Suppose a bank gives an interest of 10% per annum compounded annually for a fixed deposit for a period of two years. What should be the simple interest rate per annum if the maturity amount after two years is to remain the same?

Correct — B, 10.5%. Compounding annually at 10% for two years multiplies the deposit by (1 + 10/100)² = 1.21, so the interest earned over the two years is 0.21 of the principal. To reach the same maturity amount, simple interest must deliver that same 21% across two years, and simple interest is spread evenly: P × R × 2 ÷ 100 = 0.21P gives R = 10.5. The extra half a percent is precisely the interest that compounding earns in year two on the first year's interest.

• (a)10% — The two schemes agree only over a single year. From the second year onwards compounding pays interest on interest, so 10% simple would fall short of the 1.21 factor.
• (c)11% — Two years at 11% simple gives 22% total interest, which overshoots the 21% that compounding at 10% actually produces.
• (d)12% — Two years at 12% simple gives 24% total interest — well above the required 21%, and closer to what three years of compounding would yield.

Simple interest is a fixed percentage of the original principal every year, so the amount grows linearly as P(1 + nR/100). Compound interest is charged on the running balance, so the amount grows geometrically as P(1 + r/100)ⁿ. Asking what simple rate matches a given compound rate means equating the two amounts over the stated period.

For two years the equivalence has a tidy closed form. Expanding (1 + r/100)² gives 1 + 2r/100 + r²/10000, and setting the total interest equal to 2R/100 yields R = r + r²/200. At r = 10 that is 10 + 0.5 = 10.5, which is the answer without any principal being assumed at all. The instinct the question is testing against is the belief that 'the rate is the rate' — that if the bank quotes 10%, then 10% simple must match. It does for one year and never again. The gap widens quickly with time: over three years at 10% compound the deposit grows by 33.1%, so the equivalent simple rate rises to about 11.03% a year.

• Compound amount is P(1 + r/100)ⁿ; simple amount is P(1 + nR/100).
• At 10% compounded annually the two-year factor is 1.21, so total interest is 21% of the principal.
• Spread evenly over two years, 21% total becomes 10.5% a year as simple interest.
• The general two-year equivalence is R = r + r²/200, the extra term being interest earned on the first year's interest.
• For one year simple and compound interest are identical; the divergence begins in year two.

• Assuming that equal quoted rates give equal maturity amounts, which holds only for one period.
• Applying the simple-interest rate to the compounded balance instead of to the original principal.
• Reading 'compounded annually' as if the compounding period matched the two-year term.

Asked as an equivalence item — two schemes, one maturity amount, and the rate of the second is the unknown.

• practice — not a real PYQ

  A sum is deposited at 20% per annum compounded annually for two years. What simple interest rate per annum would give the same maturity amount?

  • (a)20%
  • (b)21%
  • (c)22%
  • (d)24%

  Answer(c) 22% — the compound factor is 1.20² = 1.44, so total interest is 44% over two years, that is 22% a year simple.

• practice — not a real PYQ

  The difference between compound and simple interest on ₹8,000 for 2 years at 5% per annum is

  • (a)₹10
  • (b)₹20
  • (c)₹40
  • (d)₹50

  Answer(b) ₹20 — for two years the difference is P(r/100)² = 8000 × 0.0025 = ₹20.