B-1 Maths Student Resource Feb 2024 PDF

Student Resource Subject B-1: Mathematics Feb., 2024 BUSINESS...

Student Resource Subject B-1: Mathematics Feb., 2024 BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Feb., 2024 B-1 Mathematics Page 2 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  CONTENTS CONTENTS 3 DEFINITIONS 5 STUDY RESOURCES 6 INTRODUCTION 7 TOPICS 1.1 – Arithmetic 1.2 – Algebra 1.3 - Geometry Feb., 2024 B-1 Mathematics Page 3 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  This Page Intentionally Left Blank Feb., 2024 B-1 Mathematics Page 4 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  DEFINITIONS Define  To describe the nature or basic qualities of.  To state the precise meaning of (a word or sense of a word). State  Specify in words or writing.  To set forth in words; declare. Identify  To establish the identity of. List  Itemise. Describe  Represent in words enabling hearer or reader to form an idea of an object or process.  To tell the facts, details, or particulars of something verbally or in writing. Explain  Make known in detail.  Offer reason for cause and effect. Feb., 2024 B-1 Mathematics Page 5 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  STUDY RESOURCES B-1 Student Resource IMPORTANT NOTICE:  NO ELECTRONIC DEVICES ARE ALLOWED IN ANY OF THE TESTS OR THE FINAL EXAMINATION.  ALL CALCULATIONS MUST BE CARRIED OUT BY MANUAL METHODS ONLY. Feb., 2024 B-1 Mathematics Page 6 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  INTRODUCTION The purpose of this subject is to give you a fundamental knowledge of the mathematical theory required to assist you in your study of the more specific aviation maintenance theory. On completion of the following topics you will be able to: Topic 1.1 Arithmetic Arithmetical terms and signs, methods of multiplication and division, fractions and decimals, factors and multiples, weights, measures and conversion factors, ratio and proportion, averages and percentages, areas and volumes, squares, cubes, square and cube roots. Topic 1.2 Algebra Evaluating simple algebraic expressions, addition, subtraction, multiplication and division, use of brackets, simple algebraic fractions. Linear equations and their solutions. Indices and powers, negative and fractional indices. Binary and other applicable numbering systems. Simultaneous equations and second degree equations with one unknown. Logarithms. Topic 1.3 Safety Precautions Associated with Aircraft Maintenance Operations Geometry Simple geometrical constructions. Graphical Representation: Nature and uses of graphs, graphs of equations/functions. Simple Trigonometry: Trigonometrical relationships, use of tables and rectangular and polar coordinates. Feb., 2024 B-1 Mathematics Page 7 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  B-1 Mathematics Table of Contents CONTENTS................................................................................................................................. 3 DEFINITIONS.............................................................................................................................. 5 STUDY RESOURCES..................................................................................................................... 6 INTRODUCTION.......................................................................................................................... 7 1.1 Arithmetic....................................................................................................................... 14 1.1.1 - Common Mathematical Signs................................................................................................. 14 Whole Numbers....................................................................................................................... 15 Addition of Whole Numbers.............................................................................................................. 15.............................................................................................................................................. 16 Subtraction of Whole Numbers......................................................................................................... 17.............................................................................................................................................. 18 Multiplication of Whole Numbers..................................................................................................... 19.............................................................................................................................................. 20 Division of Whole Numbers............................................................................................................... 21.............................................................................................................................................. 22 Averages............................................................................................................................................. 23.............................................................................................................................................. 23 Estimation.......................................................................................................................................... 24.............................................................................................................................................. 24 Decimals.................................................................................................................................. 25 Addition of Decimals.......................................................................................................................... 25.............................................................................................................................................. 26 Subtraction of Decimals..................................................................................................................... 27.............................................................................................................................................. 27 Multiplication of Decimals................................................................................................................. 28.............................................................................................................................................. 28 Division of Decimals........................................................................................................................... 29............................................................................................................................................ 30............................................................................................................................................ 31 Rounding Off...................................................................................................................................... 32............................................................................................................................................ 32 Feb., 2024 B-1 Mathematics Page 8 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Fractions............................................................................................................................................. 33 Conversion......................................................................................................................................... 34............................................................................................................................................ 35............................................................................................................................................ 36 Basic Operations with Fractions................................................................................................ 37 Addition and Subtraction................................................................................................................... 37............................................................................................................................................ 39 Solve the following and leave your answer as an improper fraction.................................................... 39 Multiplication of Fractions................................................................................................................. 40 Division of Fractions........................................................................................................................... 41............................................................................................................................................ 42 Converting Fractions to Decimals...................................................................................................... 43 (Round answers to 2 decimal places as required)......................................................... 43 Converting Decimals to Fractions...................................................................................................... 44............................................................................................................................................ 44............................................................................................................................................ 45 Conversions........................................................................................................................................ 45............................................................................................................................................ 45 Conversions........................................................................................................................................ 45 Order of Operations........................................................................................................................... 46............................................................................................................................................ 48 Directed Numbers.................................................................................................................... 51 Addition and Subtraction................................................................................................................... 51............................................................................................................................................ 53............................................................................................................................................ 55 Review Homework..................................................................................................................................... 55 Percentages............................................................................................................................. 60............................................................................................................................................ 61 Expressing a Number as a Percentage of another Number.............................................................. 62............................................................................................................................................ 62 Finding a Percentage of a Number.................................................................................................... 63............................................................................................................................................ 63 Ratios:..................................................................................................................................... 64 Rules:.......................................................................................................................................................... 64............................................................................................................................................ 65 Proportion.......................................................................................................................................... 66............................................................................................................................................ 67 Feb., 2024 B-1 Mathematics Page 9 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Direct Proportion............................................................................................................................... 67 Rules:.......................................................................................................................................................... 67............................................................................................................................................ 68 Indirect Proportion............................................................................................................................. 69............................................................................................................................................ 69 Mixtures............................................................................................................................................. 70............................................................................................................................................ 70 Weight (Mass).................................................................................................................................... 71 Metric System............................................................................................................................................. 71 Measurement.......................................................................................................................... 73 Fundamental Quantities.................................................................................................................... 73 Fundamental Units..................................................................................................................................... 73 Area............................................................................................................................................................ 73 Gallon......................................................................................................................................................... 73 Derived Quantities............................................................................................................................. 74 Conversion of Units............................................................................................................................ 75............................................................................................................................................ 77 Converting Between Measuring Systems.......................................................................................... 78............................................................................................................................................ 79 Areas and Volumes.................................................................................................................. 80 Calculating Area................................................................................................................................. 80............................................................................................................................................ 81............................................................................................................................................ 82 Calculating Volumes........................................................................................................................... 83............................................................................................................................................ 84 Square and Cube Roots............................................................................................................ 85 Square Roots.............................................................................................................................................. 85 Calculating the Square Root without a Calculator..................................................................................... 85 Cube Roots................................................................................................................................................. 86 1.2 Algebra........................................................................................................................... 87 1.2.1 - Evaluating Simple Algebraic Expressions................................................................................ 87 Principal Rules.................................................................................................................................... 87 The Associative Law.................................................................................................................................... 88 The Commutative Law................................................................................................................................ 88 The Distributive Law................................................................................................................................... 89 Feb., 2024 B-1 Mathematics Page 10 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only............................................................................................................................................. 90............................................................................................................................................ 93 Algebraic Fractions............................................................................................................................. 94............................................................................................................................................ 95 1.2.2 Linear Equations.............................................................................................................. 96............................................................................................................................................ 99 Transposition of Formulae............................................................................................................... 100.......................................................................................................................................... 100.......................................................................................................................................... 101 Using Formula (Substitution)........................................................................................................... 102.......................................................................................................................................... 102 Factorisation.................................................................................................................................... 104.......................................................................................................................................... 106.......................................................................................................................................... 108 Homework Review................................................................................................................................... 108 1.2.3 Index Notation............................................................................................................... 111.......................................................................................................................................... 112.......................................................................................................................................... 113 Conversions...................................................................................................................................... 113 Index Rules....................................................................................................................................... 114.......................................................................................................................................... 115.......................................................................................................................................... 117.......................................................................................................................................... 118 Negative Indices............................................................................................................................... 120 Zero Exponent.................................................................................................................................. 122.......................................................................................................................................... 122 Index Laws Summary................................................................................................................................ 123.......................................................................................................................................... 123 1.2.4 Binary and Other Applicable Numbering Systems........................................................... 124.......................................................................................................................................... 129 Homework:............................................................................................................................................... 129 1.2.5 Simultaneous Linear Equations...................................................................................... 130 Substitution Method........................................................................................................................ 130 Elimination Method (recommended only if substitution is too complex)...................................... 131.......................................................................................................................................... 133 Quadratic Equations......................................................................................................................... 134 Example 1................................................................................................................................................. 134 Example 2................................................................................................................................................. 135 Feb., 2024 B-1 Mathematics Page 11 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Factorisation Method....................................................................................................................... 136.......................................................................................................................................... 140.......................................................................................................................................... 141 Exponential Expressions Involving Mixed Bases.............................................................................. 142.......................................................................................................................................... 143 1.2.6 Scientific Notation......................................................................................................... 144.......................................................................................................................................... 146.......................................................................................................................................... 147.......................................................................................................................................... 149 Significant Figure.............................................................................................................................. 150 Rounding Off.................................................................................................................................... 150.......................................................................................................................................... 152 Logarithms........................................................................................................................................ 153 Logarithmic Scales............................................................................................................................ 155.......................................................................................................................................... 156 1.3 Geometry...................................................................................................................... 157 1.3.1 Angles...................................................................................................................................... 157.......................................................................................................................................... 159 Circles............................................................................................................................................... 161 Properties of a Circle................................................................................................................................ 161 Pi............................................................................................................................................................... 164.......................................................................................................................................... 165 Radians............................................................................................................................................. 166 Sine Waves....................................................................................................................................... 167 Types of Triangle.............................................................................................................................. 168 Pythagoras' Theorem and Basic Trigonometry................................................................................ 169 Pythagoras' Theorem is:........................................................................................................................... 169.......................................................................................................................................... 171 Trigonometric Ratios........................................................................................................................ 173 Table of Trigonometric Functions.................................................................................................... 175 The Application of Trigonometric Ratios......................................................................................... 176 3, 4, 5 Triangle.......................................................................................................................................... 179.......................................................................................................................................... 179.......................................................................................................................................... 182 1.3.2. Types of Graphs............................................................................................................ 185 Pictographs....................................................................................................................................... 185 Feb., 2024 B-1 Mathematics Page 12 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only........................................................................................................................................... 185 Circle Graph or Pie Chart.................................................................................................................. 186.......................................................................................................................................... 186 The Bar Graph.................................................................................................................................. 188 Interpretation........................................................................................................................................... 188 Histogram................................................................................................................................................. 189.......................................................................................................................................... 190 The Line Graph................................................................................................................................. 191 Interpolation.................................................................................................................................... 192 Extrapolation.................................................................................................................................... 192.......................................................................................................................................... 192 Constructing Graphs........................................................................................................................ 193.......................................................................................................................................... 194 1.3.3 The Rectangular Co-ordinate System.............................................................................. 201 Locating Points on the Cartesian Plane........................................................................................... 202.......................................................................................................................................... 203 Equation of a Straight Line............................................................................................................... 204.......................................................................................................................................... 206 Polar Coordinates............................................................................................................................. 207.......................................................................................................................................... 209 Vectors............................................................................................................................................. 210.......................................................................................................................................... 211 Answers:........................................................................................................................................... 213 Feb., 2024 B-1 Mathematics Page 13 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  1.1 Arithmetic 1.1.1 - Common Mathematical Signs + add addition sum − minus subtract difference ÷ divided by division quotient × multiplied by times product = equals ≠ not equal to > greater than ≥ greater than or equal to < less than  less than or equal to ± plus or minus ∝ proportional to  therefore Feb., 2024 B-1 Mathematics Page 14 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Whole Numbers Addition, subtraction, multiplication and division require practice and to this end, the following exercises will refresh your memory and ability. Addition of Whole Numbers If numbers are large or there are many of them, it is generally best to write them in a column. Take care to keep the numbers of the same place value in the same column from right to left. Example Add 452 and 317 452 4 hundreds + 5 tens + 2 unit + 317 3 hundreds + 1 ten + 7 unit =7 hundreds + 6 tens + 9 units =769 Therefore, 452 + 317 = 769 Exercise on next page Feb., 2024 B-1 Mathematics Page 15 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Add the following sets of numbers: a) 36 + 46 + 531 if 613 b) 82 + 189 + 2473 2744 c) 463 + 39 + 2975 3477 34 d) 1192 + 2 749 + 14 298 1422198 18239 2749 e) 6752 + 4097 + 208 k 6752 1 057 4097 Feb., 2024 B-1 Mathematics Page 16 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Subtraction of Whole Numbers To subtract two whole numbers, take away the bottom number from the top number in each column from right to left. Where the bottom number is larger than the top a factor of ten needs to be borrowed and then returned to the next bottom number on the left. Example Subtract 235 from 586 586 5 hundreds + 8 tens + 6 units - 235 2 hundreds + 3 tens + 5 units = 351 3 hundreds + 5 tens + 1 unit Therefore, 586 - 235 = 351 Exercise on next page Feb., 2024 B-1 Mathematics Page 17 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Subtract the following sets of numbers: 3 a) 93 – 46 4 47 b) 6023 – 5145 c) 853 – 442 411 84 d) 50 063 – 3294 46769 8 46769 e) 856 – 37 819 66 8 Feb., 2024 B-1 Mathematics Page 18 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Multiplication of Whole Numbers In multiplication, the numbers multiplied together are called factors; the result is called a product. To multiply any whole number by say one-digit number, multiply each digit of the whole number from right to left by the one-digit number, two-digit number, etc. Then add the products together. Example Recall that 376 = 300 + 70 + 6. When multiplying by 376, you in fact multiply by 300 and 70 and 6 and then add these results together. 854 x 376 854 x 300 = 256 200 854 x 70 = 59 780 854 x 6 = 5124 = 321 104 This is usually done as follows: 854 (multiplied by) 376 5124 (multiply by 6) 59 780 (multiply by 70) 256 200 (Multiply by 300) 321 104 Exercise on next page Feb., 2024 B-1 Mathematics Page 19 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Multiply the following numbers: a) 46 x 6 276 b) 142 x 5 442g 710 To c) 67 x 14 267 938 12 d) 104 x 15 1560 2 e) 2398 x 13 31174 2398 Ti Feb., 2024 B-1 Mathematics Page 20 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Division of Whole Numbers "12 divided by 3" means "the number that results in 12 when multiplied by 3". In other words, if 12 is a product and 3 is one factor, then "12 divided by 3" is the other factor. The result is 4. Symbolically, "12 divided by 3" is written: 12  3 = 4 12 is called the dividend, 3 is called the divisor and the result is called the quotient. Example Divide 38500 by 67 Working from left to right, 385 is the smallest number that can be divided by 67. 5 67 38500 - 335 By trial and error, you establish that 5 x 67 is just 50 less than 385, the difference being 50. 5 67 38500 Bring the next digit down (zero in this case). 335 The question now is: 'how many times - 500 will 67 go into 500? The original process is repeated. 574 67 38500 335 500 469 310 268 42 Answer: 574 with a remainder of 42 or (42/67) Feb., 2024 B-1 Mathematics Page 21 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Divide the following, giving quotient with the remainder as a whole number. a) 742  5 148 sit b) 1385  4 41585 321 c) 2496  62 4 Éo d) 74 938  34 22 to doublecheck 2204 34 7493672 Feb., 2024 B-1 Mathematics Page 22 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Averages The average of a set of numbers is found by adding the samples together and dividing by the number of samples. Example Find the average of 12, 15, 23, 21, 40, and 19 12 + 15 + 23 + 21 + 40 + 19 = 130 130  6 = 21.67 (2dp) a) Find the average of 34, 25, 28, 31, 40, and 29 18 31.160 affif b) Find the average of 121, 157, 230, 217, 303, and 196 1284 204 c) Find the average of 12.6, 15.2, 23.1, 21.3, 40.7, and 19.8 I 22 42.6 22.125 15.2 21.3 1 40.7 24 19.8 6 132.7 K Feb., 2024 B-1 Mathematics Page 23 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Estimation Without a calculator, long multiplication and division can use up too much time and be prone to errors. Rounding to very simple numbers can provide an estimate and help with locating the decimal point. Estimation may be enough in a Multi-Choice exam. Example 1. 987.62 x 77.85 Estimate 1000 x 80 = 80 000 Therefore, 987.62 x 77.85 = 76.886.22 (2dp) 2. 7,703.65 ÷ 42.98 Estimate 8000 ÷ 40 = 200 Therefore 7703.65 ÷ 42.98 = 179.24 (2dp) Provide estimates for the following calculations: 388 a) 296.438 x 61.87 18888 300 60 18000 sheet b)wronganswerinthe 496.3179 ÷ 23.91 500 20 25 f c) 789.48329 ÷ 0.0187 0.0187 1 8 800 00 789.48329 10000 800 1000 80050 40.000 789.48429 1g 200 Feb., 2024 B-1 Mathematics Page 24 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Decimals Addition of Decimals The most important thing about adding decimals is that when you write them in a column you keep numbers of the same place value in the same column. This can be done easily with decimals by ensuring that the decimal points are in a column one under the other. Example 25.2 + 3.546 + 0.31 + 16.02 becomes: 25.200 3.546 0.310 16.020 45.076 Note: The decimal points are lined up in a column. As you add them, use the same method that was used for adding whole numbers and make sure that the decimal point in the answer is in the correct column under all the other decimal points. If there is no number in a column then treat it as a zero. Feb., 2024 B-1 Mathematics Page 25 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Add the following decimal numbers: a) 3.46 + 4.72 b) 4.801 + 0.76 c) 19.89 + 1.35 + 20.001 w̅ d) 24.36 + 13.52 e) 8.2 + 0.102 + 104.082 1104.082 8.283 112 384 Feb., 2024 B-1 Mathematics Page 26 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Subtraction of Decimals This is done in the same way as subtracting whole numbers and again we ensure the correct place value of the numbers by putting the decimal points under one another. Example 28.37 - 9.526 can be expressed as: 28.370 -9.526 18.844 Note: If the decimal points are lined up, a zero is implied if there is no number in a column. a) 6.96 – 5.85 6.96 b) 6.39 – 2.17 839 4 22 c) 303.9 – 18.72 285.18 d) 24.3 – 12.74 11 56 e) 7.003 – 3.924 88313 3 924 3 079 Feb., 2024 B-1 Mathematics Page 27 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Multiplication of Decimals To multiply numbers which have a decimal point, first carry out the multiplication as if neither number had a decimal point, for example to multiply 7.962 by 21.4 you would multiply 7962 x 214. Example 7962 7.962 has three figures after the decimal x 214 point and 21.4 has one figure after the 31 848 decimal point so the answer must be (3 + 1) 79 620 figures after the decimal point. Hence the 1 592 400 + answer is 170.3868 170 386 844 a) 0.2 x 0.3 0.06 a 82 b) 1.103 x 0.2 b 1183 0.2206 c) 0.8316 x 4.3 C 883 3.57588 9 i d) 37.96 x 4.5 2 170.820 D 34 37948 e) 124.03 x 12 18980 e 12403 12 24806 124030 1488.36 Feb., 2024 B-1 Mathematics Page 28 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Division of Decimals Step 1: Count the number of decimal places in the divisor. Step 2: Move the decimal point to the right in both the divisor and the dividend as the result of Step l; that is, the number of decimal places required to have the decimal point after the last digit in the divisor. Step 3: Write the divisor and the dividend in long division form and divide as if they were whole numbers. Step 4: The decimal point in the result should be aligned with the decimal point of the dividend. Example Divide 6.356 by 2.8 Step 1: Note that there is one decimal place in the divisor. Step 2: Move decimal to the right one decimal place. (63.56 and 28) 2.27 Step 3 and 4: 28 63.56 56 7.5 5.6 1.96 1.96 0.00 Feb., 2024 B-1 Mathematics Page 29 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Calculate the following to the number of decimal places (dp) where indicated: a) Wrondanswer inthe 6.27 sheet ÷2 b) 8.94 ÷ 0.9 (to two decimal points) 9.53 8 c) 0.747 ÷ 0.06 12.45 d) 24.276 ÷ 5.1 e) 0.7252 ÷ 0.37 196 222 Feb., 2024 B-1 Mathematics Page 30 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Cancel or simplify the following: Example: To simplify the equation, divide the top and bottom numbers by a common factor (this is a number that will divide into the numerator and denominator evenly), for example, 10 x 3 can be written as 10 x 3 5x3 5 3 common factor is 5 and 3 because 5 will go into 10 twice and 3 cancel each out =2x1 = 2 1x1 20 × 2 7×3 a I 4 42 10 × 4 I b II 14 × 6 2 2 4 1202 4.2 × 6.3 4 c 600 × 4 d 2.1 × 2 f 2,434 4 5 2 0.3 4 × 3 × 6.4 9.6 × 7.5 3 s e 2×3 f 3.2 × 5 I as 12.8 decimit cancelthe7.5 becausewecanceld decimwith are9.0and3.2 decimilwhich Feb., 2024 B-1 Mathematics Page 31 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Rounding Off You would have found in exercises that some long divisions can go on and on before arriving at an exact answer (some do not have an exact answer; they are called irrational numbers). To save time we sometimes round-off our answer to a specified number of decimal places. Rounding off is carried out as follows: Say we must round-off 687.6760563 to 2 decimal places. Look at the third figure after the decimal point (6). Ignore all figures after (i.e. ignore 0563). If the third figure after the decimal point is less than 5 ignore it too. If the third is 5 or more then the second figure is increased by 1. Thus, the answer is written as 1687.68. Round off the following to the specified number of places: a) 2.96438 (2 dp) 296 b) 496.3179 (3 dp) 496318 c) 769.48329 (1 dp) 769.5 d) 769.48329 (4 dp) 769.4833 Feb., 2024 B-1 Mathematics Page 32 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Fractions A fraction is another way of writing a quotient. "5 divided by 12" is written in fractional form as 5/12 and read, "5 over 12". It also means, “5 out of 12” The dividend is placed over the line and is called the numerator. The divisor is placed below the line and is called the denominator of the fraction. A vulgar fraction is a fraction having the numerator above and the denominator below a horizontal or diagonal line. 5  numerator 12  denominator A fraction is called a proper fraction when the numerator is smaller than the denominator. A fraction is called an improper fraction when the numerator is greater than the denominator. 12 5 is an improper fraction, is a proper fraction. 5 12 Improper fractions may be expressed as a combination of a whole number and a proper fraction. When expressed in this form, they are called mixed numbers. 1 4 is a mixed number. 2 Feb., 2024 B-1 Mathematics Page 33 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Conversion To convert a mixed number to an improper fraction, multiply the denominator by the whole number and add the numerator. The product is then placed over the denominator. The result is called an improper fraction. 1 Convert 4 to an improper fraction. 2 1 2 4 1 9 4   2 2 2 To convert improper fractions to mixed numbers divide the numerator by the denominator. 4 1 9  2 9 and 1 remainder = 4 2 2 To convert vulgar fractions to fractions in lowest terms, find a common divisor between the numerator and the denominator and divide both. 21 21  7 3   3 7 77 1 or 7 77 1   21 21  7 3 Feb., 2024 B-1 Mathematics Page 34 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Express the following mixed number as improper fractions: 1 a) 10 2 10 21 21 13 b) 3 16 3 12 3 c) 8 8 8 8,1 3 d) 1 100 1 e) 62 6 1 1 Feb., 2024 B-1 Mathematics Page 35 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Use cancelling to simplify, (reduce) the following fractions. Where possible express as a mixed number. 18 a) 36 33 b) 121 75 c) 175 312 d) 56 5 3 s 180 e) 32 32989 5 58 Feb., 2024 B-1 Mathematics Page 36 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Basic Operations with Fractions Addition and Subtraction The example below shows how you can add and subtract fractions. If you have a mixed number, always change it to an improper fraction first, before you add, subtract, multiply or divide fractions. Examples 3 2 a. + (Find the Lowest Common Denominator - LCD) 4 3 9+8 17 = 12 12 5 1 12 (always leave as a mixed number or proper fraction) 1 1 b. - (LCD = 12) 4 6 3-2 12 1 12 1 1 3 1 c. 12+4= 2+4 (LCD = 4) 6+1 7 3 = =1 4 4 4 1 1 1 d. 3 2 -1 4 -2 2 7 5 5 - - (LCD = 4) 2 4 2 14-5-10 4 1 - 4 Feb., 2024 B-1 Mathematics Page 37 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  This page intentionally blank. Feb., 2024 B-1 Mathematics Page 38 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Solve the following and leave your answer as an improper fraction. 7 1 a) + 8 2 1 1 1 b) + + 2 4 8 f I I 3 5 c) 2 4 +1 8 5 2 d) - 6 3 3 1 e) 2 - 4 32 38 1 5 f) 2 - 16 32 2 16 1 1 3 g) 5 -2 2 4 Feb., 2024 B-1 Mathematics Page 39 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Multiplication of Fractions Examples 2 4 2 4 8    a) 3 5 3  5 15 6 14 6  2 12 b)    (Note how 14 was cancelled with 7) 7 19 119 19 2 6 9   c) 3 14 12 We can cancel 6 and 12 (by 6), 3 and 9 (by 3), and 2 and 14 (by 2), giving: 1 1 3 3    d) 1 7 2 14 Feb., 2024 B-1 Mathematics Page 40 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Division of Fractions To divide by a fraction, we invert the second fraction then multiply. Examples 9 2 9 3 27 a) ÷ becomes × = 10 3 10 2 20 6 13 6 14 6 2 12 b) ÷ = × = × = 7 14 7 13 1 13 13 1 7 9 8 3 2 6 1 c) 2 ÷1 = × = × = =1 4 8 4 15 1 5 5 5 7 3 1 7 3 2 7 d) × ÷1 = × × = 8 4 2 8 4 3 16 9 1 3 9 5 4 6 e) ÷ ÷ = × × = =6 10 5 4 10 1 3 1 1 1 3 3 17 16 17 f) 1 ×2 ÷ = × × = = 17 2 8 16 2 8 3 1 Feb., 2024 B-1 Mathematics Page 41 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Calculate the following, giving your answers as a fraction in its simplest form or a mixed number where applicable. 2 1 a) × 3 4 5 12 b) × 6 14 1 5 c) 1 ×2 3 8 8 5 84 51 or 4 1 5 d) 2 ×3 × out 7 2 6 9 9 4 1 7 e) ÷ 10 8 s 5 1 f) 1 ÷2 9 3 2 3 1 1 7 5 g) 1 ×1 ÷ 5 4 9 6 Feb., 2024 B-1 Mathematics Page 42 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Converting Fractions to Decimals Remember that a fraction can be regarded as a division. To convert a fraction to a decimal we simply perform that division. Example 3  34 4 0.75 4 3.00 28 20 20 00 3 Thus is the same as 0.75 4 (Round answers to 2 decimal places as required) Convert the following fractions to decimals. 1 a) 2 D 50 291 5 b) 9 off So 3 c) 10 0 30 103T 4 d) 7 D 57 o Feb., 2024 B-1 Mathematics Page 43 of 228 Training Material Only BUSINESS DOCUMENT This document is intended for business use and should be distributed to intended recipients only.  Converting Decimals to Fractions Decimals are fractions with a denominator of 10 (or 100, 1 000, etc.). Example 1 0.1 = 10 7 0.07 = 100 3 0.003 = 1000 15 3 0.15 = 100 = 20 486 243 0.486 = 1000 = 500 Convert the following decimals to fractions or mixed numbers in their simplest form: a) 0.79 b) 0.925 To wife c) 2.875 d) 0.4 ÉIE