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…

Question

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?

1. A. 18%
2. B. 20%
3. C. 16% (Correct answer)
4. D. 8%

Correct Answer

Option C — 16%

Detailed Solution & Explanation

The correct answer is 16%.

Key Points

• Compounded half-yearly over 1 year 6 months means 3 periods, each at half the annual rate.
• Set up the ratio and look for a cube:
  • 19,683 / 15,625 = (1 + r)³
  • 19,683 = 27³ and 15,625 = 25³
• So (1 + r)³ = (27/25)³, giving 1 + r = 27/25 and r = 2/25 = 8% per half-year.
• The annual rate is therefore 2 × 8% = 16%.

Additional Information

• For half-yearly compounding, use rate ÷ 2 over 2 × time periods; for quarterly, rate ÷ 4 over 4 × time.
• Recognising 15,625 and 19,683 as perfect cubes is what makes this quick — worth memorising 25³ = 15,625, 27³ = 19,683, 21³ = 9,261 and 24³ = 13,824.
• The general formula is A = P(1 + R/100)ⁿ, and here it is solved backwards for R rather than forwards for A.
• Note the answer asked for is the annual rate; stopping at 8% is the intended trap, and it appears as option D.