In aptitude (Simple Interest vs Compound Interest), the simple interest (SI) and compound interest (CI) on the same principal for 2 years at the same annual rate are ₹1,000 and ₹1,040 respectively. What is the annual rate of interest (in % per annum)?

Introduction / Context:
This problem compares simple interest and compound interest on the same principal and time period and asks you to determine the rate of interest from the difference between them. Such questions are popular in aptitude exams because they highlight the extra amount generated by compounding and test your ability to use known relationships between simple interest and compound interest over two years.

Given Data / Assumptions:

• Simple interest SI for 2 years = ₹1,000.
• Compound interest CI for 2 years = ₹1,040.
• The principal P is the same in both cases.
• The annual rate of interest r% is the same for SI and CI.
• Time period t = 2 years.
• We need to find r in percent per annum.

Concept / Approach:
For 2 years, the simple interest is: SI = (P * r * 2) / 100 For compound interest compounded annually, the interest for 2 years can be written as: CI = P * ((1 + r / 100)^2 - 1) There is a useful relationship: CI = SI + extra interest where the extra interest for 2 years is: extra = P * (r / 100)^2 The question gives SI and CI, so the difference CI - SI directly equals this extra term, which depends on P and r. At the same time, SI gives us P * r through its own formula. We combine both facts to solve for r.

Step-by-Step Solution:
Step 1: Use the SI formula for 2 years: SI = (P * r * 2) / 100 = 1,000. Step 2: Simplify to get P * r = 1,000 * 100 / 2 = 50,000. Step 3: Now compute the difference between CI and SI: CI - SI = 1,040 - 1,000 = 40. Step 4: This difference equals P * (r / 100)^2. Step 5: Write this as P * r^2 / 10,000 = 40. Step 6: Substitute P from Step 2: P = 50,000 / r. Step 7: Then (50,000 / r) * r^2 / 10,000 = 40. Step 8: Simplify the expression: 50,000 * r / 10,000 = 5 * r = 40. Step 9: Solve for r: 5 * r = 40 gives r = 8. Step 10: Therefore the annual interest rate is 8% per annum.

Verification / Alternative check:
Choose P so that the SI calculation is simple using r = 8%. We know P * r = 50,000 from Step 2, so: P = 50,000 / 8 = 6,250. Check SI for 2 years: SI = (6,250 * 8 * 2) / 100 = (6,250 * 16) / 100 = 100,000 / 100 = 1,000. Check CI for 2 years: Amount = 6,250 * (1 + 8 / 100)^2 = 6,250 * 1.08^2 = 6,250 * 1.1664 = 7,290. CI = Amount - Principal = 7,290 - 6,250 = 1,040. The values match the given SI and CI, confirming that r = 8% is correct.

Why Other Options Are Wrong:
10%, 9%, 11%, and 7%: For each of these, if you compute SI and CI for 2 years on any principal, the difference CI - SI will not equal ₹40 under the given SI = ₹1,000 condition. They fail to satisfy both equations simultaneously.

Common Pitfalls:
Students sometimes try to guess the principal and test rates randomly, which is slow and error prone. Another typical mistake is ignoring the specific 2 year relationship and treating CI as if it grows linearly like SI. Remember that for 2 years, the extra compound interest over simple interest is P * (r / 100)^2, which leads directly to r when combined with the SI expression.

Final Answer:
The annual rate of interest that produces SI = ₹1,000 and CI = ₹1,040 in 2 years is 8% per annum.