Question

The difference between compound interest (CI) and simple interest (SI) on a sum for $4$ years is ₹ $1282$. Find the sum.
I. Amount of simple interest accrued after $4$ years is ₹ $4000$.
II. Rate of interest is $10\%$ per annum. 

• A. I alone sufficient; II alone not.
• B. II alone sufficient; I alone not.
• C. Either I alone or II alone sufficient.
• D. Even I + II together not sufficient.
• E. I + II together necessary.

Hint:

When $r$ is given, CI-SI over multiple years becomes a simple multiplier of $P$. Use it to solve for $P$ directly from the given difference.

Answer 1

The Correct Option is B

For $n = 4$ years at rate $r$, the difference (CI – SI) equals  

\[\Delta = P \left[ \left(1 + \tfrac{r}{100}\right)^4 - \left(1 + \tfrac{4r}{100}\right) \right].\]

With $r = 10\%$ (II),  
\[\Delta = P(1.1^4 - 1.4) = P(1.4641 - 1.4) = 0.0641P.\]

Given $\Delta = 1282 \Rightarrow P = \tfrac{1282}{0.0641} = \text{₹}\,20000.$  
So II alone is sufficient.  

I alone gives SI (simple interest) = ₹4000 = $P \cdot \tfrac{4r}{100}$, but $r$ is unknown $\Rightarrow$ not sufficient.
 

Answer 2

Rather than substituting directly into the general \( n \)-year CI-SI difference formula in one shot, build up the year-by-year gap between compound and simple interest, assuming the rate found in Statement II, and see which statement(s) let this be pinned down.

With rate \( r=10\% \), track the cumulative difference \( D_n=P\big[(1.1)^n-1-0.1n\big] \) year by year: \[ D_1=0,\quad D_2=0.01P,\quad D_3=0.031P,\quad D_4=0.0641P. \]

1. Option A (I alone sufficient, II not): Statement I only gives the total simple interest, \( ₹4000 \), equal to \( P\cdot\tfrac{4r}{100} \); with both \( P \) and \( r \) unknown in this single equation, the sum \( P \) cannot be isolated. So I alone is not sufficient; this option is rejected.
2. Option B (II alone sufficient, I not): Statement II fixes \( r=10\% \), so the year-by-year buildup above applies directly: \( D_4=0.0641P=1282\Rightarrow P=20{,}000 \). This uniquely determines the sum. So II alone is sufficient, and I alone (as shown) is not. This matches.
3. Option C (either alone sufficient): Since I alone leaves two unknowns unresolved, this option is rejected.
4. Option D (even together not sufficient): Combining I and II would also work (giving \( P \) via II regardless of I), so it is false that even together they fail; rejected.
5. Option E (I + II together necessary): Since II alone already suffices without needing I, this together-necessary claim is too strong; rejected.

The incremental year-by-year buildup of the CI-SI gap, using only the rate from Statement II, pins down the sum on its own.

Hence, the correct answer is If II alone is sufficient but I alone is not sufficient.