Maths & Science Principles Lesson 4 PDF

ENGINEERING MATHEMATICS & SCIENCE PRINCIPLES UNIT CODE -EOK2-003 3.UNDERSTAND FUNDAMENTAL NUMERACY APPLIED ENGINEERING Identify the techniques used for calculating approximation Add, subtract, multiply and divide: whole numbers, fractions and decimals Convert...

ENGINEERING MATHEMATICS & SCIENCE PRINCIPLES UNIT CODE -EOK2-003 3.UNDERSTAND FUNDAMENTAL NUMERACY APPLIED ENGINEERING Identify the techniques used for calculating approximation Add, subtract, multiply and divide: whole numbers, fractions and decimals Convert fractions to decimals and decimals to fractions Calculate average, mean, median and mode Calculate ratio, proportion and percentages Calculate area, surface area, mass, volume, capacity Calculate probability Calculate the square and square root of a number Transpose simple formulae Calculate speeds and feeds Construct simple graphs Calculate values for similar triangles  BASIC ARITHMETIC OPERATIONS Arithmetic (a term derived from the Greek word arithmos, “number”) refers generally to the elementary aspects of the theory of numbers, arts of measurement, and numerical computation. The four basic arithmetic operators are add (+) subtract (−) multiply (×) divide (÷) MATHS SYMBOLS 4 SIMPLIFYING FRACTIONS STARTER How many fractions can you think of that are the same as a 1/2? SIMPLIFYING FRACTIONS When ‘cancelling’ any Fraction to its simplest form, you must always look for numbers that divide into both the numerator (top number) and denominator (bottom number) ÷4 4 1 You can divide both 8 2 numbers by 4 ÷4 SIMPLIFYING FRACTIONS When ‘cancelling’ any Fraction to its simplest form, you must always look for numbers that divide into both the numerator (top number) and denominator (bottom number) ÷3 6 2 You can divide both 9 3 numbers by 3 ÷3 CANCELLING FRACTIONS When ‘cancelling’ any Fraction to its simplest form, you must always look for numbers that divide into both the numerator (top number) and denominator (bottom number) ÷7 21 3 You can divide both 28 4 numbers by 7 ÷7 SIMPLIFYING FRACTIONS When ‘cancelling’ any Fraction to its simplest form, you must always look for numbers that divide into both the numerator (top number) and denominator (bottom number) ÷9 18 2 You can divide both 81 9 numbers by 9 ÷9  Sometimes you can simplify more than once  make sure the fraction is fully simplified before moving on… SIMPLIFYING FRACTIONS When ‘cancelling’ any Fraction to its simplest form, you must always look for numbers that divide into both the numerator (top number) and denominator (bottom number) ÷3 ÷2 12 6 2 18 9 3 You can divide both numbers by 2..and then by 3! ÷2 ÷3 FRACTIONS AND MIXED NUMBER S  There are three types of fractional numbers 1 3 4 Numerator smaller Proper , , etc than the 2 8 7 Fractions: Denominator 3 6 9 Numerator greater Improper , , etc than the 2 5 4 Fractions: Denominator 1 2 3 A whole number plus Mixed 1 , 4 , 2 etc a proper fraction 2 5 4 Fractions: Every mixed fraction can be written as an improper fraction Every improper fraction can be written as a mixed fraction 1 3 1 2 = 2 How to convert an improper fraction into a mixed number 1.Divide the numerator by the denominator. 2.Your answer is the whole number. Your remainder becomes the numerator of the fraction. For example 17/5: 17÷5 = 3 quotient 2 remainder = 3 *2/5 How to convert a mixed number into an improper fraction 1.Multiply the whole number by the denominator, then add the numerator. 2.The answer becomes the numerator and the denominator stays the same. For 2.6/8: 2x8 = 16. Then add the numerator (16+6 = 22). The answer is 22/8. FRACTIONS AND MIXED NUMBER S Today we will be looking at changing Mixed numbers into top-heavy fractions, and the other way 13 34 25 Mixed Numbers  2 5 6 15 11 5 7 3 4  Top Heavy Fractions This will allow you to do some difficult sums entirely in your head! FRACTIONS AND MIXED NUMBER S 12 Write 4 as a top-heavy fraction How many halves would make 4 wholes?  We need 8 halves to make 4 wholes  We also have an extra ‘half’  In total we have 9 halves 9  This means ‘9 halves’ 2 FRACTIONS AND MIXED NUMBER S 23 Write 3 as a top-heavy fraction How many thirds would make 3 wholes?  We need 9 thirds to make 3 wholes  We also have an extra two ‘thirds’  In total we have 11 thirds 11  This means ‘11 thirds’ 3 ADDING AND SUBTRACTING FRACTIONS  What is a Common Denominator? When the denominators of two or more fractions are the same, they are Common Denominators. Why is it Important? Before we can add or subtract fractions, the fractions need to have a common denominator In other words the denominators must be the same. Making The Denominators the Same To make the denominators the same we can: Multiply top and bottom of each fraction by the denominator of the other. STARTER ‘Adding and Subtracting Fractions’ + = + = 1 2 3 1 1 + = + 4 4 4 3 4 4 3 7 + = 12 12 12 STARTER ‘Adding and Subtracting Fractions’ - = 4 - 2 5 3 2 12 10 - 15 = 15 15 ADDING AND SUBTRACTING FRACTIONS Work out the following sum… 3 1 The denominators must be the - same before adding or 8 4 subtracting… Multiply all by 2 3 2 1 - = 8 8 8 ADDING AND SUBTRACTING FRACTIONS Work out the following sum… 1 5 The denominators must be the + same before adding or 6 8 subtracting… Multiply all by 4 Multiply all by 3 4 15 19 + = 24 24 24 ADDING AND SUBTRACTING FRACTIONS Calculate the Perimeter of this Patio 1 3 2 3/4 yds 4 3 + 2 4 Multiply all the Multiply all the fraction by 4 fraction by 3 4 1/3 yds 4 9 4 12 + 2 12 13 1 = 6 12 = 7 12 2 The perimeter will be double this = 14 12 = 14 1 yards 6 ADDING AND SUBTRACTING FRACTIONS Work out the following sum… 2 3 The denominators must be the - same before adding or x y subtracting… Multiply all by y Multiply all by x 2y 3x 2y - 3x - = xy xy xy MULTIPLYING FRACTIONS  Multiplying & Dividing Multiplication & Division Rules of Signed Numbers: If same signs, result is positive. If different signs, result is negative. a c axc Multiplication of Fractions x  b d bxd Division of Fractions a c a d   x b d b c MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) Work out the following… When Multiplying Fractions, you multiply the tops together 1 x 1 = 1 and the bottoms together 2 2 4 MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) Work out the following… When Multiplying Fractions, you multiply the tops together 3 x 5 = 15 and the bottoms together 4 7 28 This question is effectively asking, ‘what is 5/7 ths of 3/4? MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) Work out the Area of this Rectangle in square inches… 1 1 3 1/4 inches 5 x 3 5 4 26 13 338 5 1/5 inches x = 5 4 20 338 9 = 16 10 Sq. in 20 MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) The rules are the same with Algebraic Fractions… When Multiplying Fractions, you multiply the tops together a x a = a2 and the bottoms together 3 4 12 MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) The rules are the same with Algebraic Fractions… When Multiplying Fractions, you multiply the tops together 3b x 2b = 6b2 and the bottoms together 2 7 14 6b2 3b2 14 7 Divide top and bottom by 2 MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) The rules are the same with Algebraic Fractions… When Multiplying Fractions, you multiply the tops together 5pq x 2p = 10p2q and the bottoms together 9 5 45 10p2q 2p2q 45 9 Divide top and bottom by 5 MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) Sometimes you can simplify the calculation first! Divide the top and bottom by x (This gives us an answer that is already simplified!) MULTIPLYING FRACTIONS (INCLUDING ALGEBRA) Sometimes you can simplify the calculation first! Imagine we swapped the numerators (which is fine as in multiplication, the order isn’t important!) Divide both by t (This gives us an answer that is already simplified!) DIVIDING BY A FRACTION DIVIDING BY A FRACTION What does the sum; 3 ÷ /2 1 mean? 2 halves = 1 4 halves = 2 6 halves = 3 whole wholes wholes DIVIDING BY A FRACTION What does the sum; 5 ÷ 1/2 mean? It means ‘how many halves do we need to make 5 wholes’? 2 halves = 1 4 halves = 2 6 halves = 3 8 halves = 4 10 halves = whole wholes wholes wholes 5 wholes DIVIDING BY A FRACTION What do you notice? 3 ÷ 1/2 = 6 5 ÷ 1/2 = 10  Dividing by 1/2 is the same as multiplying by 2! So 12 ÷ 1/2 = 24 And 55 ÷ 1/2 = 110 DIVIDING BY A FRACTION We can now look at more tricky situations involving fractions… 23 12 When dividing, the rule is, ‘Leave, ÷ change and flip’ Leave Change Flip 1) Leave the first Fraction 2) Change the sign to multiply 23 21 43 3) Flip the second Fraction x = = 13 1 DIVIDING BY A FRACTION Work out: 2 1 5 When dividing, the rule is, ‘Leave, 11 ÷ change and flip’ Leave Change Flip 1) Leave the first Fraction 2) Change the sign to multiply 2 5 1 10 3) Flip the second Fraction 11 x = 11 DIVIDING BY A FRACTION We can now look at more tricky situations involving fractions… 49 35 When dividing, the rule is, ‘Leave, ÷ change and flip’ Leave Change Flip 1) Leave the first Fraction 2) Change the sign to multiply 49 53 20 3) Flip the second Fraction x = 27 FRACTION TO DECIMAL FRACTION TO DECIMAL We are going to look at a non-calculator method (which may even allow you to do some divisions in your head!)  The key is ‘hundredths’ x2 12 24 = 0.24 50 100 x2 FRACTION TO DECIMAL We are going to look at a non-calculator method (which may even allow you to do some divisions in your head!)  The key is ‘hundredths’ x4 3 12 = 0.12 25 100 x4 FRACTION TO DECIMAL We are going to look at a non-calculator method (which may even allow you to do some divisions in your head!)  The key is ‘hundredths’ x 10 7 70 = 0.70 (0.7) 10 100 x 10 FRACTION TO DECIMAL We are going to look at a non-calculator method (which may even allow you to do some divisions in your head!)  The key is ‘hundredths’ x5 11 55 = 0.55 20 100 x5 FRACTION TO DECIMAL We are going to look at a non-calculator method (which may even allow you to do some divisions in your head!)  The key is ‘hundredths’ x 20 4 80 = 0.80 (0.8) 5 100 x 20 PRACTICE QUESTIONS How could you answer these divisions? Remember that a division a) 3 ÷ 20 b) 7 ÷ 25 can be written as a fraction! c) 4 ÷ 50 d) 21 ÷ 20 DECIMAL TO FRACTION DECIMAL TO FRACTION Types of number  Terminating Decimals – decimals that ‘end’  0.4, 0.023, 0.0765 etc  Recurring Decimals – decimals that never end, but have a pattern  0.333 , 0.12121 2 , 0.005675675 6 7  Irrational Numbers – decimals that never end and have no pattern  √2, π DECIMAL TO FRACTION Converting a Decimal to a Fraction… 0.7 7 10 ÷2 18 9 50 0.18 100 ÷2 ÷2 222 111 500 0.222 1000 ÷2 DECIMAL TO FRACTION Converting a Decimal to a Fraction… ÷2 0.6 6 10 35 ÷2 ÷2 ÷2 24 12 50 6 0.24 100 25 ÷2 ÷2 ÷5 ÷5 ÷5 125 25 200 5 40 1 0.125 1000 8 ÷5 ÷5 ÷5 MULTIPLYING DECIMALS MULTIPLYING DECIMALS A holiday firm books 48 coaches, which can each carry 52 people. How many people could they accept bookings for in total? x 40 8 48 x 52 50 2000 400 = 2400 2 80 16 = 96 = 2496 MULTIPLYING DECIMALS Imagine we had the same sum with decimal places… Work out 48 x 5.2 x 40 8 48 x 5.2 50 2000 400 = 2400 There is one ‘decimal place’ in the question. There must also be one 2 80 16 = 96 in the answer = 2496 = 249.6 MULTIPLYING DECIMALS Imagine we had the same sum with decimal places… Work out 12.3 x 2.7 x 100 20 3 20 2000 400 60 = 2460 12.3 x 2.7 There are two ‘decimal places’ in 7 700 140 21 = 861 the question. There must also be two in the answer = 33.21 = 3321 DIVIDING DECIMALS DIVIDING DECIMALS 20 ÷ 5 = 4 x10 x10 No change 200 ÷ 50 = 4 “In a division, as long as you change both numbers by the same amount, the answer will remain the same” DIVIDING DECIMALS 7 ÷ 0.2 = 35 x10 x10 70 ÷ 2 = 35 “In a division, as long as you change both numbers by the same amount, the answer will remain the same” DIVIDING DECIMALS 12 ÷ 0.3 = 40 x10 x10 120 ÷ 3 = 40 “In a division, as long as you change both numbers by the same amount, the answer will remain the same” DIVIDING DECIMALS 17 ÷ 0.04 = 425 x100 x100 4 1700 ÷ 4 = 425 8 12 16 20 24 0 4 2 5 28 1 1 2 32 4 1 7 0 0 36 40 DIVIDING DECIMALS 19.8 ÷ 0.25 = 79.2 x100 x100 25 1980 ÷ 25 = 79.2 50 75 100 125 150 0 0 7 9. 2 175 1 19 23 5 200 2 5 1 9 8 0. 0 225 250 NOTE: Operations with Decimals: To add or subtract: line up dec. pts. To multiply: number of dec. places in the product is the sum of the number of dec. places in the factors. To divide: if divisor is whole number, bring decimal pt. up. If divisor is not, move decimal point as needed.  Adding & Subtracting 1 1 9 2 2   2  2 9 18 18 Remember to find common denominators first. Did you forget the 2  Adding & Subtracting 1 5  2 1  − 2+ 4 4 1 3 3 1 A. 3 B. C. - D. -1 4 4 4 4 It is subtraction! Subtract smaller from larger and give same sign as larger. (Thus result is negative) Adding & Subtracting 2 2 -  ( 1)  1 3 3 5 1 1 A. - B. - 1 C. D. - 3 3 3 First let us change the -(-1) to +1 Remember: bigger minus smaller, sign bigger! (result must be positive) Multiplication & Division  1  2 1 3 3  -     ×   5  3 5 2 10 10 3 3 10 A. B. C. - D. - 3 10 10 3 Same signs means positive result!! Remember to invert the second fraction!  If a unit of water costs $1.82 and 40.435 unit were used, which is a reasonable estimate? (Water is sold…) A. $80,000 B. $800 C. $8000 D.$80  500 students took an algebra test. All scored less than 92 but more than 63. Which of the following could be a reasonable estimate of the avg. score? A. 96 B. 63 C. 71 D. 60  Decimal Examples 14.22 - 1.761= 14.220 -1.761 A.12.459 It is smaller than 14.22 - 1.22=13 B.13.459 C.11.459 It is larger than 14.22 -2=12.22 D.12.261  Decimal Examples 3.43 x 2.8 A. 0.9604 Estimate 3 x 3 = 9 B. 8.504 Larger than 3 x 2.8 = 8.24 C. 7.1344 D. 9.604  Decimal Examples. 735 36.75 0.05  0.05 36.75 A. 735 Dividing by a number between 0 and 1 will cause the result to B. 73.5 be larger than original number C. 7.35 D. 0.0735 EQUIVALENCE. Rational numbers can be written as fractions, mixed numbers, dec. or % 1 Example .25 25% 4  Equivalence Examples 0.19= 19 100 19 % 9 19 19 A. B. 1 C. D. 100 10 10 100 0.19 is not greater than 1 % “means divided by 100” 19/100 %=0.19/100=0.0019 Equivalence Examples 350%= A. 0.350 B. 3.50 C. 350.0 D. 3500  Equivalence Examples 92 3. 100 A. 0.92 B. 0.092 C. 9.2% D. 0.92%  Percent Examples If 30 is decreased to 6, % decrease? 4 (diff.) 24 p  (orig.) 30 100 5 5p = 400 p = 80 A. 8% B. 24% C. 20% D. 80%  Percent Examples What is 120% of 30? x 120 10x = 360  30 100 x = 36 A. 0.25 B. 25 C. 36 D. 3.6  Word Problems A car rents for $180 per week plus $0.25 per mile. Find the cost of renting this car for a two week trip Of 400 miles for a family of 4. D. $760 A. $280 B. $380 C. $460 Bracke ts 85 INDICES Indices is the mathematical term for ‘power’. It is the plural for index. 5 This is the index or This is the Base 4 the power number What are Indices? 4³ is an example of a number in index form. It can be described as ‘4 cubed' or ‘4 to the power of 3'. The ³ is the index in the example above (also called 4³ =or4 exponent). power ×4×4 4³ = 64 22 is another example of a number in index form. Index here 2 is It can be described as ‘ 2 squared ’ or 2 to ‘ the ’. 2 × 2 power of 2 22 = 4 22 =  Examples 3 4 (7 )(6 )  (7  7  7) (6  6  6  6) 90 91 92 93 95 96 98  MEAN, MEDIAN AND MODE  Mean Consider the following set of numbers 8, 2, 3, 5, 5, 7, 12. The Mean: To find the mean, add up all the data, and then divide this total by the number of values in the data. Sum of data 8+2+ 3+5+5+ 7+12 42 𝑀𝑒𝑎𝑛= = =6 7 7 Number of data  Median Let’s now find the median for the same set of numbers 8, 2, 3, 5, 5, The7, Median: 12. To find the median, put the values in order, and then find the middle value. If there are two values in the middle then the median is calculated by the mean of these two values. Median 8, 2, 3, 5, 5, 7, 12 5 2, 3, 5, 5, 7, 8, 12  Median Calculation of the location of the median in a set of n ordered numbers can be given using the following formula: value Let’s apply this formula on the example in the previous slide: 8, 2, 3, 5, 5, 7, 12 2, 3, 5, 5, 7, 8, 12 value Median is 5  Mode Let’s now find the mode for the same set of numbers 8, 2, 3, 5, 5, 7, The12.Mode: The mode is the value which appears the most often in the data. It is possible to have more than one mode if there is more than one value which appears the most. 8, 2, 3, 5, 5, 7, 12 There is only one value which appears most often The(number 5). Mode is 5 because it appears more times than any of the other data values. QUESTION 1 SOLUTION 106 QUESTION 2 107 SOLUTION 108 PRACTICE QUESTIONS 109  Examples 2. Find the mode, median, mean of the following: a) 3, 12, 11, 7, 5, 5, 6, 4, 10 b) 16, 19, 10, 24, 19 c) 8, 2, 8, 5, 5, 8 d) 28, 39, 42, 29, 39, 40, 36, 46, 41, 30 e) 133, 215, 250, 108, 206, 159, 206, 178 f) 76, 94, 76, 82, 78, 86, 90 g) 52, 61, 49, 52, 49, 52, 41, 58 THANK YOU