Training Manual Pre-Employment Math Version 1.1 Created April 2012 1
Table of Contents Item # Training Topic Page # 1. Operations with Whole Numbers... 3 2. Operations with Decimal Numbers... 4 3. Operations with Integers (Positive And Negative Numbers)... 6 4. Order of Numbers... 7 5. Decimals / Fractions / Percent Conversions... 8 6. Angle Conversions... 10 7. Percent Problems... 11 8. Finding Tolerances... 13 9. Practice Evaluation... 16 10. Practice Evaluation Answers... 21 2
Operations With Whole Numbers Objective Calculate each of the following: A. The sum (addition) of up to five whole numbers. B. The difference (subtraction) between any two whole numbers. C. The product (multiplication) of any two whole numbers. D. The quotient (division) of any two whole numbers. Whole numbers are the "counting numbers" we first learn plus the number "0." Whole numbers = { 0, 1, 2, 3, 4, 5, 6,... } 0, 1, 2, 5, 27, 92, 104, and 5,238 are examples of whole numbers. Whole numbers do NOT include any numbers with fractions or decimals like 2.8 or 3½ or negative numbers like 2 or 35. Addition: The sum is the result of adding two or more numbers. Example: The sum of 2 and 6 and 7 = 2 plus 6 plus 7 = 2 + 6 + 7 = 15 Subtraction: The difference is the result of subtracting the smaller number from the larger number. Example: The difference between 3 and 7 = 7 minus 3 = 7 3 = 4 Multiplication: The product is the result of multiplying two or more numbers. Example: The product of 2, 3 and 7 = 2 times 3 times 7 = 2 x 3 x 7 = 2 3 7 = ( 2 ) ( 3 ) ( 7 ) = 42 Notes: 2, 3 and 7 are called factors of 42. The dot symbol (.) and parentheses ( ) ( ) both mean "multiply." 3
Division: The quotient is the result of dividing one number by another number. Example: The quotient of 10 and 5 = 10 divided by 5 = 5 divided into 10 = 10 5 = 10 / 5 = 5 10 = 2 Operations With Decimal Numbers Objective Calculate each of the following: A. The sum (addition) of up to five decimal numbers. B. The difference (subtraction) between any two decimal numbers. C. The product (multiplication) of any two decimal numbers. D. The quotient (division) of any two decimal numbers. Addition: Line up the decimal points, add zeros to the right (if needed), add the numbers, and place decimal point in the answer directly below the other decimals. Example: 823.4 + 927 + 21.992 = 823.400 927.000 21.992 1,772.392 Subtraction: Line up the decimal points, add zeros (if needed), subtract numbers as usual, and place the decimal point in the answer directly below the other decimals. Example: 21.05 19.309 = 21.050 19.309 1.741 Multiplication: Ignore the decimal points and multiply the numbers as if they were whole numbers. To place the decimal in the answer, add the number of places to the right of the decimal in each factor, and then count over (from the right) the same number of places in the answer. 1.23 x 4.4 Multiply 123 x 44 to get 5412 Since there are 2 decimal places in 1.23 and 1 decimal place in 4.4, there should be a total of 3 (2 + 1) decimal places in the answer: 5.412 4
Division: When dividing by a whole number, divide as usual and place the decimal point in the answer directly above the decimal point of the number you are dividing into. Example: 143.75 50 2.875 50 143.75 100 437 400 375 350 250 250 When dividing by a decimal number: (1) move the decimal to the right until it becomes a whole number (2) move the decimal point in the number you are dividing into the same number of places to the right, (3) divide as usual (4) place the decimal point in the answer directly above the decimal point of the number you are dividing into. Example: 2.565.15 17.1.15 2.565 = 15. 256.5 15 106 105 15 15 Numbers with no decimals: A number with no decimal is assumed to have a decimal at the (right) end of the number with as many zeros after the decimal as may be needed. 5 can be written as 5. or 5.0 or 5.00 or 5.000 1,523 can be written as 1,523. or 1,523.0 or 1,523.0000 5
Operations With Integers (Positive And Negative Numbers) Objective Calculate each of the following: A. The sum (addition) of up to five integers (positives and negatives). B. The difference (subtraction) between any two integers (positives and negatives). C. The product (multiplication) of any two integers (positives and negatives). D. The quotient (division) of any two integers (positives and negatives). Integers are all whole numbers plus their "negative opposites." Integers = {... 3, 2, 1, 0, 1, 2, 3, 4, 5, 6,... } Integers include such numbers as 0, 1, 12, 34, 52, 921, and 4,106. Integers do not include numbers with fractions or decimals like 52.3, or 10 ¾. Numbers with no signs are always assumed to be positive. +1 = 1 and +5 = 5 Addition: The "rules" depend upon the signs of the numbers. 2 positive integers: add the numbers and the answer will be positive. + 2 + + 6 = + 8 + 13 + + 31 = + 44 2 negative integers: ignore the signs and add. Answer will be negative. 4 + 7 = 11 41 + 17 = 58 A positive and negative integer: ignore the signs and subtract the smallest from the largest. Then give the answer the sign of the "largest" of the two numbers. 2 + + 9 = + 7 + 6 + 9 = 3 Subtraction: To subtract, take the following steps: 1. Change the sign of the number being subtracted (from positive to negative or from negative to positive). 2. Change the subtraction sign to an addition sign. 3. Follow the rules for addition. 6
+ 2 6 = + 2 + + 6 = + 8 3 9 = 3 + + 9 = + 6 + 10 + 4 = + 10 + 4 = + 6 19 + 8 = 19 + 8 = 27 + 9 5 = + 9 + + 5 = + 14 8 7 = 8 + + 7 = 1 + 8 + 15 = + 8 + 15 = 7 12 + 25 = 12 + 25 = 37 Multiplication: The "rules" depend upon the signs of the numbers. Signs are the same: multiply the numbers and the answer will be positive. + 2 x + 6 = + 12 9 x 4 = + 36 + 10 x + 7 = + 70 2 x 12 = + 24 Signs are different: multiply and the answer will be negative. + 2 x 6 = 12 9 x + 4 = 36 10 x + 7 = 70 + 2 x 12 = 24 Division: The "rules" depend upon the signs of the numbers. Signs are the same: divide the numbers and the answer will be positive. + 24 + 6 = + 4 18 3 = + 6 + 10 + 2 = + 5 63 7 = + 9 Signs are different: divide the numbers and the answer will be negative. + 24 6 = 4 18 + 3 = 6 10 + 2 = 5 + 63 7 = 9 Order Of Numbers Objective Arrange any set of up to 6 numbers in order from smallest to greatest. Symbols: The relative size of two numbers is shown by the signs: =, <, and >. = means "is equal to" Example: 3 = 6 / 2 means "3 is equal to 6 halves" > means "is greater than" Example: 6 > 2 means "6 is greater than 2" 7
< means "is less than" Example: 4.13 < 4.17 means "4.13 is less than 4.17" The order of any two numbers: For any two given numbers, the first one must be greater than, less than or equal to the second one. If A and B are two numbers, then either A > B, A < B, or A= B. Arranging numbers with decimals: To list decimals in order (smallest to largest or largest to smallest) it may help to write them all with the same number of decimal places. (e.g., 23.4 can be equal to 23.40 or 23.400) Example: Arrange from largest to smallest: 1.23, 1.002 1.3, 2, 1.09, and 2.203 Since 1.002 and 2.203 both include three decimal places, write all 6 numbers with 3 decimal places: 1.230, 1.002, 1.300, 2.000, 1.090, and 2.203 Now we can arrange them as 2.203 > 2.000 > 1.300 > 1.230 > 1.090 > 1.002 or 1.002 < 1.090 < 1.230 < 1.300 < 2.000 < 2.203 Arranging positive and negative numbers: The order of numbers will sometimes appear to be reversed when negatives are included. (Think about a thermometer. Since the greater number is the warmer temperature, 5 (5 below zero) will be greater (warmer) than 11(11 below zero). 5 > 2 2 > 3 5 > 7 Decimal Fraction Percent Conversions Objective Write (a) a given decimal number as a fraction or a percent, (b) a given fraction as a decimal or a percent, and (c) a given percent as a decimal or a fraction. Meaning of percent: Percent means "per hundred" and has been created to help make comparisons. You could compare the quality of two piston orders by saying that order A had 230 scrap pistons out of a total of 20,000 and that order B had only 168 scrap pistons out of a total of 8,000. But a better way would be to compare the number of scrap pistons per 100. Order A has a scrap rate of 1.15% while order B had a scrap rate of 2.1%. 73% means 73 per one 100 or 73 hundredths 231% means 231 per 100 or 231 hundredths 8
Numbers that represent the same quantity: There are always several different ways to represent the same quantity and many times there is a need to change a number from one form to another. The most common conversions are between (a) decimals, (b) percents, and (c) fractions. 1/2 =.5 = 50%.75 = 75% = 3/4 25% = 1/4 =.25 Converting fractions to decimals: To convert a fraction to a decimal, divide the numerator (top) by the denominator (bottom). 3/5 = 3 5 =.6 5/6 = 5 6 =.833 17/20 = 17 20 =.85 36/8 = 36 8 = 4.5 For 21 3 / 5, convert 3/5 to.6 to get 21.6 Converting decimals to fractions: To convert a decimal to a fraction, write the decimal as a fraction with a denominator of 10, 100, 1,000, 10,000, 100,000 etc. (This number is often simplified by dividing the numerator and denominator by a common factor.).2 = 2/10 To simplify, divide both 2 and 10 by 2 to get 1 / 5.65 = 65/100 To simplify, divide both 65 and 100 by 5 to get 13 / 20.003 = 3/1000 Cannot be simplified 23.50 = 2350/100 To simplify, divide both 50 and 100 by 50 to get 23 1 / 2 9.0040 = 940/10000 To simplify, divide 40 and 10,000 by 40 to get 9 1 / 250 Converting decimals to percents and percents to decimals: Since "percent" means "per 100" and a decimal basically means "per 1" (or "a part of one whole") we can convert any decimal to a percent by multiplying by 100. And we can convert any percent to a decimal by dividing by 100. (This process can be shorted by simply moving the decimal point 2 places to the right or 2 places to the left). For example, 50% means 50 per hundred just as.50 means 50 hundredths of 1. The only difference between 50% and.50 is the % symbol and the location of the decimal point. 50% =.50 7.1 = 710% 125% = 1.25.71 = 71% 39% =.39.071 = 7.1% 1230% = 12.30.0071 =.71% 9
Fractions and percents: Instead of trying to convert a fraction to a percent, convert the fraction to a decimal and the decimals to a percent. To convert a percent to a fraction, convert it to a decimal and the decimals to a fraction. Angle Conversions Objective Convert a given measurement in (a) decimal degrees to degrees/minutes/seconds and (b) degrees/minutes/seconds to decimal degrees. Measuring the size of an angle: The size of an angle (how wide it "opens") is measured by degrees ( ). For example, an angle formed by two perpendicular lines contains 90 degrees (90 ). Each degree can be divided into 60 smaller units called minutes ( ' ) and each minute can be divided into even smaller units called seconds ( " ). Example: 21 5' 16" is the measure of an angle co ntaining 21 degrees, 5 minutes and 16 seconds Angles can also be measured in terms of decimal degrees. For example, 23.5 means 23 and ½ degrees. The same angle can be written as either decimal degrees or degrees/minutes/seconds. 73.5 = 73 30' 31 15' = 31.25 Changing decimal degrees to degrees/minutes/seconds: To convert decimal degrees to degrees/minutes/seconds, multiply the decimal portion by 60 minutes. If the answer is decimal minutes, multiply the decimal portion by 60 seconds. See next page Convert 63.8 to degrees/minutes/seconds Multiply.8 x 60 minutes to get 48' (48 minutes) 63.8 = 63 48' Convert 9.27 to degrees/minutes/seconds Multiply.27 x 60 minutes to get 16.2' (16.2 minutes) 9.27 = 9 16.2' Then multiply.2 x 60 seconds to get 12" (12 seconds) = 9 16'12" Changing degrees/minutes to decimal degrees: To convert degrees/minutes to decimal degrees, divide the number of minutes by 60. Convert 25 45' to decimal degrees Divide 45 by 60 to get.75. 25 45' = 25.75 Convert 9 17' to decimal degrees Divide 17 by 60 to get.283 9 17' = 9.283 10
Changing degrees/minutes/seconds to decimal degrees: To convert degrees/minutes/seconds to decimal degrees, (a) divide the number of seconds by 60 to get decimal minutes & (b) divide that number by 60 to get decimal degrees. Convert 67 5' 48" to decimal degrees (a) divide 48 by 60 to get.8 (5' 48" = 5.8') (b) divide 5.8 by 60 to get.097 (67 5' 48" = 67.097 ) Percent Problems Objective Convert any problem involving percents to "A% of B is C" form with two of the three variables known, and calculate the value of the unknown variable. Two methods from which to choose: There are two basic approaches to solving most types of percent problems, but you only need to learn one method the one that makes the most sense to you. The algebraic method is described in the next section. The proportion method is used and explained in our interactive videodisc program. It starts with writing a proportion like the following: Is = Percent t Of 100 We then use "cross multiplication" and solve for the unknown. What is 7% of 300? First write: X x = 7 7 300 100 Then cross multiply to get: 100X = 2,100 Then divide 2,100 by 100 to get 21. Answer: 21. What percent is 280 of 350? First write: 280 = X x 350 100 Then cross multiply to get: 350X = 28,000 Then divide 28,000 by 350 to get 80. Answer: 80% 11
24 is 6% of what number? First write: 24 = 6 x X 100 Then cross multiply to get: 6X = 2,400 Then divide 2,400 by 6 to get 400. Answer: 400. Algebraic method: The algebraic method includes the following 3 steps: 1. Write a statement of the problem in the form A% of B is C. Note: There are three basic types of percent problem problems depending on which of these three factors (A, B or C) are known. In all cases, one of the three will not be known. 2. Change this statement into a math equation of the form A x B = C 3. Solve for A, B or C as indicated (A = C B or B = C A or C = A x B). A. If 18% of our 20,000 sq. ft. storage warehouse is now empty, how many square feet of space are in use? 1. Let X equal the unknown number of square feet that are empty and write the statement: 18% of 20,000 sq. ft. is X 2. Write a math equation by changing 18% to.18, "of" to x and "is" to =..18 x 20,000 = X 3. Find the value of X by multiplying.18 x 20,000 sq. ft. to get 3,600 sq. ft. Since the empty space is 3,600 sq. ft. and the total space is 20,000 sq. ft., the space that is in use will be equal to 20,000 3,600 = 16,400 sq. ft. B. What is the scrap rate if 45 pistons were scrap in an 3,000 piston order? 1. Let X equal the percent scrap rate and write the statement: X% of 3,000 pistons is 45 pistons 2. Write a math equation by changing "of" to x and "is" to =. X x 3,000 = 45 3. Find the value of X by dividing 45 by 3,000 to get.015 Change the decimal.015 to a percent to get 1.5% 12
C. Another order produced 225 scrap pistons. It that represented 3% of a total order, how many pistons were in the total order? 1. Let X equal the unknown number and write the statement: 3% of X pistons is 225 pistons 2. Write a math equation by changing 3% to.03, "of" to x and "is" to =..03 x X = 225 3. Find the value of X by dividing 225 by.03 to get 7500 pistons. Finding Tolerances Objective Calculate minimum, maximum, and nominal tolerances Find the tolerances for 8.234 ±.003 (can also be written 8.234 +.003 /.003): Maximum: Add the largest tolerance value (+.003 in this case) to the original number. Example: 8.234 + (+.003) = 8.237 Minimum: Add the smallest tolerance value (.003 in this case) to the original number. Example: 8.234 + (.003) = 8.231 Nominal: Add the minimum and maximum numbers together and divide by 2. Example: 8.237 + 8.231 = 16.468 16.468 2 = 8.234 1. 6.537 ±.500 Maximum: 6.537 + (+.500) = 7.037 Minimum: 6.537 + (.500) = 6.037 Nominal: 7.037 + 6.037 = 13.074 2 = 6.537 2. 7.651.023 /.017 Maximum: 7.651 + (.017) = 7.634 Minimum: 7.651 + (.023) = 7.628 Nominal: 7.634 + 7.628 = 15.262 2 = 7.631 13
3. 5.454 +.022 / +.030 Maximum: 5.454 + (+.030) = 5.484 Minimum: 5.454 + (+.022) = 5.476 Nominal: 5.484 + 5.476 = 10.960 2 = 5.480 4. 4.980.015 / +.012 Maximum: 4.980 + (+.012) = 4.992 Minimum: 4.980 + (.015) = 4.965 Nominal: 4.992 + 4.965 = 9.957 2 = 4.9785 5. 8.6 +.2 **If only given one tolerance, assume the other is zero.** Maximum: 8.6 + (+.2) = 8.8 Minimum: 8.6 + (0) = 8.6 Nominal: 8.8 + 8.6 = 17.4 2 = 8.7 6. 8.6.2 Maximum: 8.6 + (0) = 8.6 Minimum: 8.6 + (.2) = 8.4 Nominal: 8.6 + 8.4 = 17 2 = 8.5 14
PRE-EMPLOYMENT MATH PRACTICE EVALUATION 15
PRE-EMPLOYMENT PRACTICE EVALUATION Name: Graded by: Test date: Instructions: 1. The time limit is one (1) hour.. 2. Write your answers in the correct blank and, where applicable, show all your work. 3. Use a calculator. If you don't have a calculator, please ask for one. 4. Round any answer with four (4) or more decimal places to three (3) places, unless instructions state otherwise. Do not round any number until your final answer. PART A: Perform each of the following operations. 1. 21-9 = 2. 16 + 6 = 3. 19 + 22 + 14 = 4. 12 x 6 = 5. 105 7 = 6. 4 x 9 = 7. 78-24 = 8. 15 + 120 = 9. 50-12 - 3-2 = 10. 14 x 21 = 11. 35 + 6 = 12. 456 2 = 13. 91 + 41 + 8 = 14. 12 x 32 = 15. 65 5 = 16. 93-17 = 17. 64 2 = 18. 521-46 = 19. 8 x 14 = 20. 40 8 = 16
PART B: Perform each of the following operations. 21. 12.3-10 = 22. 7.9 + 7.1 = 23. 8.23 3.1 = 24. 6.1 x 9.8 = 25. 14 + 1 = 26. 6.4-2 = PART C: Solve the following problems. 27. If a box contains 12 layers of pistons and 15 pistons are stacked on a layer, how many pistons are packaged in the box? The number of pistons packaged = 28. You are allowed to stack 440 pistons on a pallet. You stack 8 layers containing 40 pistons per layer on the pallet. How many more layers of pistons could be placed on the pallet. The additional number of layers that could be placed on the pallet = 29. The original measurement of a part is 8.45 mm. If the part is decreased in size by 0.08 mm, calculate the new measurement of the part. The new measurement of the part = 30. The target size of a part is 3.42 mm. If the part measures 0.07 mm smaller than the target size, what is the actual size of the part? Actual size of the part = 17
PART D: Perform the following decimal / fraction / percent conversions. 31. 0.475 written as a percent is 32. 18/24 written as a decimal is 33. 0.41 written as a fraction is PART E: Solve the following percent problem. 34. Last year a local industry produced 550,000 parts. If 1.5% of these parts were defective, how many defective parts were produced? The number of defective parts produced = PART F: For each problem below, identify the smallest number. 35. Given the numbers 0.04 and 0.7, the smallest number is 36. Given the numbers 3.6 and 4.8, the smallest number is 37. Given the numbers 5 and 2, the smallest number is 38. Given the numbers 12 and 15, the smallest number is PART G: Complete the following table for the dimensions and tolerances given. Do not round your answers. Tolerances must be exact. Answers that have been rounded will be considered incorrect. Dimension Minimum Nominal Maximum 41. 6.755 ±.015 42. 5.34-0.04 18
PART H: Perform the following angle conversions: 45. 24.42 = ' " 46. 17 18'35" = 19
PRE-EMPLOYMENT MATH PRACTICE EVALUATION ANSWER KEY 20
PRE-EMPLOYMENT PRACTICE EVALUATION Name: Graded by: Test date: Instructions: 1. The time limit is two (2) hours. 2. Write your answers in the correct blank and, where applicable, show all your work. 3. Use a calculator. If you don't have a calculator, please ask for one. 4. Round any answer with four (4) or more decimal places to three (3) places, unless instructions state otherwise. Do not round any number until your final answer. PART A: Perform each of the following operations. 1. 21-9 = 12 2. 16 + 6 = 22 3. 19 + 22 + 14 = 55 4. 12 x 6 = 72 5. 105 7 = 15 6. 4 x 9 = 36 7. 78-24 = 54 8. 15 + 120 = 135 9. 50-12 - 3-2 = 33 10. 14 x 21 = 294 11. 35 + 6 = 41 12. 456 2 = 228 13. 91 + 41 + 8 = 140 14. 12 x 32 = 384 15. 65 5 = 13 16. 93-17 = 76 17. 64 2 = 32 18. 521-46 = 475 19. 8 x 14 = 112 20. 40 8 = 5 21
PART B: Perform each of the following operations. 21. 12.3-10 = 22.3 22. 7.9 + 7.1 = 15.0 23. 8.23 3.1 = 2.655 24. 6.1 x 9.8 = 59.78 25. 14 + 1 = 15 26. 6.4-2 = 8.4 PART C: Solve the following problems. 27. If a box contains 12 layers of pistons and 15 pistons are stacked on a layer, how many pistons are packaged in the box? The number of pistons packaged = 180 28. You are allowed to stack 440 pistons on a pallet. You stack 8 layers containing 40 pistons per layer on the pallet. How many more layers of pistons could be placed on the pallet. The additional number of layers that could be placed on the pallet = 3 29. The original measurement of a part is 8.45 mm. If the part is decreased in size by 0.08 mm, calculate the new measurement of the part. The new measurement of the part = 8.37 30. The target size of a part is 3.42 mm. If the part measures 0.07 mm smaller than the target size, what is the actual size of the part? Actual size of the part = 3.35 22
PART D: Perform the following decimal / fraction / percent conversions. 31. 0.475 written as a percent is 47.5% 32. 18/24 written as a decimal is 0.75 33. 0.41 written as a fraction is 41/100 PART E: Solve the following percent problem. 34. Last year a local industry produced 550,000 parts. If 1.5% of these parts were defective, how many defective parts were produced? The number of defective parts produced = 8,250 PART F: For each problem below, identify the smallest number. 35. Given the numbers 0.04 and 0.7, the smallest number is 0.04 36. Given the numbers 3.6 and 4.8, the smallest number is 4.8 37. Given the numbers 5 and 2, the smallest number is 5 38. Given the numbers 12 and 15, the smallest number is 15 PART G: Complete the following table for the dimensions and tolerances given. Do not round your answers. Tolerances must be exact. Answers that have been rounded will be considered incorrect. Dimension Minimum Nominal Maximum 41. 6.755 ±.015 6.74 6.755 6.77 42. 5.34-0.04 5.3 5.32 5.34 23
PART H: Perform the following angle conversions: 45. 24.42 = 24 25' 12" 46. 17 18'35" = 17.31 24