A sum of Rs 4000 becomes Rs 5800 in 3 years, when invested in a scheme of simple interest. If the same sum is invested in a scheme of compound interest with same yearly interest rate (compounding of interest is done yearly), then what will be the amount (in Rs) after 2 years?

Options

A. 443
B. 5450
C. 5290
D. 4970

Answer

To solve the problem step by step, we will first calculate the simple interest to find the rate of interest, and then use that rate to calculate the amount after 2 years under compound interest.

### Step 1: Calculate Simple Interest (SI)
Given:
- Principal (P) = Rs 4000
- Amount after 3 years (A) = Rs 5800

We can calculate the simple interest using the formula:
\[ \text{SI} = A - P \]
\[ \text{SI} = 5800 - 4000 = 1800 \]

### Step 2: Calculate the Rate of Interest (R)
We know that:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Rs 1800
- P = Rs 4000
- T = 3 years

Rearranging the formula to find R:
\[ R = \frac{\text{SI} \times 100}{P \times T} \]
Substituting the values:
\[ R = \frac{1800 \times 100}{4000 \times 3} \]
\[ R = \frac{180000}{12000} = 15\% \]

### Step 3: Calculate Amount under Compound Interest (CI)
Now, we will use the compound interest formula to find the amount after 2 years:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- P = Rs 4000
- R = 15%
- T = 2 years

Substituting the values:
\[ A = 4000 \left(1 + \frac{15}{100}\right)^2 \]
\[ A = 4000 \left(1 + 0.15\right)^2 \]
\[ A = 4000 \left(1.15\right)^2 \]

Calculating \( (1.15)^2 \):
\[ (1.15)^2 = 1.3225 \]

Now substituting back:
\[ A = 4000 \times 1.3225 = 5290 \]

### Final Answer:
The amount after 2 years when Rs 4000 is invested at compound interest at a rate of 15% is Rs 5290.

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