE ADDING OR SUBTRACTING NUMBERS:

IF YOU HAVE MORE POSITIVES THAN NEGATIVES NUMBERS YOUR ANSWER WILL BE A PLUS ANSWER.

Example: -4 + 7 equals +3

3. WHEN YOU ARE ADDING OR SUBTRACTING NUMBERS:

IF YOU HAVE MORE NEGATIVES THAN POSITIVES NUMBERS YOUR ANSWER WILL BE A MINUS ANSWER

Example: -7 + 4 equals -3

4. WHEN YOU ARE MULTIPLYING OR DIVIDING NUMBERS

LIKE SIGNS ARE POSITIVE AND UNLIKE SIGNS ARE MINUS

Example: (+ and + or -+- +) equal a plus sign (- and +) equals minus

5. WHEN ADDING OR SUBTRACTING EXPONENTS

LIKE EXPONENTS CAN ONLY BE ADDED TOGETHER

Example: x to the second power can be combined With another x to the second power only

6. WHEN YOU ARE MULTIPLYING WHOLE NUMBERS

7. THEY ARE MULTIPLIED, AND EXPONENTS ARE ADDED TOGETHER

Example: 3x to the third power times 2x to the second power equals 6x to the fifth power

8. WHEN YOU DIVIDE NUMBERS THEY ARE DIVIDED AS USUAL AND EXPONENTS ARE SUBTRACTED FROM EACH OTHER

Example: 16m to the third power divided by 4m equals 4m to the second power

34

35

Formulas 1Gear Ratio = Number of Teeth on the Driving

Gear

Number of Teeth on the DrivenGear

Reduce to Lowest Terms

Pulley Ratio = Diameter of Pulley A Diameter of Pulley B

Reduce to Lowest Terms

Compression Ratio = Expanded Volume Compressed Volume

Reduce to Lowest Terms

A Proportion is 2 Ratios that are = Example 1/3 = 4/12

Cross Product Rule A/B = C/D or A x D = B x C

Pitch = Rise Run

Changing a Decimal to a % Multiply by 100

Changing a Fraction to a % Divide the Numerator by the Denominator and Multiply by 100

Changing a % to a Decimal Divide by 100

P/B = R/100

When P is unknown

When R is unknown

When B is unknown

Changing a decimal to a fraction .375 hit 2nd hit prb hit enter

Sales Tax Sales Tax = Tax Rate Cost 100

Interest Annual Interest = Annual Interest Rate

36

Principal 100

Commission Commission Sales = Rate Sales 100

Efficiency Output = Efficiency Input 100

Tolerance Tolerance = % of ToleranceMeasurement 100

% of Change Amount of Increase = % of Increase Original Amount 100

Discounts Sales Price = List Price –

37

Discount

38

39

40

PERCENT PROBLEMSThe Percent (%)The Whole (OF)The Part (IS)

41

Trig Formulas

1. Change an angle to radians = angle times pie divided by 180

2. Change an angle to degrees = radians times 180 divided by pie

3. 30 deg., 60 deg., 90 deg., triangle; the short end is equal to ½ the hypotenuse or the hypotenuse = 2 times the short end

4. 45 deg., 45 deg., 90 deg., triangle – the 2 shorter sides are the same length and the hypotenuse is 1.4114 times the leg

5. Find trig value – put in SIN, COS, or TAN followed by degrees and hit enter

6. Find acute angle X – hit 2nd button, then SIN, COS, or TAN; enter number and hit equals. Hit RP move arrow to DMS hit enter twice

You would use this when you need an answer in degrees, minutes, and or seconds

7. Find acute angle X – hit 2nd button, then SIN, COS, or TAN; enter number and hit equals. You would use this when you need an answer in degrees.

42

Applied Mathematics Formula Sheet

Distance1 foot = 12 inches1 yard = 3 feet1 mile = 5,280 feet1 mile ≈ 1.61 kilometers1 inch = 2.54 centimeters1 foot = 0.3048 meters1 meter = 1,000 millimeters1 meter = 100 centimeters1 kilometer = 1,000 meters1 kilometer ≈ 0.62 miles

Area1 square foot = 144 inches1 square yard = 9 square feet1 acre = 43,560

Volume1 cup = 8 fluid ounces1 quart = 4 cups1 gallon = 4 quarts1 gallon = 231 cubic inches1 liter ≈ 0.264 gallons1 cubic foot = 1,728 cubic inches1 cubic yard = 27 cubic feet1 board = 1 inch by 12 inches by 12 inch

Weight1 ounce ≈ 28.3501 pound = 16 ounces1 pound ≈ 453.592 grams1 milligram = 0.0001 grams1 kilogram = 1,000 grams1 kilogram ≈ 2.2 pounds1 ton = 2,000 pounds

Rectangleperimeter = 2(length + width)area = length x width

Rectangle Solid (Box)volume = length x width x height

Cubevolume = (length of side)3

Trianglesum of angles = 180o

area = ½(base x height)

Circlenumber of degrees in a circle = 360o

circumference ≈ 3.14 x diameterarea ≈ 3.14 x (radius)2

Cylindervolume ≈ 3.14 x (radius)2 x height

Conevolume ≈ 3.14 × (radius)2 × height

3

Sphere (Ball)volume ≈ 4/3 x 3.14 x (radius)3

Electricity1 kilowatt-hour = 1,000 watt-hoursAmps = watts ÷ volts

TemperatureoC = 0.56(oF-32) or 5/9(oF-32)oF = 1.8(oC) + 32 or (9/5 x oC) + 32

43

44