Algebra

Definition: A percentage is a fraction with a denominator of 100.
- Formula: 𝑥 = 𝑥% * 100.
Expressing fractions and decimals as percentages:
- Method: Multiply the fraction or decimal by 100.
- Examples:
  - (i) To express 2/5 as a percentage:
    - 2/5 × 100 = 200/5 = 40%
  - (ii) To express 0.63 as a percentage:
    - 0.63 × 100 = 63%
Exercises:
- (i) Express 1/16 as a percentage.
- (ii) Express 5/40 as a percentage.
- (iii) Express 0.13 as a percentage.
- (iv) Express 0.457 as a percentage.
Determining percentages of numbers:
- Method: To find x% of a number, multiply the number by x/100.
- Example:
  - (i) What is 20% of 350?
    - 350 × 20/100 = 70
Exercises:
- (i) What is 45% of 165?
- (ii) What is 65% of 1053?
- (iii) What is 0.5% of 240?
- (iv) What is 10% of 0.0132?

Determining percentage increases and decreases:
- Method:
  - For an increase by x%: Multiply the number by (100 + x)/100.
  - For a decrease by x%: Multiply the number by (100 - x)/100.
- Examples:
  - (i) Result of increasing 350 by 20%:
    - 350 × 120/100 = 420
  - (ii) Result of decreasing 420 by 20%:
    - 420 × 80/100 = 336
Exercises:
- (i) What is the result if 185 is increased by 65%?
- (ii) What is the result if 1255 is decreased by 85%?
- (iii) What is the result if 260 is increased by 0.25%?
- (iv) What is the result if 0.01522 is decreased by 12%?
Further Reading and Exercises:
- From Engineering Maths – K Stroud (pages 24 - 25)
- From Fundamental Maths – Mark Breach (pages 36 - 37)
- From Foundation Maths – Anthony Croft and Robert Davison (pages 34 - 39)

Definition: An average is a measure of central tendency to summarize a set of data.
- Types of averages:
  1. Arithmetic Mean:
    - Method: Add all values and divide by the number of values.
      - Formula: mean = sum of values / total number of values.
      - Example: For scores 10, 5, 7, 6, 2:
        mean = (10 + 5 + 7 + 6 + 2) / 5 = 6.
  2. Median:
    - The middle value when values are in order.
    - For an even number of values, the average of the two middle values is taken.
      - Example: For the list 1, 7, 3, 3, 2, 6, 4, 7, 8, 9:
        Sorted: 1, 1, 2, 3, 3, 4, 6, 7, 7, 8, 9 = Median = 4.
  3. Mode:
    - The most frequently occurring value.
    - If multiple values occur most frequently, the set is bimodal.
      - Example: For the list 1, 3, 3, 4, 5, 4, 2, 1, 3, 3, 3, 6, 4, 2:
        Mode = 3 (occurs 5 times).
Exercises:
- Calculate the mean, median, and mode for datasets:
  - (i) 10, 14, 16, 2, 15, 8, 8, 10, 20, 18
  - (ii) 0, 10, 4, 7, 2, 9, 5, 5, 5

From Engineering Maths – K Stroud (pages 1055 - 1062)
- Test Exercise (page 1073)
From Fundamental Maths – Mark Breach (pages 359 - 361)
- Exercises (page 361)
From Foundation Maths – Anthony Croft and Robert Davison (pages 360 - 367)
- Test and Assignment Exercises (page 372 - 373)

Definition: Algebra uses letters or symbols to represent unknown numbers.
Rules of Algebra:
- Commutative: x + y = y + x, xy = yx
- Associative: x + (y + z) = (x + y) + z, x(yz) = (xy)z = xyz
- Distributive: x(y + z) = xy + xz
Algebraic Powers:
- Follow laws similar to arithmetic powers:
  - First Law: a^m × a^n = a^(m+n)
  - Second Law: a^m / a^n = a^(m-n)
  - Third Law: (a^m)^n = a^(m*n)
  - Special cases: a^1 = a, a^0 = 1.
Algebraic Fractions:
- Follow similar rules to numerical fractions.
- Example: a/b + c/d = (ad+bc)/bd, b ≠ 0, d ≠ 0.
Expressions and Formulae:
- An algebraic expression includes numbers, letters, and operators.
- A formula relates quantities, e.g., s = d/t (speed = distance/time).
Example:
- Evaluate: 2x - 5 when x = 2 → 2(2) - 5 = -1.
- Exercises:
  - (i) Find value of x^4 when x = 2.
  - (ii) Calculate area A = lb for rectangle with sides 40cm and 35cm.

Method: Rearrange equation to isolate unknown.
- Rules:
  - Add, subtract, multiply, or divide the same number on both sides.
  - Perform algebraic operations (e.g., square or square root both sides).
Examples:
- (i) Solve: 3x + 4 = 7
  - 3x = 7 - 4 → 3x = 3 → x = 1.
- (ii) Solve: 1/2x^2 – 9/7 = 9
  - 1/2x^2 = 63 + 9 → 1/2x^2 = 72 → x^2 = 144 → x = ±12.
- (iii) Solve: 5x + 3 = 2x - 7
  - 5x + 3 - 2x = -7 → 3x + 3 = -7 → 3x = -10 → x = -10/3.
Exercises:
- Find x for:
  - (i) x + 1 = 3
  - (ii) x + 1/5 = 5
  - (iii) x^2 – 1/5 = 3
  - (iv) 3x - 7/4 = 5
  - (v) 2x + 8 = 7x - 5
  - (vi) x - 4/2 = x - 5/3.

Method: Remove brackets and collect like terms.
- Examples:
  - (i) Simplify: 2(3x + 1) = 6x + 2.
  - (ii) Simplify: y(2x - 5) = 2xy - 5y.
Exercises:
- (i) Simplify: 3(2x + 8)
- (ii) Simplify: y(6x + 7)
- (iii) Simplify: 4(8x - 5).

Method:
- (a + b)(c + d) = ac + bc + ad + bd.
Examples:
- (i) Simplify (x + 5)(y + 1) → xy + x + 5y + 5.
- (ii) Simplify (2x - 1)(x + 3) → 2x² + 5x - 3.
Exercises:
- (i) Simplify: (x + 1)(y + 2)
- (ii) Simplify: (3x - 1)(y + 1)
- (iii) Simplify: (x + 4)(x - 1).

From Engineering Maths – K Stroud (pages 65 - 87)
- Multiple frames covering exercises and problems pertaining to algebra, simplifications, and averages.

Introduction to Algebra

Algebra uses letters or symbols to represent unknown numbers and allows for the manipulation of these symbols to solve problems.

Commutative Property:
- Addition: x + y = y + x
- Multiplication: xy = yx
Associative Property:
- Addition: x + (y + z) = (x + y) + z
- Multiplication: x(yz) = (xy)z = xyz
Distributive Property:
- This property states that multiplying a number by a sum equals the sum of that number multiplied by each addend:
- Example: x(y + z) = xy + xz

Algebraic powers follow laws similar to arithmetic powers:
1. First Law: a^m × a^n = a^(m+n)
2. Second Law: a^m / a^n = a^(m-n)
3. Third Law: (a^m)^n = a^(m*n)
Special Cases:
- a^1 = a
- a^0 = 1

An algebraic expression consists of numbers, letters (variables), and operators (such as +, −, ×, ÷).
A formula expresses the relationship between quantities.
Example: Evaluate the expression 2x - 5 when x = 2:
- Step 1: Substitute 2 for x: 2(2) - 5 = 4 - 5 = -1.