Ratio, Percentage, Algebra & Geometry
1. Ratio and Proportion
Ratio
A ratio compares two quantities of the same unit. Written as a:b or a/b.
Example: Class has 20 boys, 15 girls. Ratio of boys to girls = 20:15 = 4:3.
Proportion
Two ratios are in PROPORTION if they are EQUAL. a:b :: c:d means a/b = c/d.
Unitary Method
Find the value of ONE unit first, then multiply. 'If 5 pens cost ₹60, what is the cost of 8 pens?' → 1 pen = ₹12 → 8 pens = ₹96.
2. Percentage
Percent = Per hundred. Written as %. 25% = 25/100 = 1/4.
Conversions
- Fraction to %: Multiply by 100. 3/5 = (3/5)×100 = 60%
- % to Fraction: Write over 100, simplify. 40% = 40/100 = 2/5
- Decimal to %: Multiply by 100. 0.35 = 35%
3. Introduction to Algebra
What Is Algebra?
Algebra uses LETTERS to represent unknown numbers. It is the GENERALISATION of arithmetic.
Algebraic Expressions
- 3x + 2y — 5z ('three times x, plus two times y, minus five times z')
- Term: Each part separated by + or — (3x, 2y, -5z)
- Coefficient: The number multiplying the variable (3 is coefficient of x)
- Constant: A term without a variable (-5)
Like and Unlike Terms
- Like terms: Have the SAME variable with the SAME power. 3x and 7x are like. CAN be added.
- Unlike terms: Different variables or powers. 3x and 3y are unlike.
Using Algebraic Expressions
- Write in words: 'A number increased by 7' → x + 7
- Write in algebra: 'Twice a number subtracted from 15' → 15 — 2x
4. Basic Geometry
Points, Lines, and Angles
Types of Angles:
| Angle | Measure |
|---|---|
| Acute | < 90° |
| Right | = 90° |
| Obtuse | 90° to 180° |
| Straight | = 180° |
| Reflex | 180° to 360° |
Pairs of Angles
- Complementary: Sum = 90°
- Supplementary: Sum = 180°
- Adjacent: Share a common arm and vertex
- Vertically Opposite: Equal. Formed when two lines intersect.
Triangles
| By Sides | By Angles |
|---|---|
| Equilateral — all sides equal | Acute — all angles < 90° |
| Isosceles — two sides equal | Right — one angle = 90° |
| Scalene — all sides different | Obtuse — one angle > 90° |
Quadrilaterals
| Shape | Properties |
|---|---|
| Square | All sides equal. All angles 90°. |
| Rectangle | Opposite sides equal. All angles 90°. |
| Rhombus | All sides equal. Opposite angles equal. |
| Parallelogram | Opposite sides parallel and equal. |
Circle
- Radius (r) : Distance from centre to any point on the circle
- Diameter (d) : 2 × Radius. Longest chord.
- Circumference = 2πr (π ≈ 22/7 or 3.14)
