Difficulty: Medium
Correct Answer: 14 percent
Explanation:
Introduction / Context:
This question asks you to use both simple interest and compound interest information to determine the interest rate. Knowing the simple interest for 2 years and the compound interest for 2 years on the same principal offers enough information to solve for the rate without explicitly knowing the principal.
Given Data / Assumptions:
Concept / Approach:
For simple interest: SI = P * r * t / 100. For 2 years, SI = 2P * r / 100 = 2,800.
For compound interest over 2 years: CI = P * (1 + r/100)^2 − P. Also, for 2-year CI, the difference CI − SI equals the interest on the first year's interest: CI − SI = P * r^2 / 100^2. We can use SI to relate P and r, then use CI − SI to solve for r directly.
Step-by-Step Solution:
Step 1: Write SI equation. SI = 2P * r / 100 = 2,800. So P * r = 2,800 * 100 / 2 = 140,000. Step 2: Write CI − SI equation. CI − SI = 2,996 − 2,800 = 196. Also, CI − SI = P * r^2 / 100^2. So 196 = P * r^2 / 10,000. Step 3: Use P * r = 140,000 to eliminate P. P * r^2 / 10,000 divided by P * r gives (r / 10,000). Hence, r / 10,000 = 196 / 140,000 → r = 196 * 10,000 / 140,000 = 14.
Verification / Alternative check:
Check with r = 14%. SI for 2 years: SI = 2P * 14 / 100 = 0.28P = 2,800 → P = 10,000. Now CI: Amount A = 10,000 * (1.14)^2 = 10,000 * 1.2996 = 12,996. CI = A − P = 12,996 − 10,000 = 2,996, matching the given CI.
Why Other Options Are Wrong:
Rates of 7%, 21% or 28% give SI and CI values that do not produce a difference of exactly 196. For example, at 7% the difference CI − SI would be far smaller; at higher rates it would be much larger.
Common Pitfalls:
A common error is to try to solve for P and r independently without using the simplifying relation between CI − SI and r^2. Another mistake is miscomputing the difference CI − SI or forgetting to square r when using the formula for CI − SI over 2 years.
Final Answer:
The annual rate of interest is 14 percent.
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