On a certain principal, the simple interest for 2 years is Rs 1,400, while the compound interest for the same 2 years at the same annual rate is Rs 1,449. What is the rate of interest per annum?

Introduction / Context:
This question links simple interest and compound interest on the same principal and over the same period. We know the simple interest and the compound interest for 2 years, and we need to extract the annual rate of interest. The key idea is that the extra amount in compound interest over simple interest for 2 years is due to interest on interest, and it can be expressed in a compact formula involving the principal and the square of the rate. Combining this with the simple interest information allows us to solve for the rate efficiently.

Given Data / Assumptions:

• Simple interest for 2 years (SI2) = Rs 1,400.
• Compound interest for 2 years (CI2) = Rs 1,449.
• Both are calculated on the same principal P and at the same annual rate R.
• Time period T = 2 years.
• Interest for CI is compounded annually.

Concept / Approach:
Let R be the annual rate in percent. First, simple interest for 2 years is:
SI2 = P * R * 2 / 100 = 1400So:
P * R = 70000The difference between compound interest and simple interest for 2 years on the same principal and at the same rate is:
CI2 - SI2 = P * (R / 100)^2We know CI2 - SI2 = 1449 - 1400 = 49. Thus:
P * (R / 100)^2 = 49We now have two equations in P and R, which we can solve systematically.

Step-by-Step Solution:
Step 1: From simple interest, P * R * 2 / 100 = 1400, so P * R = 1400 * 100 / 2 = 70000.Step 2: From the difference, CI2 - SI2 = 49, and CI2 - SI2 = P * (R / 100)^2.Step 3: Substitute P from the first relation into the second. We have P = 70000 / R.Step 4: Then 49 = (70000 / R) * (R / 100)^2.Step 5: Simplify (R / 100)^2 = R^2 / 10000, so 49 = 70000 * (R^2 / 10000) / R.Step 6: This simplifies to 49 = 70000 * R / 10000 = 7 * R.Step 7: Solve 49 = 7 * R, giving R = 7.Step 8: Therefore, the rate of interest is 7% per annum.

Verification / Alternative check:
We can verify quickly by finding P from P * R = 70000. With R = 7, P = 70000 / 7 = 10000. Simple interest for 2 years at 7% is SI2 = 10000 * 7 * 2 / 100 = 1400, which matches the given SI. For compound interest, the amount after 2 years at 7% is A = 10000 * (1.07)^2 = 10000 * 1.1449 = 11449, so CI2 = 11449 - 10000 = 1449, which matches the given CI. This confirms that R = 7% is correct.

Why Other Options Are Wrong:
Rates of 3.5%, 5%, 10%, or 14% would not produce the given pair of SI2 and CI2. For example, at 10% per annum, SI2 would be 2000 on a principal of 10000, not 1400. At 14% per annum, the difference CI2 - SI2 would be much larger than 49. These inconsistent outcomes show that such rates cannot satisfy both conditions at the same time. Only 7% per annum fits all the data exactly.

Common Pitfalls:
Some students try to guess the rate instead of using the relationships systematically and might not test their guesses thoroughly. Others forget or do not know the formula for CI2 - SI2 and therefore attempt longer computations that are more error prone. Remembering that CI2 - SI2 = P * (R / 100)^2 for 2 years greatly simplifies problems of this type.

Final Answer:
The rate of interest per annum that satisfies the given simple and compound interest values is 7 percent per annum.