Surds, Indices and Commercial Mathematics
What the domain tests
This domain combines surds and rationalisation, laws of indices, and commercial arithmetic including percentage, profit and loss, simple interest and compound interest.
Core rules and concepts
- Surds can be added or subtracted only when they have the same radical part: 3√5 + 2√5 = 5√5.
- To rationalise 1/(√a + √b), multiply numerator and denominator by √a − √b.
- Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ᵐ = 1/aᵐ.
- Simple interest = P × R × T / 100.
- Compound interest uses A = P(1 + R/100)ᵗ, so CI = A − P; for half-yearly compounding, halve the rate and double the time.
Worked example
Find the compound interest on ₹8,000 at 10% per annum for 2 years. A = 8000(1 + 10/100)² = 8000 × 1.1² = 8000 × 1.21 = ₹9,680. CI = 9680 − 8000 = ₹1,680. The first year’s interest is ₹800 and the second year’s interest is ₹880, which shows compounding.
Traps that cost marks
Do not add unlike surds: √2 + √3 is not √5. When rationalising with a conjugate, change the sign between the terms. Compound interest is not simple interest doubled; interest earns interest. Convert the rate to a decimal or divide by 100 before substituting, and match the time unit to the rate period.