Algebra, Equations, Ratio & Commercial Mathematics
1. Algebraic Expressions — Operations
Addition and Subtraction of Algebraic Expressions
Combine LIKE TERMS (same variable, same power).
- (3x + 2y — z) + (5x — 3y + 4z) = 8x — y + 3z
Multiplication of Algebraic Expressions
- Monomial × Monomial: Multiply coefficients. Add exponents of same base. (3x²)(—2x³) = —6x⁵.
- Monomial × Binomial: Use DISTRIBUTIVE LAW. 2x(3x + 4) = 6x² + 8x.
- Binomial × Binomial: (x + a)(x + b) = x² + (a+b)x + ab.
Standard Identities
| Identity | Expanded Form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a — b)² | a² — 2ab + b² |
| (a + b)(a — b) | a² — b² |
2. Linear Equations in One Variable
What Is an Equation?
A statement that two expressions are EQUAL. It has an EQUAL SIGN (=). Solving = finding the VALUE of the variable that makes the equation TRUE.
Solving Linear Equations
- Simplify both sides (remove brackets, combine like terms)
- TRANSPOSE: Move terms with variables to ONE side. Move constants to the OTHER. 'When you move a term across the =, CHANGE ITS SIGN.'
- Divide by the coefficient of the variable.
Example: 3x + 5 = 2x + 11 → 3x — 2x = 11 — 5 → x = 6.
Word Problems
- 'The sum of three consecutive integers is 36.' Let numbers be x, x+1, x+2. x + (x+1) + (x+2) = 36 → 3x + 3 = 36 → x = 11. Numbers: 11, 12, 13.
3. Ratio and Proportion
Ratio in Simplest Form
Divide by the HCF. 36:24 = (36÷12):(24÷12) = 3:2.
Proportion
If a:b :: c:d, then ad = bc (product of extremes = product of means).
Unitary Method
Find value of ONE unit first. 'If 8 pens cost ₹96, 1 pen = ₹12. 15 pens = ₹180.'
4. Percentage and Its Applications
Percentage
% = 'per hundred.' To convert fraction to %: multiply by 100.
Profit and Loss
- Profit = SP — CP (when SP > CP). Profit % = (Profit/CP) × 100.
- Loss = CP — SP (when CP > SP). Loss % = (Loss/CP) × 100.
- SP = CP × (100 + Profit%) / 100. SP = CP × (100 — Loss%) / 100.
Simple Interest
SI = (P × R × T) / 100
- P = Principal. R = Rate of interest per annum. T = Time in years.
- Amount = P + SI = P + (P×R×T)/100 = P(1 + RT/100).
