Find the compound interest on ₹10,000 at 10% per annum for 2 years 6 months, compounded annually.

Correct — B. Compound the 2 full years, then add the half year at half the rate.

Year 1, add 10%: 10,000 × 1.10 = ₹11,000
Year 2, add 10% again: 11,000 × 1.10 = ₹12,100

Last 6 months: rate = 10% × 6⁄12 = 5%
Add 5% of 12,100 = ₹605, so amount = ₹12,705

Subtract the principal: CI = 12,705 − 10,000 = ₹2,705 → option (b)

Why the others are wrong

• (a)
  ₹2,000 — ₹2,000 is simple interest for 2 years: ₹1,000 a year on ₹10,000. It ignores both the compounding and the extra 6 months.
• (c)
  ₹2,700 — It is ₹5 short. The last half-year's 5% is charged on ₹12,100, the amount after two years, which gives ₹605, not ₹600.
• (d)
  ₹2,500 — ₹2,500 is simple interest for 2½ years, ₹1,000 a year. Compounding adds ₹205 on top: ₹100 in year 2 and ₹105 in the last half-year.

With annual compounding, interest joins the principal only at the end of each full year. A leftover part-year earns simple interest, at the proportional rate, on the amount reached so far.

So 2 years 6 months means (1.1)² for the full years, then × (1 + 10%⁄2) for the half: 10,000 × 1.21 × 1.05 = 12,705.

Key facts

• Amount for n full years plus a fraction f of a year, compounded annually = P × (1 + r)ⁿ × (1 + f × r), with r as a decimal.
• ₹10,000 at 10% compounded annually becomes ₹11,000 after 1 year and ₹12,100 after 2.
• At 10%, CI for 2 years is 21% of the principal and SI is 20%.

Common traps

• Charging the full 10% for the last 6 months: that treats half a year as a whole one and gives ₹3,310.
• Raising 1.1 to the power 2.5 gives about ₹2,691: the part-year earns simple interest, so the power stays at 2.

No directly related past PYQ was found.