1 Unit 1. Basic Numeracy Skills Lecturer: A’lam Asadov Office hrs: Friday pm Room 103 QUANTITATIVE METHODS FOR BUSINESS UGB108

Objectives Scientific notation; Use symbols to represent relationships ; Simplify expressions containing symbols. Add and subtract, multiply and divide fractions; Understand positive and negative powers and related operations; Understand and manipulate fractional powers Plot a straight line on a graph; Calculate the slope or gradient of a line and the intercept with the y axis. 2 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Essential Maths A few basic concepts are enough for most of the topics in this book. Without these basic ideas you will come across lots of things you don’t understand in the rest of your course, career and life. There won’t be any proofs here, everything will be explained fully and we start at the very beginning. 3 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Multiplying Numbers the same signs gives a + different signs gives a – +  + = +2  3 = 6 +  – = –2  (-3) = -6 –  + = –(-2)  3 = -6 –  – = +(-2)  (-3) = 6 4 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Dividing Numbers the same signs gives a + different signs gives a – + ÷ + = +6 ÷ 3 = 2 6 / 3 = 2 + ÷ – = –6 ÷ (-3) = -2 6 / (- 3) = -2 – ÷ + = –(-6) ÷ 3 = -2 (-6) / 3 = -2 – ÷ – = +(-6) ÷ (-3) = 2 (-6) / (-3) = 2 5 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Combining Operations In order: BEDMAS Brackets Exponentials (Powers or Roots) Divide and Multiply (from left to right) Add and Subtract (from left to right) 2  2  (27 ÷ 3) + (1 – 20) = 2  2  (9) + (–19) = 36 – 19 = 17 6 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Scientific Notation a  10 b ◦ Where 1 ≤ a < 10 ◦ B is an integer = 1.2  = 1.2  Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Working With Symbols The rules for addition, subtraction, multiplication and division are exactly the same as for arithmetic with numbers  -(-a) = a, -(+a) = -a, etc  a  (-b) = -ab, (-a)  (-b) = ab, etc  a/(-b) = -a/b, (-a)/(-b) = a/b, etc Remember explicit multiplication 2a = 2  a 8 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Collecting Terms Together Often we have to collect like terms together  2pq + pq - 5pq = -2pq  s/2r +4s/2r = 5s/2r 9 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Fractions A fraction is a: numerator denominator A fraction keeps the same value when you do the same thing to both the numerator and denominator 10 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Multiplying Fractions Multiply the numerators and the denominators 2/3  4/5 = 8/15 2¼  4/5 = 9/4  4/5 = 36/20 = 9/5 = 1 4/5 (-3/2)  (5/7) = -15/14 a/b  c/d = ab/cd 11 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Dividing Fractions To divide one fraction by a second, turn the second upside down and multiply 1/2 ÷ 2/3 = 1/2  3/2 = 3/4 5/7 ÷3 = 5/7  1/3 = 5/21 a/b ÷ c/d = a/b  d/c = ad/bc 12 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Percentages Percentages are the number of hundredths 5% = 5/100 20% = 20/100 6% of 300 = 6/100  300 = 18 25/400 = 6.25/100 = 6.25% 13 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Expanding Brackets The value before a bracket is multiplied by everything inside the brackets a(b + c) = ab + ac x(y + z) – xy = xy +xz – xy = xz (b + c)/a = b/c + c/a 14 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Product of Brackets Everything inside the first bracket is multiplied by everything inside the second bracket (a + b)(c + d) = ac + bc + ac + ad (x – 2)(y + 1) = xy – 2y + x Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Powers Multiplying a value by itself a number of times a squared = a  a = a 2 2 cubed = 2  2  2 = 2 3 = 8 -2 to the power 4 = (-2) 4 = 16 (2/3) to the power 5 = 2 5 /3 5 = 32/243 a 0 = 1 for any value of a 16 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Negative Powers Take the reciprocal of the positive power b -m = 1/b m 2 -2 = 1/2 2 = ¼ 3 -4 = 1/3 4 = 1/81 (1 + a) -2 = 1/(1 + a) 2 17 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Multiplying Powers To multiply powers of the same number, add the power b m b n = b m+n 2 4  2 -5 = = 2 -1 = ½ 18 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Dividing Powers To divide powers of the same number, subtract the power b m / b n = b m-n 2 4 ÷ 2 -5 = = 2 9 = Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Powers of a Power To raise a power to a power, multiply the powers together (b m ) n = b mn (3 2 ) 3 = 3 6 = Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Power of a Product (a  b) n = a n  b n (3  4) 3 = 3 3  4 3 = 27  64 = 1728 (2ab) m = 2 m  a m  b m Similarly: (a/b) n = a n / b n (5/2) 3 = 5 3 / 2 3 = 125/8 21 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Fractional Powers The square root of a = √a = a 1/2 The cube root of 64 = 64 1/3 = 4 The q th root of a is a 1/q Similarly: b n/q = (b 1/q ) n (27) 2/3 = (27 1/3 ) 2 = 3 2 = /4 = (16 1/4 ) 3 = 2 3 =8 22 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Equations Show the relationship between constants and variables They form quantitative models of problems When there is a single unknown value, rearrange the equation so that this value is on one side of the equals sign, and all the known values are on the other side. Then doing the calculations gives the unknown value – or solves the equation 23 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Examples C = (60 + 5h) / n When h = 10 and n = 5 the solution is: C = ( ) / 5 = 22 V = 1000(1+i) n When I = 0.05 and n = 10 V = 1000(1.05) 10 = Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Inequalities = is equal to ≠ is not equal to < is less than ≤ is less than or equal to > is greater than ≥ is greater than or equal to 25 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Manipulating Inequalities Inequalities can be manipulated in the same way as equations Doing the same thing to both sides give an equivalent result EXCEPT When you multiply by a negative number you must reverse the sign of the inequality (, ≤ becomes ≥, etc) 26 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Examples 5x + 2 > 3x – 1 means 2x > -3, or x > -3/2 3/x > 2means x < 3/2 3 – x < 1 < 5 – x means 2 < x < 4 But beware negative values: -2x > 6means –x > 3 or x < -3 p/(-2) > 3meansp/2 < -3 or p < Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Straight line graphs 28 Y = 2x + 1 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Y = -3x +2 X Y

Linear Equations Y = ax + b Y = 2x + 1 3y – 3 = x 2x – 4y = 5 30

b is the intercept of the y axis 31

a is the gradient (slope) 32 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

Positive and Negative Gradients 33 Quantitative Methods for Business, Management and Finance (3rd Ed) by Louise Swift and Sally Piff

QUESTIONS ??? 34