The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is…

Question

The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is Re. 1. The sum (in Rs.) is

1. A. 625 (Correct answer)
2. B. 635
3. C. 645
4. D. 655

Correct Answer

Option A — 625

Detailed Solution & Explanation

The correct answer is 625.

Key Points

• For 2 years, the difference between compound and simple interest is given by:
  • $D = P\left(\frac{r}{100}\right)^2$
• Substituting $D = 1$ and $r = 4$:
  • $1 = P\left(\frac{4}{100}\right)^2 = P \times \frac{16}{10000}$
  • $P = \frac{10000}{16} = 625$

Shortcut Trick

• For a 2-year difference, $P = \dfrac{D \times 100^2}{r^2}$. With $D = 1$ and $r = 4$: $P = \dfrac{10000}{16} = 625$ in a single step.

Alternate Method

• Verify by direct computation on $P = 625$ at 4% for 2 years:
  • Simple interest $= \frac{625 \times 4 \times 2}{100} = 50$
  • Amount under compounding $= 625 \times (1.04)^2 = 625 \times 1.0816 = 676$, so CI $= 51$
  • Difference $= 51 - 50 = 1$ ✓

Additional Information

• Where the formula comes from: the difference for two years is exactly the *interest earned on the first year's interest*. First-year interest is $\frac{Pr}{100}$; interest on that amount for one more year is $\frac{Pr}{100} \times \frac{r}{100} = P\left(\frac{r}{100}\right)^2$.
• For 3 years the corresponding formula is:
  • $D = P\left(\frac{r}{100}\right)^2 \times \left(\frac{r + 300}{100}\right)$