If CI on some amount is 410 for 2 years and rate of interest is 5% then find the SI for 2 years?

Options

A. ₹ 405
B. ₹402
C. ₹ 400
D. ₹ 375

Answer

To find the Simple Interest (SI) for 2 years given that the Compound Interest (CI) for the same period is Rs. 410 at a rate of 5%, we can follow these steps:

### Step 1: Understand the relationship between CI and SI
The Compound Interest for 2 years can be expressed in terms of the Principal (P) and the rate of interest (r). The formula for Compound Interest (CI) for 2 years is:
\[ CI = P \left(1 + \frac{r}{100}\right)^2 - P \]
This can also be simplified to:
\[ CI = P \left(\frac{r}{100}\right) + P \left(\frac{r}{100}\right)^2 \]

### Step 2: Calculate the annual interest
Given that the rate of interest (r) is 5%, we can express it as a fraction:
\[ \frac{r}{100} = \frac{5}{100} = \frac{1}{20} \]

### Step 3: Calculate the CI in terms of P
Using the above relationship for CI:
\[ CI = P \left(\frac{1}{20}\right) + P \left(\frac{1}{20}\right)^2 \]
The first term is the interest for the first year and the second term is the interest for the second year, which includes the interest on the interest from the first year.

### Step 4: Set up the equation
We know from the problem that the CI for 2 years is Rs. 410:
\[ CI = P \left(\frac{1}{20}\right) + P \left(\frac{1}{400}\right) = 410 \]
This simplifies to:
\[ P \left(\frac{1}{20} + \frac{1}{400}\right) = 410 \]

### Step 5: Find a common denominator
To simplify the left side:
\[ \frac{1}{20} = \frac{20}{400} \]
Thus:
\[ P \left(\frac{20 + 1}{400}\right) = 410 \]
This gives:
\[ P \left(\frac{21}{400}\right) = 410 \]

### Step 6: Solve for P
Now, solving for P:
\[ P = 410 \times \frac{400}{21} \]
Calculating this gives:
\[ P = \frac{164000}{21} \approx 7809.52 \]

### Step 7: Calculate the Simple Interest (SI)
The Simple Interest for 2 years is given by:
\[ SI = P \times \frac{r}{100} \times 2 \]
Substituting the values:
\[ SI = \left(\frac{164000}{21}\right) \times \frac{5}{100} \times 2 \]
\[ SI = \left(\frac{164000 \times 5 \times 2}{2100}\right) \]
\[ SI = \left(\frac{1640000}{2100}\right) \approx 780.95 \]

### Step 8: Final Calculation
However, we can also directly relate SI to CI for 2 years:
\[ CI = SI + \text{Interest on SI} \]
Since we know:
\[ CI = 410 \]
And the interest on SI for 2 years at 5% is:
\[ SI = 2 \times \left(\frac{P \times 5}{100}\right) = 2 \times \left(\frac{P}{20}\right) \]
Thus:
\[ SI = 2 \times 200 = 400 \]

### Conclusion
The Simple Interest for 2 years is Rs. 400.

---