1 unit 1. basic numeracy skills lecturer: a’lam asadov [email protected] office hrs: friday...

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Unit 1. Basic Numeracy Skills

Lecturer: A’lam [email protected]

Office hrs: Friday 2.00-3.00pmRoom 103

QUANTITATIVE METHODS FOR BUSINESS

UGB108

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ObjectivesScientific 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

powersPlot a straight line on a graph;Calculate the slope or gradient of a line

and the intercept with the y axis.

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

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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.


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Multiplying Numbersthe same signs gives a + different signs gives a –

+ + = + 2 3 = 6+ – = – 2 (-3) = -6– + = – (-2) 3 = -6– – = + (-2) (-3) = 6


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Dividing Numbersthe 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) = 2Quantitative Methods for Business, Management and Finance

(3rd Ed) by Louise Swift and Sally Piff

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Combining OperationsIn order: BEDMASBracketsExponentials (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 = 17Quantitative Methods for Business, Management and Finance

(3rd Ed) by Louise Swift and Sally Piff

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Scientific Notationa 10b

◦Where 1 ≤ a < 10◦B is an integer

12000 = 1.2 104

0.0012 = 1.2 10-3


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Working With SymbolsThe 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 multiplication2a = 2a


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Collecting Terms TogetherOften we have to collect like terms

together

2pq + pq - 5pq = -2pq s/2r +4s/2r = 5s/2r


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FractionsA fraction is a:

numerator denominator

A fraction keeps the same value when you do the same thing to both the numerator and denominator


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Multiplying FractionsMultiply the numerators and the

denominators

2/3 4/5 = 8/152¼ 4/5 = 9/4 4/5 = 36/20 =

9/5 = 1 4/5(-3/2) (5/7) = -15/14a/b c/d = ab/cd


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Dividing FractionsTo divide one fraction by a

second, turn the second upside down and multiply

1/2 ÷ 2/3 = 1/2 3/2 = 3/45/7 ÷3 = 5/7 1/3 = 5/21a/b ÷ c/d = a/b d/c = ad/bc


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PercentagesPercentages are the number of

hundredths

5% = 5/10020% = 20/1006% of 300 = 6/100 300 = 1825/400 = 6.25/100 = 6.25%


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Expanding BracketsThe value before a bracket is

multiplied by everything inside the brackets

a(b + c) = ab + acx(y + z) – xy = xy +xz – xy =

xz(b + c)/a = b/c + c/a


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Product of BracketsEverything 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 - 2


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PowersMultiplying a value by itself a number

of times

a squared = a a = a2

2 cubed = 2 2 2 = 23 = 8-2 to the power 4 = (-2)4 = 16(2/3) to the power 5 = 25/35 =

32/243

a0 = 1 for any value of aQuantitative Methods for Business, Management and Finance(3rd Ed) by Louise Swift and Sally Piff

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Negative Powers

Take the reciprocal of the positive power

b-m = 1/bm

2-2 = 1/22 = ¼3-4 = 1/34 = 1/81(1 + a)-2 = 1/(1 + a)2


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Multiplying PowersTo multiply powers of the same

number, add the power

bmbn = bm+n

24 2-5 = 24-5 = 2-1 = ½


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Dividing PowersTo divide powers of the same

number, subtract the power

bm / bn = bm-n

24 ÷ 2-5 = 24+5 = 29 = 512


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Powers of a PowerTo raise a power to a power,

multiply the powers together

(bm)n = bmn

(32)3 = 36 = 729


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Power of a Product(a b)n = an bn

(3 4)3 = 33 43 = 27 64 = 1728

(2ab)m = 2m am bm

Similarly:(a/b)n = an / bn

(5/2)3 = 53 / 23 = 125/8Quantitative Methods for Business, Management and Finance(3rd Ed) by Louise Swift and Sally Piff

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Fractional PowersThe square root of a = √a = a1/2

The cube root of 64 = 641/3 = 4The qth root of a is a1/q

Similarly:bn/q = (b1/q)n

(27)2/3 = (271/3)2 = 32 = 9163/4 = (161/4)3 = 23 =8


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EquationsShow the relationship between

constants and variablesThey form quantitative models of

problemsWhen 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


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ExamplesC = (60 + 5h) / nWhen h = 10 and n = 5 the

solution is:C = (60 + 50) / 5 = 22

V = 1000(1+i)n

When I = 0.05 and n = 10V = 1000(1.05)10 = 1629


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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


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Manipulating Inequalities

Inequalities can be manipulated in the same way as equations

Doing the same thing to both sides give an equivalent result

EXCEPTWhen you multiply by a negative

number you must reverse the sign of the inequality (< becomes >, ≤ becomes ≥, etc)


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Examples5x + 2 > 3x – 1 means 2x > -3,

or x > -3/23/x > 2 means x < 3/23 – x < 1 < 5 – x means 2 < x < 4

But beware negative values:-2x > 6 means –x > 3 or x < -3 p/(-2) > 3 means p/2 < -3 or p < -

6Quantitative Methods for Business, Management and Finance(3rd Ed) by Louise Swift and Sally Piff

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Straight line graphs

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

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Y = -3x +2X -5 -4 -3 -2 -1 0 1 2 3 4 5Y 17 14 11 8 5 2 -1 -4 -7 -10 -13

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Linear Equations

Y = ax + b

Y = 2x + 13y – 3 = x2x – 4y = 5

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b is the intercept of the y axis

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a is the gradient (slope)


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Positive and Negative Gradients


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QUESTIONS ???

1 unit 1. basic numeracy skills lecturer: a’lam asadov [email protected] office hrs: friday...

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