For a certain principal, the differences between compound interest and simple interest for 3 years and for 2 years are in the ratio 23 : 7 respectively. Based on this information about the gap between compound interest and simple interest, what is the annual rate of interest in percent for this aptitude problem on interest?

Introduction / Context:
This question tests your understanding of how the difference between compound interest and simple interest behaves over time for the same principal and the same annual rate. Instead of giving the actual principal and interest amounts, the problem provides a ratio between the differences over 3 years and 2 years. Using algebra and standard compound interest relationships, we can deduce the rate of interest per annum in percent without ever knowing the principal explicitly.

Given Data / Assumptions:

• Principal amount is some fixed sum P (not numerically specified).
• Annual rate of interest is r percent per annum.
• Time periods considered are 2 years and 3 years.
• The difference (Compound Interest − Simple Interest) for 3 years and for 2 years are in the ratio 23 : 7.
• Interest is compounded annually when calculating compound interest.

Concept / Approach:
For a principal P and annual rate r percent, we use x = 1 + r/100 for convenience. The formula for simple interest is SI = (P * r * t) / 100. The formula for compound interest is CI = P * (x^t − 1). The difference between CI and SI for t years, denoted Dₜ, can be simplified algebraically. For 2 years, D₂ becomes P * (x − 1)^2. For 3 years, D₃ becomes P * (x − 1)^2 * (x + 2). Because P and (x − 1)^2 are common factors, the ratio D₃ : D₂ simplifies nicely, allowing us to solve directly for x and hence for r.

Step-by-Step Solution:
Step 1: Let x = 1 + r/100. For t years, SIₜ = (P * r * t) / 100 = P * t * (x − 1). Step 2: For 2 years, CI₂ = P * (x^2 − 1). So D₂ = CI₂ − SI₂ = P * (x^2 − 1 − 2(x − 1)). Step 3: Simplify D₂: x^2 − 1 − 2x + 2 = x^2 − 2x + 1 = (x − 1)^2. So D₂ = P * (x − 1)^2. Step 4: For 3 years, CI₃ = P * (x^3 − 1). SI₃ = 3P * (x − 1). Then D₃ = P * (x^3 − 1 − 3(x − 1)). Step 5: Simplify inside: x^3 − 1 − 3x + 3 = x^3 − 3x + 2. This factors as (x − 1)^2 * (x + 2). So D₃ = P * (x − 1)^2 * (x + 2). Step 6: The ratio D₃ : D₂ is therefore [P * (x − 1)^2 * (x + 2)] : [P * (x − 1)^2] = x + 2. Step 7: Given that D₃ : D₂ = 23 : 7, we set x + 2 = 23/7. Step 8: Solving, x = 23/7 − 2 = (23 − 14) / 7 = 9/7. Step 9: Recall x = 1 + r/100, so 1 + r/100 = 9/7 implies r/100 = 9/7 − 1 = 2/7. Step 10: Therefore r = (2/7) * 100 = 200/7 percent per annum.

Verification / Alternative check:
As a quick check, r ≈ 200/7 ≈ 28.57 percent. Then x = 1 + r/100 ≈ 1.2857. Compute x + 2 ≈ 3.2857, which is exactly 23/7. Since the simplified ratio D₃ : D₂ equals x + 2, this confirms that the difference ratio is indeed 23 : 7 when r = 200/7 percent. This back substitution confirms the correctness of our algebraic reasoning.

Why Other Options Are Wrong:
100/7 %, 300/7 %, 400/7 %, and 50/7 % are all obtained by guessing different fractional forms without respecting the relationship x + 2 = 23/7. For each of these alternative rates, if we compute x = 1 + r/100 and then x + 2, we do not obtain 23/7, so the given ratio 23 : 7 would not hold. Only 200/7 % satisfies the ratio condition derived from the exact formulas for the differences between compound and simple interest over 2 years and 3 years.

Common Pitfalls:
Many learners try to plug in numerical values for the principal or use trial and error with the given options directly in interest formulas, which is time consuming. Another pitfall is to attempt to compute actual amounts of compound and simple interest for 2 and 3 years separately and then compare, instead of using the neat algebraic simplifications. Forgetting that the principal P and the factor (x − 1)^2 cancel out in the ratio also leads to unnecessary complexity. Careful factorization and understanding of how the difference grows with time make this question much simpler and faster to solve.

Final Answer:
The annual rate of interest that makes the ratio of the differences between compound and simple interest for 3 years and 2 years equal to 23 : 7 is 200/7 %.