Lufthansa Technical Training Mathematics PDF

This document satisfies the requirements of the “DGCA KCASR 1 Part 66 Appendix I” and is approved for use at AU Fundamentals M1 MATHEMATICS Rev.−ID: 1APR2023...

This document satisfies the requirements of the “DGCA KCASR 1 Part 66 Appendix I” and is approved for use at AU Fundamentals M1 MATHEMATICS Rev.−ID: 1APR2023 Author: SCP FOR TRAINING PURPOSES ONLY LTT Release: Jun. 09, 2023 Topics for Cabin Base ElectricianMechanic In compliance with: EASA Part-66; UAE GCAA CAR 66; CAAS SAR−66 Category B1/B2 M01−B12 E Training Manual Export Control For training purposes and internal use only. This Lufthansa Technical Training (LTT) technical data does not  Copyright by Lufthansa Technical Training GmbH (LTT). contain Export Administration Regulations (EAR) controlled LTT is the owner of all rights to training documents and information. training software. Any use outside the training measures, especially reproduction and/or copying of training documents and software − also extracts there of − in any format at all (photocopying, using electronic systems or with the aid of other methods) is prohibited. 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Lufthansa Technical Training Dept HAM US Lufthansa Base Hamburg Weg beim Jäger 193 22335 Hamburg Germany E-Mail: [email protected] Internet: www.LTT.aero Revision Identification:  The revision-tag given in the column ”Rev-ID” on the face  Dates and author’s ID, which may be given at the  The LTT production process ensures that the Training of this cover is binding for the complete Training Manual. base of the individual pages, are for information about Manual contains a complete set of all necessary pages in the latest revision of the content on that page(s) only. the latest finalized revision. Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 M01 MATHEMATICS FOR TRAINING PURPOSES ONLY! HAM US/O54 TrJ Aug 03, 2022 ATA DOC Page 1 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Arithmetic Terms and Signs M1.1 M01.01 ARITHMETIC ARITHMETICAL TERMS AND SIGNS INTRODUCTION PRIORITY RULES GENERAL OPERATIONS Just as studying a new language begins with learning basic words, the study of When a term is to be calculated, there is a priority rule for operations. mathematics begins with arithmetic, its most basic branch. Arithmetic uses real Multiplications and divisions must be done first. and non−negative numbers, which are also known as counting numbers, and Then additions and subtractions are done. consist of only four operations:  addition Example  subtraction 5 + 3  2 = 11  multiplication If you do the addition first and then multiply by 2, you get a wrong result.  division. While you have been using arithmetic since childhood, a review of its terms and operations will make learning the more difficult mathematical concepts much easier. DIGITS Numbers are represented by symbols which are called digits. There are nine digits which are 1, 2, 3, 4, 5, 6, 7, 8 and 9. We also use the symbol 0 (i.e. zero) where no digits exists. Digits and zero may be combined together to represent any number. TERMS A single number or a single variable or a combination of numbers, variables FOR TRAINING PURPOSES ONLY! and operator symbols is called term. So a term is an mathematical construction of different values or variables HAM US/OF2 SCP May 12, 2023 01|Terms and Signs|L1 Page 2 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Arithmetic Terms and Signs M1.1 THIS PAGE INTENTIONALLY LEFT BLANK FOR TRAINING PURPOSES ONLY! HAM US/OF2 SCP May 12, 2023 01|Terms and Signs|L1 Page 3 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Methods of Multiplication and Division M1.1 METHODS OF MULTIPLICATION AND DIVISION ADDITION GENERAL DIGITS BEHIND DECIMAL POINT The process of finding the total of two or more numbers is called addition. This This procedure is identical if numbers have some digits after the decimal point, operation is indicated by the plus (+) symbol. but it is still necessary that the decimal points are in the same column. When numbers, called summands, are combined by addition, the resulting total How many digits are after the Decimal point is irrelevant. is called the sum. Example When adding whole numbers whose total is more than nine, it is necessary to arrange the numbers in columns so that the last digit of each number is in the Tens Ones Tenth same column. The ones column contains the values zero through nine, the 7. 8 tens column contains multiples of ten, up to ninety, and the hundreds column + 2 4. 3 consists of multiples of one hundred. + 4 6. 2 Example 7 8. 3 Hundreds Tens Ones 7 8 + 2 4 3 + 4+1 6 +1 2 7 8 3 To add the sum of the above, first add the ones column: 8 and 3 and 2 makes 13. Place the 3 in the ones column of the answer and carry the 1 forward to the tens column. Adding this we add: 1 and 7 and 4 and 6 is 18. Place the 8 in the tens column of the answer and carry the 1 forward to the hundreds column which we now FOR TRAINING PURPOSES ONLY! add. 1 and 2 and 4 is 7. Place the 7 in the hundreds column of the answer. We see that the answer (sum) to the addition is 783. HAM US/OF2 KrA Apr 19, 2023 02|Calculations|L1 Page 4 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Methods of Multiplication and Division M1.1 SUBTRACTION GENERAL HOW TO PROCEED The process of finding the difference between two numbers is known as There are two methods by which subtraction can be performed. subtraction and is indicated by the minus (−) sign. Consider 15 − 8 = 7 Subtraction is accomplished by taking the quantity of one number away from another number. The number which is being subtracted is known as the Method 1 subtrahend (smaller number), and the number from which the quantity is Take 8 from 15. taken is known as the minuend (larger number). We have 7 left. To find the difference of two numbers, arrange them in the same manner used This method is the most common method. for addition. With the minuend on top and the subtrahend on the bottom, align the vertical columns so the last digits are in the same column. Beginning at the Another way is also used to write the calculation, especially when the numbers right, subtract the subtrahend from the minuend. Repeat this for each column. are greater. 15 Example: – 8 Hundreds Tens Ones 7 4 (1)4 3 Method 2 – 2+1 6 2 Add various numbers to the 8. When the result is 15, you found the correct 1 8 1 number. If we add 8 to 7 then we obtain 15. 7 is therefore the difference between 15 Place 262 under 443. and 8. 2 from 3 leaves 1. This method is a kind of “try and error“. Write 1 in the ones column of the answer. DECIMAL NUMBERS 6 from 4 is clearly impossible, so the 4 is increased in value to 14 by taking 1 from the hundreds column leaving 3. For this procedure all decimal points need to be in the same column. 14 from 6 leaves 8. Write 8 in the tens column. FOR TRAINING PURPOSES ONLY! Finally, 3 from 2 in the hundreds columns leaves 1. To check a subtraction problem, you can add the difference to the subtrahend to find the minuend. HAM US/OF2 KrA Apr 19, 2023 02|Calculations|L1 Page 5 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Methods of Multiplication and Division M1.1 MULTIPLICATIONS GENERAL PROCEDURE Multiplication is a special form of repetitive addition. Like addition and subtraction, when multiplying large numbers it is important When a given number is added to itself a specified number of times, the that they have to be aligned vertically. process is called multiplication. When multiplying numbers greater than nine, multiply the multiplicand by each The sum of 4 + 4 + 4 = 12 is expressed by multiplication as digit in the multiplier. Once all multiplicands are used as a multiplier, the products of each multiplication operation are added to arrive at a total product. 4  3 = 12. The numbers 4 and 3 are called factors and the answer 12, represents the Example product. Multiplicand is 532 The number multiplied (4) is called the multiplicand, and the multiplier Multiplier is 24 represents the number of times the multiplicand is added to itself, in this case 3. Write: SIGN 532  24 Multiplication is typically indicated by the symbol  Calculation: In many countries the middle dot is used, which is also called interpunct. 532  24 An operation with the middle dot looks like this: 2  2 = 4 1064 (first partial product) In some cases (e.g. algebra and/or brackets), it is denoted by the lack of any + 2128 (second partial product) other operation sign. 12768 (product) Examples The product is 12,768. 5  3 = 15 5  3 = 15 5ab=5ab SEQUENCE When multiplying, the order in which numbers are multiplied does not change FOR TRAINING PURPOSES ONLY! the product. Examples 3  4 = 12 4  3 = 12 HAM US/OF2 KrA Apr 19, 2023 02|Calculations|L1 Page 6 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Methods of Multiplication and Division M1.1 DIVISIONS GENERAL CALCULATION Just as subtraction is the reverse of addition, division is the reverse of 416 : 8 = 52 multiplication. Division is a means of finding out how many times a number is 40 contained in another number. The number divided is called dividend, the 16 number you are dividing by is the divisor, and the result is the quotient. 16 REMAINDER 0 With some division problems, the quotient may include a remainder. A 1. 4 : 8 = Not possible remainder represents that portion of the dividend that cannot be divided by the 2. 41 : 8 = 5 Remainder 1 divisor. 3. 16 : 8 = 2 Remainder 0 SIGN 1. Take the first digit, the 4. Division is indicated by the use of the division signs () or ( : ) with the Try to divide 4 by 8. This is not possible. dividend to the left and the divisor to the right of the sign, or with the dividend inside the sign and the divisor to the left. 2. Take the first two digits, the 41. Try to divide 41 by 8. This is 5 and the remainder is 1. FRACTIONS 3. Take the remainder 1 and the third digit, the 6. An other way of representing Divisions is the use of fractions Combine to 16. For example in the fraction 3 the 3 is the dividend, also called numerator, and Try to divide 16 by 8. This is 2 and the remainder is 0. 8 8 is divisor, also known as denominator. When the division is performed our 4. Take the first digit you got (in step 2) and the second digit you got (in step result is the quotient 0,375. 3). This is 52, and this is the result. LARGE QUANTITIES The quotient is 52. The process of dividing large quantities is performed by breaking the problem CHECK down into a series of operations, each resulting in a single digit quotient. To check a division problem for accuracy, multiply the quotient by the divisor This is best illustrated by an example. and then add the remainder (if any). FOR TRAINING PURPOSES ONLY! Example If the operation is carried out properly, the result equals the dividend. Let us divide 416 by 8. Dividend is 416 Divisor is 8 Write: 4168 = 52 or 416 : 8 = 52 HAM US/OF2 KrA Apr 19, 2023 02|Calculations|L1 Page 7 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Methods of Multiplication and Division M1.1 SIGNED NUMBERS ADDING SIGNED NUMBERS MULTIPLYING SIGNED NUMBERS When adding two or more numbers with the same sign, ignore the sign and find Multiplication of signed numbers is accomplished in the same manner as the sum of the values and then place the common sign in front of the answer. multiplication of any other number. However, after multiplying, the product must In other words, adding two or more positive numbers always results in a be given a sign. There are three rules to follow when determining a products positive sum, whereas adding two or more negative numbers results in a sign. negative sum. Examples When adding a positive and negative number, find the difference between the two numbers and apply (+ or −) of the larger number. In other words, adding 1. The product of two positive numbers is always positive. negative number is the same as subtracting a positive number. The result of 6 x 2 = 12 adding or subtracting signed numbers is called algebraic sum of those 2. The product of two negative numbers is always positive. numbers. (−6) x (−2) = 12 Example 3. The product of a positive and a negative number is always negative. Add 25 + (−15) (-6)  2 = –12 25 + (−15) = 10 6 (-2) = –12 25 – 15 = 10 DIVIDING SIGNED NUMBERS SUBTRACTING SIGNED NUMBERS Like multiplying signed numbers, division of signed numbers is accomplished in When subtracting numbers with different signs, change the operation sign to the same manner as dividing any other number. plus and change the sign of the subtrahend. Once this is done, proceed as you The sign of the quotient is determined using the rules identical to those used in do in addition. multiplication. For example +3 − (−4) is the same as +3 + 4. Examples There is no difference if the subtrahend is larger than the minuend, since the 12  3 =4 operation is done as though the two quantities are added. 12  (–3) = –4 Example (–12)  3 = –4 FOR TRAINING PURPOSES ONLY! Subtract 48 from −216 (–12)  (–3) =4 Step 1: Set up the subtraction problem −216 − 48 Step 2: Change the operation sign to a plus sign and change the sign of the subtrahend. Now add: (−216) + (−48) = −264 HAM US/OF2 KrA Apr 19, 2023 03|Signed No|L2|B12 Page 8 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Methods of Multiplication and Division M1.1 THIS PAGE INTENTIONALLY LEFT BLANK FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Apr 19, 2023 03|Signed No|L2|B12 Page 9 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Fractions and Decimals M1.1 FRACTIONS AND DECIMALS GENERAL PRINCIPLE GENERAL ADDING DECIMALS Working with fractions is typically time consuming and complex. One way you The addition of decimals is done in the same manner as the addition of whole can eliminate fractions in complex equations is by replacing them with decimal numbers. However, care must be taken to correctly align the decimal points fractions or decimals. A common fraction is converted to a decimal fraction by vertically. dividing the numerator by the denominator. Example As result we get numbers were behind the ones come a decimal point and some more digits. Add the following 25.78 + 5.4 + 0.237 For example,  is converted to a decimal by dividing the 3 by the 4. Rewrite with the decimals aligned and add. The decimal equivalent of  is 0.75. Improper fractions are converted to 25.78 decimals in the same manner. However, whole numbers appear to the left of + 5.4 the decimal point. + 0.237 In a decimal, each digit represents a multiple of ten. The first digit represents 31.417 tenths, the second hundredths, the third thousandths. Once everything is added, the decimal point in the answer is placed directly Examples below the other decimal points. 0.5 means 5 x  and is read as five tenths SUBTRACTING DECIMALS  Like adding, subtracting decimals is done in the same manner as with whole 0.05 means 5 x  and is read as five hundredths numbers. Again, it is important that you keep the decimal points aligned.  0.005 means 5 x  and is read as five thousandths Example  If you have 325.25 kilogram of ballast on board and remove 30.75 kilogram, When writing decimals, the number of zeros to the right of the decimal does not how much ballast remains? affect the value as long as no other number except zero appears. In other 325.25 kg FOR TRAINING PURPOSES ONLY! words, numerically, 2.5, 2.50 and 2.500 are the same. – 30.75 kg DECIMAL PLACES 294.50 kg The number of digits after the decimal point is called decimal places. Examples 27.6 one decimal place 27.16 two decimal places 27.026 three decimal places and so on. HAM US/OF2 KrA Apr 19, 2023 04|Decimals|L1 Page 10 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Fractions and Decimals M1.1 MULTIPLYING DECIMALS DIVIDING DECIMALS When multiplying decimals, ignore the decimal points and multiply the resulting When dividing decimals, the operation is carried out in the same manner as whole numbers. division of whole numbers. However, to ensure accurate placement of decimal Once the product is calculated, count the number of digits to the right of the point in the quotient, two rules apply: decimal point in both the multiplier and multiplicand. This number represents 1. When the divisor is a whole number, the decimal point in the quotient aligns how many digits must be right of the decimal point in the product. vertically with the decimal in the dividend when doing long division. If there ist an zero before decimal point it can be omitted, when decimal point is 2. When the divisor is a decimal fraction, it should first be converted to a set. whole number by moving the decimal point to the right. However, when the decimal in the divisor is moved, the decimal in the dividend must also move Example in the same direction and the same number of spaces. Multiplicand is 26.757 (3 decimal places) Example Multiplier is 0.32 (2 decimal places) Divide 37.26 by 2.7. Write: Dividend is 37.26 26.757 x 0.32 Divisor is 2.7 Calculation: Move the decimal in the divisor to the right to convert it to a whole number. 26 757  32 Move the decimal in the dividend the same number of places to the right. 80271 Calculation: + 53514 372.6 : 27 = 13.8 856224 27 The Product is 856224. 102 Count 5 decimal places to the left of the last digit. Then set the decimal point. 81 The result is 8.56224. 216 216 0 FOR TRAINING PURPOSES ONLY! 1. 3727 = 1 Remainder 10 2. 10227 = 3 Remainder 21 3. Take 6 and set the point 4. 21627 = 8 Remainder 0 HAM US/OF2 KrA Apr 19, 2023 04|Decimals|L1 Page 11 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Factors and Multiples M1.1 FACTORS AND MULTIPLES FACTORS INTRODUCTION MULTIPLES Factors can be used to count the units. Multiples are the simple product of a multiplication. If something weights 5 times one kilogram so 5 is a factor. A soccer team has 11 sportsmen on the field. The calculation is a multiplication. During the ceremonies at the beginning of Olympic Games you may see many Sometimes short expressions for factors are used for units. These are called teams the same time. So you have 44, 99 or even more sportsmen. prefixes. This is used to shorten long expressions like 80 000 metres to 80 km. This is a multiple. 99 sportsmen on the lawn, that is 9 times a team. The k is for kilo and means the factor 1 000. DIVIDING MULTIPLES Kilo, hecto, deca, centi and milli are the most common metric prefixes in daily When you divide a multiple by 2, 3, 4 etc. you will get a natural number. use. All other show either values too big or too small for daily use. When you see multiple sportsmen on the field and you divide this number by In the metric world – all over the world except USA and Myanmar – metric the sportsmen in a team, you get the number of teams. prefixes are used. For example 99 sportsmen divided by 11 sportsmen for a team has a result of They all are based on the factor 10. 9 teams. METRIC PREFIXES Prefix Symbol Factor Decimal Power tera T 1 000 000 000 000 x1012 giga G 1 000 000 000 x109 mega M 1 000 000 x106 kilo k 1 000 x103 hecto h 100 x102 FOR TRAINING PURPOSES ONLY! deca da 10 x101 deci d Divide by 10 x10−1 centi c Divide by 100 x10−2 milli m Divide by 1 000 x10−3 micro m Divide by 1 000 000 x10−6 nano n Divide by 1 000 000 000 x10−9 HAM US/OF2 KrA Jun 07, 2023 05|Multiples/Fractions|L1 Page 12 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Factors and Multiples M1.1 COMMON FRACTIONS INTRODUCTION LOWEST TERMS A common fraction represents a portion or part of a quantity. A fraction is said to be in its lowest terms when it is impossible to find a For example, if a number is divided into three equal parts, each part is number which will divide exactly into both its numerator and denominator.  one−third 1 of the number. 3 The fractions 5 and 11 are both in their lowest terms but the fraction 6 is not 7 19 10 A fraction consists of two numbers, one above and one below a line, or in its lowest terms because it can be reduced to 3 by dividing top and bottom 5 fraction bar. The fraction bar indicates division of the top number, or numbers by 2. numerator, by the bottom number, or denominator. Example For example, the fraction 3 indicates that three is divided by four to find the 4 decimal equivalent of 0.75. Reduce 21 to its lowest terms. 35 When a fractions numerator is smaller than the denominator, the fraction is 21 = 21  7 = 3 called a proper fraction. A proper fraction is always less than 1. If the 35 35  7 5 numerator is larger than the denominator, the fraction is called an improper fraction. In this situation the fraction is greater than 1. If the numerator and the denominator are identical, the fraction is equal to 1. A mixed number is the combination of a whole number and a proper fraction. Mixed numbers are expressed as 1 5 and 29 9 and are typically used in place 8 16 of improper fractions. The numerator and denominator of a fraction can be changed without changing the fractions value. A mixed number can be converted into an improper fraction and vice versa. Example 1 Express the mixed number 8 2 as a proper fraction. 3 FOR TRAINING PURPOSES ONLY! (8  3)  2 24  2 26 82 = = = 3 3 3 3 Example 2 Express the improper fraction 27 as a mixed number. 4 27 = 24  3 = 24  3 = 6 3 4 4 4 4 4 (since 27  4 = 6 remainder 3) HAM US/OF2 KrA Jun 07, 2023 05|Multiples/Fractions|L1 Page 13 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Fractions M1.1 HANDLING FRACTIONS COMPARING THE SIZE OF FRACTIONS ADDING & SUBTRACTING FRACTIONS When the values of two or more fractions are to be compared, express each of Two fractions which have the same denominator can be added together by the fractions with the same denominator. This common denominator should be adding their numerators. the lowest common denominator of the denominator of the fractions to be compared. It is sometimes called LCM (lowest common denominator). Example 1 3  5 = 35  8 Example 11 11 11 11 Arrange the fractions 5 , 8 and 7 in order of size beginning with the smallest. When two fractions have different denominators they cannot be added together 6 9 8 directly. However, if we express the fractions with the same denominator they The LCM of the denominators 6, 8, and 9 is 72. can be added. 5 ist gleich 5  12 = 60 6 6  12 72 Example 2 8 ist gleich 8  8 = 64 Add 2 and 3 9 9  8 72 5 7 7 ist gleich 7  9 = 63 The lowest common denominator of 5 and 7 is 35. 8 8  9 72 2  3 = 2  7  3  5 = 14  15 Because all the fractions have been expressed with the same denominator all 5 7 5  7 7  5 35 35 that we need to do is to compare the numerators. = 14  15 35 Therefore the order of size is 60 , 63 and 64 or 5 , 7 and 8 72 72 72 6 8 9 29 = 35 FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Jun 07, 2023 06|Fractions|L2|B12 Page 14 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Fractions M1.1 ADDITION OF MIXED NUMBERS SUBTRACTION OF MIXED NUMBERS When mixed numbers are to be added together, the whole numbers and the To subtract two fractions which do not do have the same denominator, a fractions are added separately. method similar to that for addition is used. Example Example 1 Add 42 and 23 Subtract 3 from 5 3 5 4 6 The lowest common denominator is 12. 42 23 = 6 2  3 3 5 3 5 5 3 = 10 9 = 6  10  9 6 4 12 12 15 15 (10 9) = = 6  19 12 15 =  = 61 4  15 When mixed numbers are involved first subtract the whole numbers and then = 7 4 deal with the fractional parts. 15 Example 2 Subtract 4 1 from 6 3 3 4 6 3 41 = 2 3 1 4 3 4 3 (9 4) = 2 12 =    = 2 5 12 FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Jun 07, 2023 06|Fractions|L2|B12 Page 15 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Fractions M1.1 MULTIPLICATION OF FRACTIONS MULTIPLICATION OF MIXED NUMBERS Multiplication of fractions is performed by multiplying the numerators of each Mixed numbers must be converted into improper fractions before multiplying. fraction to form the product numerators, and multiplying the individual denominators to form the product denominator. The resulting fraction is then Example reduced to its lowest terms. Multiply 1 3  2 1 8 3 Example 1 1 3  2 1 = 11  7 Multiply 3  5 8 3 8 3 8 7 11  7 = 3  5 = (3  5) 83 8 7 (8  7) = 77 24 = 15 56 = 3 5 If any factors are common to a numerator and a denominator they should be 24 cancelled before multiplying. In problems with fractions the word “of” is frequently used. It should always be taken as meaning “multiply”. Example 2 Find the value of 2  5  21 3 7 32 2  5  21 = 1  5  1 3 7 32 1  1  16 = 5 16 FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Jun 07, 2023 06|Fractions|L2|B12 Page 16 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Fractions M1.1 DIVISION OF FRACTIONS Division of common fractions is accomplished by inverting, or turning over, the divisor and then multiplying. However, it is important that you invert the divisor only and not the dividend. Once the divisor is inverted, multiply the numerators to obtain a new numerator, multiply the denominators to obtain a new denominator. Then reduce the quotient to its lowest terms. Example Divide 3 by 7 5 8 37 = 38 5 8 5 7 = 38 57 = 24 35 Mixed numbers must be converted into improper fractions before dividing. FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Jun 07, 2023 06|Fractions|L2|B12 Page 17 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Weights and Measures M1.1 WEIGHTS AND MEASURES WEIGHTS GENERAL Weight is defined as the gravitational pull of the earth on a given body. This is a force. The direction of this force is regarded toward the geometrical center of the earth. Physicists are very careful to distinguish between “mass & weight”. But in normal life, people do not distinguish. We mean mass but say weight. On the market, we buy an amount of potatoes with an exact weight. Due to this fact in the following we speak correctly about the mass. So we should say: ”we buy potatoes with an mass of one kilogram”. SYSTEM OF MEASUREMENT The system of measurement is based mainly on the International System of Units, usually abbreviated as SI. In french: Système international d’unités. UNIT The kilogram (kg) is the SI unit for mass. All SI units today are base on physical constants, such as the kilogram which refers to the Planck-constant. In 1889 the kilogram was defined by a platinum – iridium alloy cylinder. This original or prototype kilogram is stored in a safe near Paris, the French capital. It was used until 2019 for reference. Using the decimal system, based on number of 10 we are able to calculate bigger and smaller values. In accordance to the decimal system 1 kg consists FOR TRAINING PURPOSES ONLY! of 1000 g (gram) and 1 g in turn consists of 1000 mg (milligram). Multiplying 1 kg by 1000 we reach 1 Mg (mega gram) usually called 1 metric ton. In daily life we use the units between gram and ton. In th US the Pound (lb) is used. The unit lb is a abbreviation for the latin word ”libra”. The conversion is: 1 lb = 0.45359237 kg HAM US/OF2 SCP Jun 05, 2023 07|Weights Measures|L1 Page 18 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Weights and Measures M1.1 Prefix Symbol Factor Decimal Power tera T 1 000 000 000 000 x1012 giga G 1 000 000 000 x109 mega M 1 000 000 x106 kilo k 1 000 x103 hecto h 100 x102 deca da 10 x101 deci d Divide by 10 x10−1 centi c Divide by 100 x10−2 milli m Divide by 1 000 x10−3 micro m Divide by 1 000 000 x10−6 nano n Divide by 1 000 000 000 x10−9 1kg FOR TRAINING PURPOSES ONLY! Figure 1 Decimal System Kilogram HAM US/OF2 SCP Jun 05, 2023 07|Weights Measures|L1 Page 19 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Weights and Measures M1.1 MEASURES GENERAL Measures are used to give an exact impression of a distance. This may be the length of a street or the length of an object. UNIT The meter (m) is the SI unit for length. Since 1983 the scholars use natural constants to define some basic units. Therefore the meter is defined as the distance travelled by light in a vacuum in 1 of a second. 299, 792, 458 The speed of light is a natural constant and defined with 299,792,458 meter per second. Natural constants remain unchanged and can be reproduced all over the world, at least theoretically. In 1799 the original or prototype meter was defined as the 10millionths part of 1/4 of an earth meridian. A meridian quadrant is approximately the distance from the geographic pole to the equator. Also the prototype meter is made of a platinum – iridium alloy. It was used until 1960. Using the decimal system, based on number of 10 we are also able to calculate bigger and smaller values. In accordance to the decimal system 1 m consists of 10 dm or 100 cm or 1000 mm. Multiplying 1 m by 1000 we reach 1 km. In daily life we use the values between kilometer and millimeter. In th US the Metric system is not used, for length measurement foot (ft) is used. The conversion is: FOR TRAINING PURPOSES ONLY! 1 ft = 0.3048 m HAM US/OF2 SCP Jun 05, 2023 07|Weights Measures|L1 Page 20 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Weights and Measures M1.1 FOR TRAINING PURPOSES ONLY! Figure 2 Decimal System Metric HAM US/OF2 SCP Jun 05, 2023 07|Weights Measures|L1 Page 21 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Conversions M1.1 CONVERSIONS GENERAL As already mentioned, the majority of the countries in Europe and most of the countries in the world are using the SI units to define the basic units and their derived units. This simplifies international communication and the description of standards and technical data. But there are also other unit systems. For example, quantities with units from the Middle Ages or the Anglo−American system of measurement, which is used in the USA, can be converted into metric units. Some units are often still used in the imperial system of measurement, e.g. speeds or heights in aviation. Here are some conversion factors that are used to express a quantity in a different unit than the original one. FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Apr 20, 2022 08|Unit Conversions|L1|B12 Page 22 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Conversions M1.1 Length Water - Mass and Volume 1 in (inch) = 25.4 mm 1 Liter Water = 1 kg 1m = 39.37 in or 3.281 ft 1 pint Water = 1 lb 12 in = 1 ft Force & Weight 1 ft (foot) = 12 in or 0.3048 m 1N = 0.2248 lbf = 3.5969 ozf 1 yd (yard) = 3 ft or 36 in or 0.9144 m 1 km = 0.621 miles Torque 1 mile = 1760yd = 5280 ft or 1.61 km 1 daNm = 10 Nm 1 n.m. = 1.151 miles or 1.852 km 1 Nm = 8.851 lbf in 1 lbf ft = 12 lbf in Area 1 m2 = 10,000 cm2 = 10.76 ft2 Velocity 1 acre = 4840yd2 = 4046.87m2 1 km/h = 3.6 m/s 1 yd2 = 9 ft2 1 m/s = 3.281 ft/s 1 ft2 = 144 in2 or 0.0929 m2 1 mph = 1.47 ft/s 1 in2 = 6.452 cm2 1 mph = 1.61 km/h 1 knot = 1.688 ft/s = 1.151 mph = 1.852 km/h Volume 1 m3 = 1 000 dm3 = 1000 liter = 7,481 gal Pressure 1 ft3 = 1728 in3 = 0.0283 m3 1 Pa = 0.000145 lb/in2 1 liter = 1000 cm3 = 1.0576 qt (US) 1 bar = 100 000 Pa = 1000 hPa = 14.5038 PSI 1 qt (US quart) = 0.8327 qt (UK) = 0.9464 liter 1 bar = 750.0638 mm Hg = 29.53 in Hg 1 gal (US gallon) = 8 pints (US) 1 atm = 29.92 in Hg 1 gal (UK) = 4.546 liters 1 PSI = 689 kPa 1 gal (US) = 3.785 liters Power FOR TRAINING PURPOSES ONLY! Mass 1 HP (metric) = 735.4988 W = 0.9863 HP (UK) 1 amu = 1.66 x10−27 kg = 542.4760 lbf ft/s 1 kg = 1000 g 1 HP (UK) = 745.6999 W = 550 lbf ft/s 1 metric ton = 1000 kg 1W = 0.738 ft lb/s 1 lb = 16 oz = 0.4536 kg = 453.6 g 1 Btu/h = 0.293 W 1 oz = 28.3495 g 1 slug = 14.59 kg HAM US/OF2 KrA Apr 20, 2022 08|Unit Conversions|L1|B12 Page 23 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Conversions M1.1 Temperature (conversion formula may required) D1C = D1 K (Kelvin) 1 C = 33.8 F (Fahrenheit) = 493.47 R (Rankine) 1 F = – 17.22 C 1 R = – 272.59 C = – 458.67 F = 0.5556 K 0K = – 273.15 C = – 459.67 F = 0 R F = C  95  32 C = (F 32)  5 9 Energy 1J = 0.738 ft lb 1 cal = 4.186 J 1 Btu = 252 cal Time 1 year = 365 days 1 day = 24 h = 1440 min 1 Hour = 3600 s FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Apr 20, 2022 08|Unit Conversions|L1|B12 Page 24 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Conversions M1.1 THIS PAGE INTENTIONALLY LEFT BLANK FOR TRAINING PURPOSES ONLY! HAM US/OF2 KrA Apr 20, 2022 08|Unit Conversions|L1|B12 Page 25 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Ratio and Proportion M1.1 RATIO AND PROPORTION RATIOS GENERAL A ratio is a comparison between two quantities. Example 1 A truck fuel tank has a Volume of 120 liter. The tank contains 30 liter of fuel. So it contains 30 l of the maximum possible 120 l. 30 l 30 l : 120 l =  30  1 120l 120 4 The tank is  full! Example 2 Write the following ratio as a fraction and truncate as much as possible: 40 mm at 2.2 m 2.2 m = 2 200 mm 40 mm 40 mm : 2 200 mm= = 1 2200 mm 55 Example 3 800 g at 1.6 kg 1.6 kg = 1 600 g 800 g =1 FOR TRAINING PURPOSES ONLY! 800 g : 1 600 g = 1600 g 2 HAM US/OF2 KrA May 09, 2023 09|Ratio|L1 Page 26 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Ratio and Proportion M1.1 ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎÎÎÎÎÎÎ FOR TRAINING PURPOSES ONLY! Figure 3 Fuel Tank Indication HAM US/OF2 KrA May 09, 2023 09|Ratio|L1 Page 27 Lufthansa Technical Training MATHEMATICS EASA PART-66 M01 ARITHMETIC Ratio and Proportion M1.1 MOTOR/PROPELLER RATIO For example, if an engine turns at 4000 revolutions per minute or rpm in short, and the propeller turns at 2400 rpm, the ratio of the two speeds is 4000 to 2400, or 5 to 3, when reduced to lowest terms. This relationship can also be expressed as 5/3 or 5:3. n Engine  5 n Propeller =  = 3 COMPRESSION RATIO One ratio you must be familiar with is compression ratio, which is the ratio of cylinder displacement when the piston is at bottom center to the cylinder displacement when the piston is at top center. For example, if the volume of a cylinder with the piston at bottom center is 400 cubic centimeters and the volume with the piston at