The compound interest on a sum at 18 % p.a. for 1 ½ years, compounded yearly, is ₹2 862. Find the principal.

Correct — A. Compounded yearly for 1½ years means one full year at 18%, then the half year at 9% (half the rate) on the amount already built up.

Growth factor = 1.18 × 1.09
= 1.2862
CI = amount − principal = 0.2862 × P

0.2862 × P = 2,862
P = 2,862 ÷ 0.2862 = ₹10,000 → option (a)

Why the others are wrong

• (b)
  ₹8 000 — Run it forward: 8,000 × 0.2862 = ₹2,289.60 of interest, not the ₹2,862 stated. The principal must be larger.
• (c)
  ₹2,000 — ₹2,000 earns 2,000 × 0.2862 = ₹572.40 in 1½ years at 18%, exactly a fifth of the stated ₹2,862.
• (d)
  ₹4,000 — 4,000 × 0.2862 = ₹1,144.80, which is 40% of the stated ₹2,862. The principal has to be 2.5 times larger.

Compounded yearly means interest joins the principal once a year. When the period ends partway through a year, the leftover fraction earns the proportional rate on the amount reached at the last full year.

So 1½ years at 18% is one year at 18% followed by half a year at 9%, and the two growth factors multiply: 1.18 × 1.09 = 1.2862.

Finding the principal runs the same product backwards: divide the interest by 0.2862.

Key facts

• Under yearly compounding, amount = P × (1 + R⁄100)ⁿ × (1 + f × R⁄100), where n is the whole years and f the leftover fraction of a year.
• 1.18 × 1.09 = 1.2862, so ₹10,000 grows to ₹12,862 in 1½ years at 18%.
• Compound interest = amount − principal.

Common traps

• Raising 1.18 to the power 1.5 instead of multiplying by 1.09 for the half year: it gives about ₹10,156.
• Reading 'compounded yearly' as half-yearly: 9% for three half-years gives about ₹9,701.
• Dividing the interest by 1.2862, the amount factor: that treats ₹2,862 as principal plus interest.

No directly related past PYQ was found.