MBA Jump Start Program

Transcription

1 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1

2 The Number Spectrum Generally we depict numbers increasing from left to right (or sometimes from bottom to top) Numbers can be positive or negative (or zero) Negative numbers can be depicted as $200 ($200) ($200) Numbers can be integers or not 3 Arithmetic Operations Operation Normal Notation Excel Addition + + Subtraction Multiplication x * Division / What is 3+4*5? 35 or 23? 4 2

3 Arithmetic Operations To avoid confusion, always follow the standard: First: Multiply and divide in the order they appear Then: Add and subtract in the order they appear Brackets override previous rules So: 3+4*5 = 23 But: (3+4)*5 = 35 When there are multiple brackets, calculate from the inside out So: [4+(4+3)*2]/6 = 3 5 Fractions Fractions and decimals are alternative ways of expressing non integer numbers Fraction = numerator /denominator Percentages are like decimals but multiplied by 100 Fractions Decimals Percentages 1/ % 1/ % 1/ % 3/ % 6 3

4 Fractions and Percentages Calculations encourage us to think in decimals, but fractions often make more intuitive sense. Which sounds better? Earnings are 1.33 times last year s earnings Earnings are up by a third over last year But use judgment when selecting fractional expressions. Which sounds better? We sold 17/4 million dollars worth of merchandise We sold 4 ¼ million dollars worth of merchandise Percentages allow apples to apples comparisons Weekly sales in our Seattle store went from $25,000 to $30,000 while sales in our Tacoma store went from $15,000 to $19,000 Weekly sales in our Seattle store went up by 20% while sales in our Tacoma store went up by 26.6% 7 Math with Fractions When denominator is the same simply add or subtract the numerators Addition Subtraction x y x y x y x y z z z z z z Examples: 1/5+2/5=3/5 3/7 4/7 = 1/7 When the denominators are different, find a common denominator x y x q y z x y x q y z z q z q q z z q z q q z Examples: 2/3 + 1/4 = 8/12 + 3/12 = 11/12 1/2 1/4 = 1/4 8 4

5 Math with Fractions (2) Multiplication: Simply multiply the numerators and the denominators x a x a y b y b Example: 1/3 * 1/2 = 1/6 Division: Cross multiply x a x b y b y a x b y a Example: 1/3 1/2 = 1/3 *2/1= 2/3 9 Decimals Remember that decimals can be converted into fractions and back, so: 0.12/3 = (12/100)/3 = (12/100)*(1/3) = 12/300 = 4/100 =

6 Rounding Different levels of accuracy are required for different purposes Some general rules: If your measure is not accurate do not provide lots of decimal points If your inputs are rounded your output should be rounded Even if your measure is accurate, do not give more information than is necessary for the decision maker Pay attention to rounding function on your calculator or spreadsheet! 11 Algebraic Expressions We use letters or variables to represent numerical quantities These variables are combined into algebraic expressions using arithmetic operations Example: Taxes paid (in $) = tax rate(in %) * taxable income (in $) or: T = t * I or: T = t I note that we do not need the * This is an algebraic expression! Be careful to specify the units 12 6

7 Algebraic Expressions Algebraic expressions are easiest to understand when simplified as much as possible Normally, we wouldn t write 4+6+7, but we would write 17 Similarly, we wouldn t write 4a+5a+5b+2ab+ab, but we would write 9a+5b+3ab Here a and b are the variables While 9, 5, and 3 are the coefficients 13 Powers Example: Suppose you must double your website s hits every month for 6 months. You expect to get 100 hits in the first month. How many will you have in six months if you meet the goal? 100*2*2*2*2*2 = 100*2 5 In general: b to the power of 1 = b = b 1 b to the power of 2 = b*b = b 2 b to the power of 3 = b*b*b = b 3. b to the power of n = b*b*(n times) = b n 14 7

8 Powers and Roots Note that: b 2 /b 4 = (b*b)/(b*b*b*b) = 1/b 2 = b 2 In general b to the power 1 = 1/b 1 = b 1 b to the power 2 = 1/b 2 = b 2 b to the power n = 1/b n = b n Note also that: b n /b n = b 0 = 1 15 Fractional Powers Powers and Roots Since: b 1/2 *b 1/2 = b 1 = b then b 1/2 must be the same as b, the square root of b In general b 1/2 = b 0.5 = square root b 1/3 = b 0.33 = cube root b 3/2 = b 1.5 = (b 1/2 ) 3 = ( b) 3 etc. 16 8

9 Powers (or Exponents) Manipulation rules with powers: 0 a 1 a n n m n mn m n mn m 1 a a a a a a a a n a mn m m m ab a b m m m ab a b m m m m ab ba b a 17 Roots (or Radicals) Square root: 2 1/2 0.5 x x x x In general: n x x 1/n Rules: n n n n n n n n n n a a a a b ab a b a b 18 9

10 Priority: Arithmetic Operations Powers and roots take precedence over all other mathematical operations Addition and Subtraction: Numbers taken to different powers are to be treated as different variables and can not be summed together, so: 2a + 5a + 3a 2 + 4a 2 + 7a 3 = 7a + 7a a 3 19 Scientific Notation A way to abbreviate large numbers often seen on calculators and spreadsheets Example: 22,430,000 = 2.243*10,000,000 = 2.243*10 7 = 2.243*10^7 = 2.243E07 Scientific notation = 2.24+E07 If rounding occurs 20 10

11 Scientific Notation Also used for small numbers: Example: = 4.63/10,000 = 4.63/10 4 = 4.63*10 4 = 4.63*10^ 4 = 4.63E 04 Scientific notation 21 Basics of Graph Plotting Most of your MBA instruction will take place in 2 dimensional space Traditionally we put the independent variable on the horizontal (x) axis and the dependent variable on the vertical (y) axis y is a function of x y depends on x Numbers on these axes can be positive or negative 22 11

12 Cartesian Axes 23 Cartesian Coordinates (x,y) 24 12

13 Polynomials Polynomial is the general term for equations that contain a variable, x, raised to some power Linear: x to the power of 1 Quadratic: x to the power of 2 Cubic: x to the power of 3 And so on! 25 Exponential Function Compounding (earning interest on interest) leads to exponential growth x The exponential function is: y f x exp x e where e = Exponential Growth y x 26 13

14 Exponential Rules Manipulation rules are as for the powers e x 1/ e e / e e x e e e x y x y x y x y e x y e xy 27 An Aside on Logarithms All numbers could potentially be described using power symbols: n = b p where b = base and p = power When n = b p, then p = log b n Common logarithms use base 10, so: y = log x means x = 10 y Natural logarithms use the base e = , so: y = ln x means x = e y 28 14

15 Logarithmic Function The natural logarithm is the inverse of the y ln x exponential: y ln x e e x x ln x y e ln y e x Key: exp and ln undo each other Logarithmic Growth y x 29 Manipulation rules: Logarithmic Rules x y xy ln ln ln x ln xln yln y ln m x mln x The last rule is very useful for bringing powers down 30 15

16 Inverse Proportions An inverse relationship is when one variable decreases as the other increases The equation is y = k/x Example: Wholesaler of footballs charges $16 per delivery regardless of how many units are ordered. What is the delivery charge per football? 31 Inverse Proportion C=16/n 32 16