At what rate of interest per annum will a sum of ₹15,625 amount to ₹19,683 in 1 year 6 months, if the interest is compounded half-yearly?

Correct — C. Compounded half-yearly for 1 year 6 months means 3 half-years, each at half the annual rate.

Amount ÷ principal = (1 + r)³, with r the half-year rate
19,683 = 27³ and 15,625 = 25³
So (1 + r)³ = (27⁄25)³ → 1 + r = 27⁄25

Half-year rate: r = 27⁄25 − 1 = 2⁄25 = 8%
Annual rate: 8% × 2 = 16% → option (c)

Why the others are wrong

• (a)
  20% — At 20% a year the half-year rate is 10%, and 15,625 × 1.1³ = ₹20,796.875, which overshoots ₹19,683.
• (b)
  8% — 8% is the half-year rate, not the annual one. Two half-years make a year, so the rate per annum is 8% × 2 = 16%.
• (d)
  18% — At 18% a year each half-year adds 9%: 15,625 × 1.09³ ≈ ₹20,234.83, still more than ₹19,683.

Half-yearly compounding changes two things: the rate for each period is half the annual rate, and the number of periods is twice the number of years.

So A = P × (1 + R⁄200)ⁿ, with R the annual rate in per cent and n = 2 × years. Here n = 2 × 1.5 = 3.

Both numbers are perfect cubes, 15,625 = 25³ and 19,683 = 27³, so the cube root of A⁄P is 27⁄25 in one step.

Key facts

• Half-yearly compounding: rate per period = annual rate ÷ 2, and periods = years × 2.
• 15,625 = 25³ = 5⁶, and 19,683 = 27³ = 3⁹.
• 16% a year compounded half-yearly grows a sum by 1.08² = 1.1664 in a year, an effective rate of 16.64%.

Common traps

• Giving the half-year rate (8%) when the question asks for the rate per annum.
• Counting 1 year 6 months as 1.5 periods instead of 3 half-years.

No directly related past PYQ was found.