Anil invests some money at a fixed rate of interest, compounded annually. If the interests accrued during the second and third year are ₹ 806.25 and ₹ 866.72, respectively, the interest accrued, in INR, during the fourth year is nearest to

Question

Bank A offers 6% interest rate per annum compounded half yearly. Bank B and Bank C offer simple interest but the annual interest rate offered by Bank C is twice that of Bank B. Raju invests a certain amount in Bank B for a certain period and Rupa invests ₹ 10,000 in Bank C for twice that period. The interest that would accrue to Raju during that period is equal to the interest that would have accrued had he invested the same amount in Bank A for one year. The interest accrued, in INR, to Rupa is

Answer 1

Let Anil invest a principal amount \( P \) at an annual interest rate of \( r\% \), compounded yearly. The interest amounts for the second and third years are provided, and the objective is to determine the interest for the fourth year.

Step 1: Interest in the second year

The interest earned in the second year is represented by: \[ P \left( 1 + \frac{r}{100} \right)^2 - P \]. This amount equals 806.25 rupees. The equation derived is: \[ P \left( 1 + \frac{r}{100} \right)^2 = P + 806.25 \quad \text{(1)} \]

Step 2: Interest in the third year

The interest earned in the third year is: \[ P \left( 1 + \frac{r}{100} \right)^3 - P \left( 1 + \frac{r}{100} \right)^2 \]. This amounts to 866.72 rupees, yielding the equation: \[ P \left( 1 + \frac{r}{100} \right)^3 = P \left( 1 + \frac{r}{100} \right)^2 + 866.72 \quad \text{(2)} \]

Substituting equation (1) into equation (2): \[ P \left( 1 + \frac{r}{100} \right)^3 = P + 806.25 + 866.72 \]

Simplified equation: \[ P \left( 1 + \frac{r}{100} \right)^3 = P + 1672.97 \]

Step 3: Determining the interest rate

The ratio of the third year's interest to the second year's interest is: \[ \frac{866.72}{806.25} = 1 + \frac{r}{100} \]

Solving for \( r \): \[ 1 + \frac{r}{100} = 1.075 \quad \implies \quad \frac{r}{100} = 0.075 \quad \implies \quad r = 7.5\% \]

Step 4: Calculating the interest for the fourth year

With \( r = 7.5\% \), the interest for the fourth year is calculated as: \[ \text{Interest in fourth year} = P \left( 1 + \frac{7.5}{100} \right)^4 - (P + 1672.97) \]

Simplified expression: \[ \text{Interest in fourth year} = P \left( 1.075^4 \right) - (P + 1672.97) \]

From the second year's interest: \[ 806.25 = P \left( 1.075^2 - 1 \right) \]

The value of \( P \) can be determined from this equation, enabling the calculation of the fourth year's interest.

Step 5: Conclusion

The calculation can be complex, potentially requiring iterative methods or algebraic manipulation. However, observing the increasing interest amounts annually, the fourth year's interest is expected to exceed the third year's interest, which in turn exceeds the second year's interest. Among the provided options, 931.72 rupees is the closest value for the fourth year's interest.

Final Answer:

The correct option is \( \boxed{(A): 931.72} \).

Anil invests some money at a fixed rate of interest, compounded annually. If the interests accrued during the second and third year are ₹ 806.25 and ₹ 866.72, respectively, the interest accrued, in INR, during the fourth year is nearest to

The Correct Option is A

Solution and Explanation