review of basic mathsreview of basic maths(ctd) a) 0. ഥthis means x= 0.66 so we have one repeating...

Review of basic Maths

1.Natural Numbers:

N = {1, 2, 3…..}

Closed for addition and multiplication

2.Whole Numbers:

N0 = {0, 1, 2, 3…..}

Closed for addition, multiplication

3.Integers:

Z = {…..-3, -2, -1, 0, 1, 2, 3…..}

Closed for addition, multiplication and subtraction but not division4.

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Review of basic Maths(ctd) 4.Rational Numbers:

• Q = {a/b : a, b, є Z; b ≠ 0}• e.g. a = 3 b = 4 a/b = ¾• closed for all four operations

have finite decimals or repeating decimals example: 2.6 , 2. 121212

Express each of the following decimals in the form of a/ba) 0.6b) 0. 35c) 0.585

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Review of basic Maths(ctd)

a) 0.ഥ𝟔 This means x= 0.66 so we have one repeating digit

i) x= 0.66ii)10x=6.6

10x-x= 6.6-0.6=9x=6X= 6/9 = 2/3

b) 0.𝟑𝟓 this means x=0.3535 so we have two repeating numbers

i) x=0.3535ii) 100x=35.3535

100x-1x=35.3535-0.353599x=35X=35/99

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Review of basic Maths(ctd) c) 0.𝟓𝟖𝟓 this means x=0.585585585 so we have two repeating numbers

i) x=0.585585585ii) 1000x=585.585585585

999x=585X=585/999x= 61/111

5.Irrational Numbers:

• Lie between the rational numbers• Have an infinite number of decimals• e.g. √2 = 1.4142135……..• e.g. π =3.1415………….[ NOTE: If the decimal place is finite or recurring the number is rational 3/4 = 0.7 ; 61/2 = 6.21

3= 0.3333= 0.ത3

82

11=7.4545=7..45

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Review of basic Maths(ctd) 6.Real Numbers:

R = {Rational numbers} + {Irrational Numbers}Some properties of Real Numbers

1. The Transitive PropertyLet a,b and c be real numbers If a = b and b = c then a = c

2. Commutative Propertya+b = b+a ↔ ab = bacan add or multiply two real numbers in any order

3. Associative Property

a + (b+c) = (a+b) + c a(bc) = (ab)c

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4.Inverse Property

i) Additive Inverse

a + (-a) = 0

-6 + 6 = 0ii) Multiplicative Inverse

𝑎 ∗ 𝑎−1 = 𝑎 ∗1

𝑎= 1

3 ∗ 3−1 = 3 ∗1

3= 1

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Operations with signed numbers

2 – 7 = 2 + (-7) = -52 – (-7) = 2 +7 = 96 (7-2) = 6 (7) – 6(2) = 42 – 12 = 30-(7+2) = -7 - 2 = -9-(2-7) = -2 + 7 = 52(0) = 0

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Fundamental Principle of Fractions

𝑎

𝑏=𝑎

𝑏∗𝑐

𝑐= 𝑎𝑐

𝑏𝑐

7

1/8=7∗81

8∗8

= 56

1=56

The order of mathematical operationsI) BracketsII) ExponentsIII) Multiplication and division (equal status but perform operations from left to right)IV) Addition and subtraction (equal status but perform operations from left to right)

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e.g. 2 + 3 * (4/2 ÷ 8/16) – 10Treat item in brackets first:

4/2 ÷ 8/16 = 2 ÷ 1/2 = 2 * 2/1 = 4Next treat the multiplication:

2 + 3 x 4 – 10 = 2 + 12 – 10 = 4Note: 5 * 4 – 10 = 10 is WRONG

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Review of basic Maths(ctd) I) Exponent𝑎4 where 4 is the exponent and 𝑎 is the base 23= 2 x 2 x 2 = 8Special exponents:𝑎1 = a41 = 4𝑞0 = 140 = 1

II)Dot for multiplication:a . b ↔ a x b see also a*b

III) Coefficient:

3X where 3 is the coefficient and χ the variable2ab where 2 is the coefficient and ab the variables

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IV)Terms of an expression:

Terms are formed by positive (+) or negative (-)e.g. 2a + 3 is an expression with two terms

2.a.3 one term6χ4 – 5y + 8 three terms

5(3χ + 4) + 4(2 – 3y) two terms

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Review of basic Maths(ctd) V) Variables and constants:

Variable: -a symbol e.g. (x;y;p)-can be assigned any numerical valueConstants: -numbers with fixed value-e.g. 18; 2; 341

Note: χy = yχ but 25 ≠ 52

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Review of basic Maths(ctd) VI) Addition and subtraction of variables

Variables with the same powers can be added or subtractede.g. 7χ – 3χ = 4χ ; 7χ2 – 4χ2 = 3χ27𝑋2 – 3X cannot be simplified 6mn + 9mn = 15mn9a + 2ab cannot be simplified

Rules of Exponents

1. Multiplication Rule

𝑘𝑚* 𝑘𝑛 = 𝑘𝑚+𝑛

e.g. 24* 25 = 24+5

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2. Division Rule

𝑥𝑚 ÷ 𝑥𝑛 = 𝑥𝑚/ 𝑥𝑛 = 𝑥𝑚−𝑛

e.g. 57 ÷ 54 = 57/ 54 = 57−4

3. Involution Rule (Raising to a power)

(𝑘𝑚)𝑛 = 𝑘𝑚∗𝑛

e.g. (𝑥4)3 = 𝑥4∗3

Negative Exponents

𝑥−5= 1

𝑥5alternatively

1

𝑥−5=𝑥5

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Review of basic Maths(ctd) Fractions as exponents (also referred as roots and radicals)

𝑥1/2 = 𝑥Note: 𝑥 * 𝑥= 𝑥1/2*𝑥1/2

e.g. 𝑥1/4=4 𝑥

e.g. χ3/4 =𝑥3/4=4𝑥4

LOGARITHMS

Intuitive approach using examples16 = 42 where 2 is the log and 4 is the base , 2 is the log of 16 to the base of 416 = 24 4 is the log of 16 to the base of 2Note: → exponential form 16 = 42 → log form 𝑙𝑜𝑔416 = 2

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i) The natural number 10-the log key on your calculator

ii) The irrational number e = 2,7182818

-the ln key on your calculator some examples:e.g. 𝑙𝑜𝑔10100 = 2 ↔ 102 = 100

𝑙𝑜𝑔101000 = 3 ↔ 103 = 1000

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THE LAWS OF LOGARITHMSI)Log of a Productln a.b = ln a + ln be.g. ln 3χ2 = ln 3 + ln χ2

II)Log of a Quotientln (a/b) = ln a – ln be.g. ln (6/5χ) = ln 6 – ln 5χ = ln 6 – (ln 5 + lnχ)

iii) Log of a Powerln aχ = χ ln ae.g. ln χ4 = 4 ln χe.g. ln (3X)2 = 2 ln 3X = 2 (ln3 + lnX)

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Further Notes on Logarithms

•The log of 0 or a negative number (-) is undefined•The log of 1 to any base = 0

100 = 1 20 = 1 𝑒0 = 1

𝑙𝑜𝑔𝑚1 = 0 ↔ 𝑚0 = 11 cannot be a base of a log

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Some application examplesa) 𝑙𝑜𝑔3𝑥 = 4 solve for x34 = x81= x

c) 𝑙𝑜𝑔𝑥49 = 2𝑥2 = 4x= 7

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Review of basic Maths(ctd) Review of basic algebra Operations that guarantee equivalence

i) Adding (subtracting) the same polynomial to both sides of an equation

e.g 3χ = 5 – 6χ3χ + 6χ = 59χ = 59χ/9 = 5/9χ = 5/9

ii) Multiplying (dividing) both sides of an equation by the same constant except 0e.g 10χ = 510χ/10 = 5/10χ = 5/10

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(iii) Replacing either side of an equation by an equal expression

e. g χ3 + 4 = y and y = a2 + b + 4 then χ3 + 4 = a2 + b + 4

Linear Equations

An equation of degree 1ax + b = 0 a and b are constants

a) e.g. 5χ – 6 = 3χ 5χ – 3χ = 6

2χ = 6

2χ/2 = 6/2

χ = 3

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b) I =Prt solve for rI/Pt = Prt/Pt (divide through by Pt) r = I/Pt

c) S = P + Prt solve for PS = P (1 + rt) (factor out P)

S / (1 + rt) = P (1 + rt) / (1 + rt) (divide through by (1+ rt) P = S/(1+rt)

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• Applications

Application 1

A certain municipality has 1,000,000 population. Of this population, 20%are women of child bearing age. A woman of child bearing age in thismunicipality has on average 3 children. One of these 3 children attendhigh school and the rest attend primary school. The fees per yea forprimary school education is R3000 and the fee per year for secondaryschool education is R10,000

a) How many women of child bearing age are in the municipality

b) How many children attending primary school and how many attending high school in the municipality

c) What is the total fees of primary school and total fees of secondary per year in the municipality

d) What the total amount of school fees in the municipality23

Solutions

a) The number of women of child bearing age is

1,000,000*0.20=200,000 women

b) Because each woman has on average 3 children, one of whom

attending high school

The municipality has 2*200,000=400,000 children attending primary

school and 1*200,000=200,000 children attending high school

c) The total fees for primary school per year is obtained by multiplying

fees by the number of children in primary school

R3000*400,000=R1200,000

d) The total fees for high school per year in the municipality is obtained

by multiplying the school fees per year with the number of high school

children R10,000*200,000=

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FractionsObjectives pursued consists of having participants

Section 1

• Distinguish between various types of fractions

• Converting improper fractions to whole or mixed

numbers

• Converting mixed numbers to improper fractions

• Reducing fractions to lowest terms

• Raising fractions to higher terms

25

Fractions(contd)

Section 2

• Adding fractions and mixed numbers

• Subtracting fractions and mixed numbers

• Multiplying fractions and mixed numbers

• Dividing fractions and mixed numbers

26

Fraction (contd)

A mathematical way of expressing a part of a whole thing. ¼ is a

fraction expressing one part out of a total of four parts.

numerator

The number on top of the division line of a fraction. It

represents the dividend in the division. In the fraction ¼,

1 is the numerator.

denominator

• The number on the bottom of the division line of a

fraction. It represents the divisor in the division. In the

fraction ¼, 4 is the denominator

27

Fraction (contd)

Various types of fractions

common or proper fraction

• A fraction in which the numerator is less than the

denominator. It represents less than a whole unit. The fraction

¼ is a common or proper fraction.

improper fraction

• A fraction in which the denominator is equal to or less than the

numerator. It represents one whole unit or more. The fraction 4/1

is an improper fraction.

mixed number

• A number that combines a whole number with a proper

fraction. The fraction 10¼ is a mixed number.

28

Fraction (contd)

• To convert improper fractions to whole or mixed number

• STEP 1 Divide the numerator of the improper fraction by

the denominator.

• STEP 2a If there is no remainder, the improper fraction

becomes a whole number.

• STEP 2b If there is a remainder, write the whole number

and then write the fraction as

𝑤ℎ𝑜𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑟𝑒𝑚𝑖𝑛𝑎𝑑𝑛𝑒𝑟

𝑑𝑖𝑣𝑖𝑠𝑜𝑟

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Fraction (contd)

Converting improper fraction to whole and mixed numbers examples25

5=5

4

3=1

1

3

66

10=6

6

10

30

Fraction (contd)

• Converting mixed numbers to improper fractions

STEP 1 Multiply the denominator by the whole number.

STEP 2 Add the numerator to the product from Step 1.

STEP 3 Place the total from Step 2 as the “new” numerator.

STEP 4 Place the original denominator as the “new”

Denominator

Examples:

3 2

5= 5∗3+2

5=

17

5

11

2=2∗1+1

2=

3

2

226

7=22∗7+6

7=

160

7

31

Fraction (contd)

Reducing Fractions to Lowest Terms

reduce to lowest terms

The process of dividing whole numbers, known as common

divisors or common factors, into both the numerator and

the denominator of a fraction.

• Raising Fractions to Higher Terms

raise to higher terms

The process of multiplying the numerator and denominator

of a fraction by a common multiple.

Needed to have fractions with differing denominators to

have the same denominator

32

Fraction (contd)

Examples

2

5=

?

10one does as follows

10÷5 =2, then 2*2 =4 so ?=4

The answer becomes:

2

5=4

10

Adding up fractions:

Like fractions: these are fractions with the same

denominator but different numerators

Example: 2

7+5

7+

3

7+

1

7=11

7= 1

4

7

. 33

Fraction (contd)

When adding like fractions you add up the numerators and

the denominator remain the same as for individual fractions

The same rule applies when subtracting fractions

unlike fractions: these are fractions with the different

denominators

Example

2

5+3

6+

1

4=

48+60+30

120=138

120=1

18

120=1

9

60

Adding unlike fractions requires fining a common

denominator. The easy way to get it is multiply all the

denominators. The numerator is found by dividing the

34

Fraction (contd)

common denominator by each individual denominators and

multiplying the result with the numerator of individual

fractions. Adding mixed numbers

STEP 1 Add the fractional parts. If the sum is an improper

fraction, convert it to a mixed number.

STEP 2 Add the whole numbers.

STEP 3 Add the fraction from Step 1 to the whole number

from Step 2.

STEP 4 Reduce the answer to lowest terms if necessary.

35

Fraction (contd)

1

2*3

4=3

8

To multiply mixed number, convert each mixed number into

an improper fraction before multiplying

Example:

21

2*3

3

4=

5

2*15

4

Dividing fractions: multiply the first fraction with the inverse

of the second fractions

Example: 3

7÷*

4

9=

3

7*9

4

36

Fraction (contd)

Fractions: applications

A certain ward received money from the government to be

spent on activities aimed at uplifting the community . The

municipality decided to spend 1/3 of the money on

assisting the disabled. It then spent 1/9 of the money on

the youth empowerment by informing them on how to get

jobs

a) What fraction of the money received did the municipality

use

b) If the municipality received R360,000 what amount was

spent on community uplifiment

37

Fraction (contd)

c) How much was spend on the disabled

d) How much was spend on the youth empowerment

see solutions at the end of the notes

38

Fraction (contd)

solutions

a)1

3+1

9=3

9+1

9=

4

9the fraction of the money is 4/9

b) The money spent on community upliftment was

R360,000* 4/9=160,000

c) The money spent on the disabled is

1/3*R360000=120,000

d) Money spend on the youth empowerment was R

1/9*360000= R40,000

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review of basic mathsreview of basic maths(ctd) a) 0. ഥthis means x= 0.66 so we have one repeating...

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