Electrician First Period Math Applications PDF

030102a ELECTRICIAN First Period Math Applications First Period Circuit Fundamentals © 2022, Northern Alberta Institute of Technology (NAIT) and Southern Alberta Institute of Technology (SAIT). All Rights Reserved. No part of this publication may be reproduced, stored in o...

030102a ELECTRICIAN First Period Math Applications First Period Circuit Fundamentals © 2022, Northern Alberta Institute of Technology (NAIT) and Southern Alberta Institute of Technology (SAIT). All Rights Reserved. No part of this publication may be reproduced, stored in or introduced into a retrieval system, transmitted or communicated in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the prior written permission of the publishers, NAIT and SAIT, except as permitted by law. For permission requests, please contact: [email protected]. NAIT and NAIT’s logos are registered trademarks of NAIT and may not be used without the prior written permission of NAIT. SAIT and SAIT’s logos are registered trademarks of SAIT and may not be used without the prior written permission of SAIT. Other logos, trademarks, registered trademarks and tradenames depicted herein are the property of their respective owners. Use of them does not imply any affiliation with or endorsement by them. All reasonable efforts have been made to ensure that this publication is accurate, reliable and free from error at the time of publication; however, there may be instances where information contained herein is not current. To the fullest extent permissible under applicable law, NAIT and SAIT disclaim any and all warranties and conditions respecting this publication, whether expressed, implied, statutory, or otherwise.  Table of Contents Objective One............................................................................................................................................... 2 Arithmetic Symbols.................................................................................................................................. 2 Objective One Exercise................................................................................................................................. 3 Objective Two............................................................................................................................................... 4 Addition.................................................................................................................................................... 4 Subtraction................................................................................................................................................ 8 Objective Two Exercise.............................................................................................................................. 11 Objective Three........................................................................................................................................... 13 Multiplication.......................................................................................................................................... 13 Division................................................................................................................................................... 16 Objective Three Exercise............................................................................................................................ 18 Objective Four............................................................................................................................................ 21 Sequence of Arithmetic Operations........................................................................................................ 21 Objective Four Exercise.............................................................................................................................. 23 Objective Five............................................................................................................................................. 24 Equations................................................................................................................................................. 24 Transposing Equations............................................................................................................................ 25 Objective Five Exercise.............................................................................................................................. 30 Objective One Exercise Answers................................................................................................................ 31 Objective Two Exercise Answers............................................................................................................... 31 Objective Three Exercise Answers............................................................................................................. 31 Objective Four Exercise Answers............................................................................................................... 32 Objective Five Exercise Answers............................................................................................................... 32  NOTES First Period Math Applications Rationale Why is it important for you to learn this skill? You must have a good understanding of basic mathematics in order to solve problems encountered on the job and when studying electrical theory. Some of your responsibilities that require an understanding of mathematics are blueprint reading, measuring distances and analyzing electrical circuits. Outcome When you have completed this module, you will be able to: Solve trade-related problems using basic mathematical skills. Objectives 1. Recognize basic arithmetic symbols. 2. Add and subtract whole, decimal and fractional numbers. 3. Multiply and divide whole, decimal and fractional numbers. 4. State the correct sequence of arithmetical operations and solve equations which use brackets. 5. Apply the math skill required for transposition of equations in relation to Ohm's Law. Introduction In this module, you will be reminded of some basic mathematical symbols and the ways to perform the operations they represent. You will review your mathematical skills using practical, trade-related questions. 1 030102a24  NOTES Objective One When you have completed this objective, you will be able to: Recognize basic arithmetic symbols. Arithmetic Symbols Arithmetic symbols are shorthand marks that represent mathematical operations. Some common symbols are shown in Table 1. Arithmetic Symbol Operation + Addition − Subtraction × Multiplication a ÷ or or a b Division b = Equality Table 1 - Arithmetic symbols. The equal (=) sign indicates the values on either side of the sign are equivalent to each other. Addition is represented by the + symbol. For example, 6 + 2 = 8 or 12 + 4 = 16. The expression 14 plus 15 can be shown as 14 + 15 or as follows. 14 +15 Subtraction is represented by the – symbol. The expression 17 minus 12 can be shown as 17 − 12 or as follows. 17 −12 Multiplication is represented by the × symbol. The expression 16 times 12 can be shown as 16 × 12. Division can be represented in any of the following ways: a b, a or b ÷. 28 The expression 28 divided by 4 can be shown as 28 4 or or 28 ÷ 4. 4 2 030102a24  NOTES Objective One Exercise 1. State the mathematical operation that is indicated by the following symbols: a) + b) - c) x d) ÷ e) = f) ⁄ 2. State the meaning of the equal sign and give an example of its use. 3 030102a24  NOTES Objective Two When you have completed this objective, you will be able to: Add and subtract whole, decimal and fractional numbers. Addition The following examples show how to add whole numbers, decimals and fractions. Whole Numbers When adding columns of numbers, keep the units (or ones) lined up under the other units: the tens under the tens and the hundreds under the hundreds. Working from the top down, add the units column first, then the tens column and so on. The result of adding is called the sum. For example, add 452 and 107 as follows. 452 +107 559 Figure 1 shows how to solve the problem using a calculator. Figure 1 - Adding with a calculator. If the sum of any column is more than 10, write only the rightmost-digit below the column, then add and carry (or add) the other digit to the next column to the left. For example, add 2157, 845 and 34 (Figure 2). Figure 2 - Adding whole numbers. To check your answer of 3036, add the numbers again, but this time work in the opposite direction from bottom to top. Figure 3 shows how to solve with a calculator. Figure 3 - Adding whole numbers on a calculator. 4 030102a24 Decimal Numbers NOTES When adding decimal numbers, write the numbers above each other so that all of the decimal points stand in a vertical line, then add the numbers as you would whole numbers. The decimal point in the answer is in the same vertical line as the other decimal points. For example, add the following numbers: 0.875 + 1.2 + 375.007 + 71.1357 + 735. Solve as follows. 0.875 1.2 375.007 71.1357 735.0 1183.2177 Figure 4 shows how to solve the problem on a calculator. Figure 4 - Adding decimals on a calculator. Fractions Fractions are parts of a whole. The number above the line is called the numerator and the number below the line is called the denominator. 3 Numerator 4 Denominator Before fractions can be added, each fraction must have the same denominator, called a common denominator. A common denominator is a number into which all of the original denominators of a group of fractions divide evenly. The common denominator can be found by multiplying the denominators together and finding their product. 1 3 To find a common denominator for the fractions and , multiply the denominators. In 2 5 this example, 2 × 5 = 10. The fractions have a common denominator of 10, since both 2 and 5 divide into 10 evenly. The value of the fraction does not change if you multiply both the top and bottom by the same number as follows. 1 5 5 1 3 2 6 3 × = equals and × = equals 2 5 10 2 5 2 10 5 5 030102a24  NOTES Example 1 1 3 2 The fractions , and have a common denominator of 2 × 4 × 3 = 24. The three 2 4 3 fractions can be rewritten with a denominator of 24. Divide the original denominator into the common denominator, and then multiply both the top and bottom by this value. 1 24 (common denominator) 1 12 12 = = 12 = × = 2 2 (original denominator) 2 12 24 3 24 (common denominator) 3 6 18 = =6= × = 4 4 (original denominator) 4 6 24 2 24 (common denominator) 2 8 16 = =8= × = 3 3 (original denominator) 3 8 24 The product of the denominators may not be the lowest common denominator (LCD), which is the smallest whole number that can be evenly divided by the original denominators. In this example you can reduce each fraction down to the lowest common denominator by dividing both the numerator and denominator by 2, giving the following results. 12 6 18 9 16 8 = = = 24 12 24 12 24 12 Twelve is the lowest common denominator that produces whole numbers in the numerator. Example 2 1 1 Add the fractions +. First, convert them to the same common denominator and 2 8 rewrite as follows. 1 8 8 1 2 2 × = and × = 2 8 16 8 2 16 Once the fractions have been converted to the same denominator, add the numerators. 1 1 8 2 8 + 2 10 5 + = + = = = 2 8 16 16 16 16 8 Figure 5 shows how to add fractions with a calculator. Figure 5 - Adding fractions with a calculator. 6 030102a24 Mixed Numbers NOTES A mixed number is a whole number and a fraction, such as 21⁄4. In order to calculate a mixed number, it must first be converted to an improper fraction. An improper fraction has a numerator that is equal to or greater than the denominator. To convert a mixed number into an improper fraction, follow these steps. 1. To get the new numerator of the improper fraction, multiply the whole number of the mixed number by the denominator and then add the numerator. 2. The new denominator of the improper fraction equals the old denominator of the mixed number. 1 For example, convert 2 to an improper fraction. The new numerator is the whole 4 1 number multiplied by the denominator, then added to the numerator. Convert 2 as 4 follows. 2x4+1=9 The new denominator equals the old denominator. In this example, the denominator is 4. 1 2×4+1 9 2 = = 4 4 4 Example 1 1 Add 2 +. First, convert mixed numbers to improper fractions as follows. 4 2 1 2×4+1 9 2 = = 4 4 4 1 1 9 1 2 + = + 4 2 4 2 Convert the fractions to a common denominator as follows. 9 1 9 2 + = + 4 2 4 4 Add the numerators as follows. 9 2 9 + 2 11 + = = 4 4 4 4 Convert the improper fraction back to a mixed number as follows. 3 11 ÷ 4 = 2 4 Figure 6 shows how to solve the problem using a calculator. Figure 6 - Adding mixed numbers with a calculator. 7 030102a24  NOTES Subtraction The following examples show the subtraction of whole numbers, decimals and fractions. Whole Numbers When subtracting whole numbers, keep the ones lined up under the ones, the tens under the tens, the hundreds under the hundreds and so on. For example, subtract 20 from 430 as follows. 430 –20 410 Figure 7 shows how to solve the problem using a calculator. Figure 7 - Subtracting whole numbers with a calculator. Example Subtract 254 from 623 as follows. 623 –254 First, subtract the ones column: 3 minus 4 is a negative number, so borrow one unit from the tens column. The ones subtraction is as follows. 61 −25 4 9 Then, subtract the tens column: 1 minus 5 is a negative number, so borrow one unit from the hundreds column. The tens subtraction is as follows. 5 −2 5 4 6 9 Finally, subtract the hundreds column: 5 minus 2 is 3. 5 623 −2 5 4 or −254 3 6 9 369 To check your answer, add the numbers from the bottom up. Adding the two bottom lines results in the top line. 8 030102a24 Figure 8 shows how to solve the problem with a calculator. NOTES Figure 8 - Subtracting whole numbers with a calculator. Decimal Numbers When subtracting decimals, arrange the numbers above each other so the decimal points stand in a vertical line, then subtract the decimal numbers as you would whole numbers. The decimal point in the answer is in the same vertical line as the other decimal points. For example, subtract 1.75 from 32.256 as follows. 32.256 – 1.750 30.506 Figure 9 shows how to solve the problem using a calculator. Figure 9 - Subtracting decimals with a calculator. Fractions To subtract fractions, first convert all fractions to a common denominator, and then subtract the numerators. 1 2 For example, subtract from as follows. 2 3 2 1 4 3 4−3 1 − = − = = 3 2 6 6 6 6 Figure 10 shows how to solve the problem using a calculator. Figure 10 - Subtracting fractions with a calculator. Mixed numbers can also be subtracted if they are first changed to improper fractions. For 5 1 example, subtract from 1 as follows. 8 8 1 5 9 5 9−5 4 1 1 − = − = = = 8 8 8 8 8 8 2 Figure 11 shows how to solve the problem using a calculator. Figure 11 - Subtracting fractions with a calculator. 9 030102a24  NOTES 1 1 For example, subtract 1 from 1 as follows. 4 2 1 1 6 5 6−5 1 1 −1 = − = = 2 4 4 4 4 4 Figure 12 shows how to solve the problem using a calculator. Figure 12 - Subtracting fractions with a calculator. 10 030102a24  NOTES Objective Two Exercise 1. Add the following whole numbers. 1024 39 101 2. Add the following decimal numbers. 9.99 87.04 1.82 100.00 62.34 3. Add the fractions. Give your answer in the lowest common denominator. 1 3 + 4 16 4. Add the mixed numbers. Give your answer in the lowest common denominator. 5 5 1 +7 8 8 5. The material costs for an electrical installation are $89.40 for electrical boxes, $250.00 for cable, $132.48 for receptacles and switches, $18.29 for connectors and $13.97 for screws and nails. Calculate the total material cost. 6. An electrical installation requires runs of cable that are 1051⁄2 m, 271⁄4 m and 983⁄4 m in length. Calculate the total length of cable required. Give your answer as a mixed number reduced to lowest terms. 11 030102a24  NOTES 7. Subtract the following whole numbers. 764 –323 8. Subtract the following decimal numbers. 7102.76 –10.63 9. Solve the following and give your answers as fractions reduced to lowest terms. 3 1 a) − 4 8 1 3 b) 5 −4 16 16 10. A spool contains 1000 metres of low voltage cable. If one run uses 327 metres and another uses 245 metres, is there enough cable left for a run of 437 metres? Explain your answer. 11. A load of 17 000 metres of conduit is delivered to a job site. If the first floor slab requires 9876 metres and the second floor and third floor slabs each require 5746 metres, how much more conduit is required? 12. Arc Electric had $21 478.50 in its bank account. They purchased a truck for $16 127.20 and a heavy-duty utility trailer for $2237.30. How much money remains in the bank account? 13. An electrical contractor charges $15 750.00 for a job. Calculate the profit for this job where: materials = 5375.00, labor = 5800.00 and expenses = 925.00. 12 030102a24  NOTES Objective Three When you have completed this objective, you will be able to: Multiply and divide whole, decimal and fractional numbers. Multiplication The following examples show multiplication of whole numbers, decimals and fractions. Whole Numbers Arrange two numbers so that one is above the other. Each digit of the top number is multiplied in turn by each digit of the bottom number (multiplier). The separate results, called partial products, are lined up, and then added. For example, multiply 27 by 5 as follows. First, calculate the ones column as follows. 27 ×5 5 (5 × 7 = 35: write down the 5 and carry the 3 to the tens column) Next, calculate the tens column as follows. 27 ×5 135 (5× 2 = 10, plus the 3 that was carried over to the tens column = 13 Write down the 13 in front of the five.) Figure 13 shows how to solve the problem using a calculator. Figure 13 - Multiplying whole numbers with a calculator. Example 1 Multiply 23 by 24 as follows. First, multiply the top number by the ones in the multiplier. 24 × 23 (3 x 4 = 12. Write down the 2 and carry the l to the tens column.) 72 (3 x 2 = 6 plus the 1 that was carried over to the tens column = 7. Write down the 7 in front of the 2.) 72 is the partial product. Next, multiply the top number by the tens in the multiplier. 24 (2 x 4 = 8. Write down the 8 in front of the placeholder.) × 23 (2 x 2 = 4. Write down the 4 in front of the 8.) 72 480 552 (Add the two partial products.) 13 030102a24  NOTES The 0 is the placeholder. By using the placeholder, the answer 24 x 2 = 48 is moved one column to the left. When multiplying large numbers, indent each additional partial product one column to the left. Figure 14 shows how to solve the problem using a calculator. Figure 14 - Multiplying whole numbers with a calculator. Example 2 Multiply 338 by 123 using the indent method with the zero placeholders as follows. 338 × 123 1014 (338 × 3 = 1014) 6760 (338 × 2 = 676 and indent one position) 33800 (338 × 1 = 338 and indent two positions) 41574 (Add the three partial products) Figure 15 shows how to solve the problem using a calculator. Figure 15 - Multiplying whole numbers with a calculator. Decimal Numbers To multiply decimal numbers, follow these steps. 1. Place the larger number over the smaller one, with the right-hand digits aligned. 2. Multiply the decimal numbers, using the multiplication of whole numbers method. 3. From the right-hand digit, count the total number of decimal places found in both the number to be multiplied and the multiplier. This is the number of decimal places to be found in the answer, also counted from the right-hand digit. For example, multiply 23.394 by 7.27 as follows. 23.394 × 7.27 163758 (23394 × 7 = 163758) 467880 (23394 × 2 = 46788 and indent one position) 16375800 (23394 × 7 = 163758 and indent two positions) 170.07438 (Add the three partial products (17007438) and then set the decimal point at 3 + 2 = 5 places from the right-hand digit.) Figure 16 shows how to solve the problem using a calculator. Figure 16 - Multiplying decimals with a calculator. 14 030102a24 Example NOTES When multiplying decimal numbers by multiples of 10, move the decimal point to the right by the same number of decimal places as there are zeroes in the multiplier. In the following example, the multiplier has one zero, so the decimal point is moved one position to the right. 3.5625 × 10 = 35.625 In the following example, the multiplier has two zeroes, so the decimal point is moved two positions to the right. 3.5625 × 100 = 356.25 In the following example, the multiplier has three zeroes, so the decimal point is moved three positions to the right. 3.5625 × 1000 = 3562.5 Fractions When multiplying fractions, follow these steps: 1. Multiply the numerators to obtain a product of the numerators. 2. Multiply the denominators to obtain a product of the denominators. 3. Reduce the answer to lowest terms. 1 1 3 2 For example, multiply × and × as follows. 2 2 4 3 1 1 1×1 1 × = = 2 2 2×2 4 3 2 3×2 6 1 × = = = 4 3 4 × 3 12 2 Figure 17 shows how to solve the problem using a calculator. Figure 17 - Multiplying fractions with a calculator. To multiply a fraction by a whole number, follow these steps. 1. Convert the whole number into an improper fraction by placing the whole number over the number 1. For example, 2 converts to 2/1. 2. Convert any mixed fractions into improper fractions. 3. Multiply the numerators to obtain a product of the numerators. 4. Multiply the denominators to obtain a product of the denominators. 5. Give the answer as a mixed number reduced lowest terms. 15 030102a24  NOTES 3 For example, multiply 4 by as follows. 8 3 4 3 4 × 3 12 4 1 4× = × = = =1 =1 8 1 8 1×8 8 8 2 Figure 18 shows how to solve the problem using a calculator. Figure 18 - Multiplying mixed numbers with a calculator. Cancellation is a process of simplifying your answer when multiplying fractions. In this process any one denominator and any one numerator are both divided evenly by the same 3 5 1 number. For example, solve × × as follows. 4 6 10 3 1 1 Divide the numerator and the denominator by 5. This produces × ×. Divide the 4 6 2 1 1 1 numerator and the denominator by 3.This produces × ×. Using cancellation, the 4 2 2 answer is reduced or simplified to lowest terms. Multiply the numerators and the 1 1 1 1×1×1 1 denominators: × × = =. 4 2 2 4×2×2 16 Division The following examples show division of whole numbers, decimals and fractions Whole Numbers When dividing whole numbers, divide the dividend by the divisor. For example, divide 912 (dividend) by 24 (divisor) as in Figure 19. Figure 19 - Dividing whole numbers with a calculator. Decimal Numbers When dividing decimal numbers, divide the dividend by the divisor. For example, divide 1.728 by 1.2 as follows in Figure 20. Figure 20 - Calculator for example 2 16 030102a24 Fractions NOTES Dividing fractions involves using the reciprocal of a fraction and multiplication. The reciprocal is found by inverting the fraction or switching the numerator for the denominator and the denominator for the numerator. For example, the reciprocal 1 2 of is. To solve fractions, follow these steps. 2 1 1. Convert whole numbers into improper fractions by placing the whole number over the number 1. 2. Convert mixed fractions into improper fractions. 3. Change the ÷ sign to a × sign and invert the last fraction. 4. Multiply the numerators to obtain a product of the numerators. 5. Multiply the denominators to obtain a product of the denominators. 6. Give the answer as a mixed number reduced to lowest terms. 3 1 For example, divide 5 by as follows. 4 3 3 23 5 = 4 4 23 1 ÷ 4 3 Change the ÷ sign to a × sign and invert the last fraction. 23 3 × 4 1 Multiply the numerators to obtain a product of the numerators, and multiply the denominators to obtain a product of the denominators. 23 × 3 69 = 4 4 Then give the answer as a mixed number reduced to lowest terms. 69 1 = 17 4 4 When dividing fractions, remember to invert and multiply. Figure 21 shows how to solve the problem using a calculator. Figure 21 - Dividing fractions using a calculator. 17 030102a24  NOTES Objective Three Exercise 1. Solve the following. 234 x 82 2. Five electrical metal tubing (EMT) bundles, each containing 33 m of EMT, are delivered to the job site every day for seven days. Calculate the total length of EMT delivered to the job site. 3. If there are 100 wire connectors in a box and there are 24 boxes to a case, how many wire connectors are in a case? 4. Solve the following. 36.75 x 1.5 5. Solve the following and give answers as fractions reduced to lowest terms. 1 1 a) × 2 4 3 2 b) × 4 3 6. As an electrical apprentice, you are paid $15.56 per hour. How much do you earn in a 40-hour work week? 7. If a coil of No. 6 AWG copper wire weighs 18.7 kilograms per 100 m, calculate the weight of a 300 m coil of this wire. 18 030102a24 8. As an electrical apprentice working for a company, you are paid $14.80 per hour. NOTES Calculate your gross pay for an 11-hour work day if anything over 8 hours is overtime. Overtime, for this company, gets paid at double the standard rate. 9. An electric motor turns at 1740 r/min (revolutions per minute). How many revolutions does it turn in one hour? 10. Solve the following and give each answer as a mixed number reduced to lowest terms. 253 a) 8 12 358 b) 4 11. Solve the following. 358.92 a) 3.6 16.6549 b) 32.98 12. Solve the following and give each answer as a fraction reduced to lowest terms. 1 3 a) ÷ 2 4 1 3 b) 1 ÷ 1 2 4 13. If a 32.4 m long extension cord is coiled into loops averaging 1.2 m in circumference, how many loops are in the coil? 19 030102a24  NOTES 14. An electrical apprentice earns $632.00 for a 40-hour work week. Calculate the hourly wage. 15. A No. 16 AWG copper wire has a resistance of 13.12 Ω (Ohms) per 1000 m. Calculate the resistance, in ohms, of 50 m of this wire. 16. A carton containing 30 fluorescent lamps costs $17.85. An electrical maintenance job requires 750 fluorescent lamps to be replaced. Calculate the material cost of replacing the lamps. 17. An electrical job is estimated to take 1800 hours to complete. If three electricians work 8 hours per day and 5 days per week, how many weeks does the job take? 18. An electrical apprentice earns $19.80 per hour, which is 3/4 of a journeyman's earnings. How much does the journeyman electrician earn? 20 030102a24  NOTES Objective Four When you have completed this objective, you will be able to: State the correct sequence of arithmetical operations and solve equations which use brackets. Sequence of Arithmetic Operations Consider the following statement: 4 + 6 × 3 − 2 = The answer could be 28, 10 or 20. To correctly solve this statement, use the rules for Order of Operations. Mathematical equations need to be solved in this order: 1. Perform the operation within the brackets first. If the statement contains two or more sets of brackets, work from the innermost bracket to the outside bracket. 2. Apply exponents. 3. Perform all multiplication and division, from left to right. 4. Perform all addition and subtraction, from left to right. Remember the rules as BEDMAS, which stands for the order in which the mathematical operations are performed: brackets, exponents, division, multiplication, addition and subtraction. Using these rules, solve 4 + 6 × 3 − 2, where: The first step is brackets, which in this case does not apply. The next step is exponents, which again is not applicable. Then do the division and multiplication. 4+6×3−2= 4 + 18 − 2 The last step is addition and subtraction, from left to right. 22 − 2 = 20 For example, calculate 80 − 5 × (4 ÷ 2 + 5) as follows. 80 − 5 × (4 ÷ 2 + 5) = 80 − 5 × (2 + 5) = 80 − 5 × 7 = 80 − 35 = 45 Figure 22 shows how to solve the problem using a calculator. Figure 22 - Using a calculator to follow the order of operations. 21 030102a24  NOTES Following the BEDMAS rules, solve everything inside the brackets before solving the rest of the mathematical statement. When more than one set of brackets is used, the inside set is called parentheses ( ), the next set is called brackets [ ] and the outside set is called braces { }. However, it is common practice to call them all brackets and use only the parenthesis symbol ( ). For example, calculate 3{200 − [33 × 2 − (4 + 16) ÷ 2]} as follows. 3{200 − [33 × 2 − (4 + 16) ÷ 2]} = 3{200 − [33 × 2 − 20 ÷ 2]} = 3{200 − [66 − 10]} = 3{200 − 56} = 3{144} = 3 × 144 = 432 Figure 23 shows how to solve the problem using a calculator. Figure 23 - Using a calculator to follow the order of operations. 22 030102a24  NOTES Objective Four Exercise Use BEDMAS to solve the following mathematical statements. 1. 2 × 4 + 6 − 3 = 2. 30 ÷ 15 + 200 + 10 − 10 × 20 = 3. [20 + (30 + 15 × 2) ÷ 40] = 4. 30(5 + 15 − 20 + 2 × 20 ) ÷ 50 = 5. 8 ÷ 4 × 2 + 6(100 ÷ 50 ) + 7 = 6. A reel contains 150 m of cable. Three runs of 15 m and two runs of 18 m of this cable are used on a job. Calculate the length of cable left on the reel. 7. A machine shop has three 3/8 hp motors, five 11/4 hp motors and seven 11/2 hp motors. Calculate the total horsepower load of all of the motors. Give the answer as a mixed number reduced to lowest terms. 23 030102a24  NOTES Objective Five When you have completed this objective, you will be able to: Apply the math skill required for transposition of equations in relation to Ohm's Law. Equations An equation is a mathematical statement that two expressions are equal in value. Think of an equation as a balance scale. As long as both sides carry equal amounts, the equation is in balance and is valid (Figure 24). If something is added or subtracted from one side, the same has to be done to the other side so the equation remains in balance. Figure 24 - An equation can be viewed as a balance scale. Example 1 Adding the same number to both sides of an equation maintains the balance to the equation. In the following equation, 6 is added to both sides. 8=2×4 8+6=2×4+6 14 = 14 (Adding 6 to each side maintains the balance.) Example 2 Multiplying the same number on to both sides of an equation also maintains the balance to the equation. In the following example, both sides of the equation are multiplied by 5. 7 = 9 –2 5 × 7 = 5 × (9 –2) Multiply each side by 5. 35 = 5 × 7 35 = 35 The result is equal. 24 030102a24 Example 3 NOTES Dividing both sides of an equation by a common term maintains the balance to the equation. In the following equation, both sides of the equation are divided by 4. 24 = 3 × 8 24 3×8 = Divide each side by 4 4 4 6=6 Transposing Equations To find an unknown value in a formula, do the same thing to each side to transpose the formula. This isolates the unknown value on one side of the equation. For example, given the equation for Ohm's Law which is E = I×R, solve for I. E IR = (Divide each side by R.) R R E I = (Cancel the Rs on the right side of the equation.) R E E = I or I = R R Example 1 E2 Given the equation P = solve for E. R E2 ×R R×P= (Multiply each side by R.) R E2 × R×P= (Cancel the Rs on the right side of the equation.) R × P = E2 √R × P = √E 2 (Take the square root of each side.) √R × P = E or E = √R × P When an equation has fractions on both sides, solve it using cross-multiplication. The numerator of one fraction is multiplied by the denominator of the other fraction and vice P 0.6 versa. For example, given the equation = , solve for P. 4 2 𝑃𝑃 0.6 Cross multiply ⟹ 2 × P = 4 × 0.6 4 2 2P = 2.4 2P 2.4 = (Divide each side by 2.) 2 2 P = 1.2 25 030102a24  NOTES If one side of an equation is not in fraction form, it can be made so by placing it over 1. 2P This makes cross-multiplication possible. For example, if = 10, find the value of P. 5 2P 10 = (Change the right side into a fraction.) 5 1 2P 10 (Cross multiply.) 5 1 1 × 2P = 10 × 5 2P = 50 2P 50 = (Divide each side by 2.) 2 2 P = 25 You can double-check your work by inserting the answer into the given equation to see if it balances. Use the equation in the previous example in which P = 25 and check as follows. 2P = 10 5 2×25 = 10 (Substitute 25 for P and calculate.) 5 50 = 10 5 10 = 10. Therefore, P equals 25. V For example, if I = , solve for V. R I V = (Change the left side into a fraction.) 1 R I × R = V × 1 (Cross multiply and calculate.) IR = V or V = IR Equations can also be solved by moving an expression from one side of the equation to the other. When this is done, the expression changes sign: positive to negative or negative to positive (+ to – or – to +). For example, If 2P + 4 = 8, find the value of P. 2P + 4 − 4 = 8 − 4 (Subtract 4 from both sides.) 2P = 8 − 4 26 030102a24 This is the same as changing the sign of the 4 on the left side and moving it to the right NOTES side of the equation (Figure 25). Figure 25 - Changing the sign from positive to negative. 2P = 4 2P 4 = (Divide both sides by 2.) 2 2 P=2 Example 2 If 10 − 3P = 2P, solve for the value of P (Figure 26). Figure 26 - Changing the sign from negative to positive. 10 = 5P 10 5P = (Divide each side by 5.) 5 5 2 = P or P = 2 Example 3 If R T = R1 + R 2 , transpose to find R2 as follows (Figure 27). R2 = RT − R1 Figure 27 - Changing a sign for formula transposition 27 030102a24  NOTES If you move an expression from one side to the other and you do not change the sign, you must change its function in order for the equation to remain true. For example, if 3A = 9 transpose to solve for A (Figure 28). 3A 9 = (Divide each side by 3 and calculate.) 3 3 9 A= (This is the same as changing the function of the 3 on the left side from 3 multiply to divide and moving it to the right side of the equation Figure 28 - Formula transposition using division. A=3 Example 4 B If = 25, solve for B (Figure 29). 10 B = 250 Figure 29 - Formula transposition using multiplication. Example 5 Taking the root of the value on to both sides of an equation by a common term maintains the balance to the equation. In the following equation, the square root of each side is calculated. 22 = 4 √22 = √4 (Square root of each side.) 2=2 28 030102a24 Example 6 NOTES If 200 = I 2 × 2, solve for I (Figure 30). Figure 30 -Transposition of a formula using division. 100 = I 2 √100 = √I 2 (square root of each side) 10 = I or I = 10 29 030102a24  NOTES Objective Five Exercise V 1. Use the equation I = to answer the questions. R a) Calculate I if V = 50 and R = 25. b) Calculate V if I = 7 and R = 20. c) Calculate R if V = 75 and I = 25. 2. Use the equation P = I 2 × R to answer the questions. a) Calculate P if I = 5 and R = 20. b) Calculate I if P = 1500 and R = 15. c) Calculate R if I = 20 and P = 2050. V2 3. Use the equation P = to answer the questions. R a) Calculate P if V = 250 and R = 20. b) Calculate V if P = 950 and R = 5. c) Calculate R if P = 2560 and V = 235. 30 030102a24  NOTES Objective One Exercise Answers 1. a) + addition b) - subtraction c) x multiplication d) ÷ division e) = equal f) / division 2. The equal sign means the values on either side of an equation are equivalent to each other, for example a = b + c. Objective Two Exercise Answers 1. 1164 2. 261.19 3. 7 ⁄16 4. 91⁄4 5. $504.14 6. 2311⁄2 m 7. 441 8. 7092.13 9. a) 5 ⁄8 b) 7 ⁄8 10. No, there are only 428 m left. The job is short by 9 m. 11. 4368 m 12. $3114.00 13. $3650.00 Objective Three Exercise Answers 1. 19 188 2. 1155 m 3. 2400 4. 55.125 5. a) 1 ⁄8 a. 1⁄2 6. $622.40 7. 56.1 kg 8. $207.20 31 030102a24  NOTES 9. 104 400 10. a) 31 5⁄8 b) 3089 1⁄2 11. a) 99.7 b) 0.505 12. a) 2 ⁄3 b) 6 ⁄7 13. 27 loops 14. $15.80 15. 0.656 Ω 16. $446.25 17. 15 weeks 18. $26.40 Objective Four Exercise Answers 1. 11 2. 12 3. 21.5 4. 24 5. 23 6. 69 m 7. 17 7⁄8 hp Objective Five Exercise Answers 1. a) I = 2 b) V = 140 c) R = 3 2. a) P = 500 b) I = 10 c) R = 5.125 3. a) P = 3125 b) V = 68.92 c) R = 21.572 32 030102a24  NOTES 33 030102a24  NOTES 34 030102a24 030102a VERSION 24 The Individual Learning Modules team is committed to continuous improvement and is interested in your comments regarding technical accuracy, content presentation, and instructional clarity. Please visit the Comments page of our website to submit your feedback: http://ilm.nait.ca. The ILM team would also like to thank the many instructors, individuals, partners, and companies for their support in the development of this content and gratefully acknowledges their valuable contribution. ILM Modules are owned/published jointly by NAIT and SAIT. NAIT and SAIT would like to acknowledge and thank the following Alberta partner institutes that support content development and updates: Grande Prairie Regional College Keyano College Medicine Hat College Olds College Lakeland College Northern Lakes College Red Deer Polytechnic Lethbridge College Portage College