A loan of ₹ 34,370 is paid back in three annual instalments. Second instalment is twice of first and third is three-fourth of second instalment. If for the remaining time, the rate of compound interest be 10% per annum on each instalment, find all three instalments.

Options

A. ₹6,000, ₹ 12,000.₹9000
B. ₹8,000, ₹14,000,₹9000
C. ₹9317,₹ 18635,₹ 13975.5
D. ₹9535, ₹19624,₹12975.5

Answer

To solve the problem step by step, we will denote the first installment as \( x \). According to the problem, we can express the second and third installments in terms of \( x \).

1. **Define the Installments:**
- First installment = \( x \)
- Second installment = \( 2x \) (twice the first)
- Third installment = \( \frac{3}{4} \times 2x = \frac{3}{2}x \) (three-fourths of the second)

2. **Calculate the Total Amount of Installments:**
The total amount paid back in installments is the sum of all three installments:
\[ \text{Total Installments} = x + 2x + \frac{3}{2}x = 3x + \frac{3}{2}x = \frac{6x + 3x}{2} = \frac{9x}{2} \]

3. **Set Up the Equation:**
We know that the total amount paid back, including compound interest, must equal the loan amount of ₹ 34,370. The formula for the future value of each installment when compounded annually at 10% for the remaining years is:
- First installment (compounded for 2 years): \( x(1 + 0.10)^2 = x(1.21) \)
- Second installment (compounded for 1 year): \( 2x(1 + 0.10)^1 = 2x(1.10) \)
- Third installment (compounded for 0 years): \( \frac{3}{2}x(1 + 0.10)^0 = \frac{3}{2}x \)

Therefore, the equation becomes:
\[ x(1.21) + 2x(1.10) + \frac{3}{2}x = 34,370 \]

4. **Combine the Terms:**
Simplifying the equation:
\[ 1.21x + 2.2x + 1.5x = 34,370 \]
\[ (1.21 + 2.2 + 1.5)x = 34,370 \]
\[ 4.91x = 34,370 \]

5. **Solve for \( x \):**
\[ x = \frac{34,370}{4.91} \approx 7,000 \]

6. **Calculate the Installments:**
Now that we have \( x \), we can find the values of all three installments:
- First installment = \( x = 7,000 \)
- Second installment = \( 2x = 14,000 \)
- Third installment = \( \frac{3}{2}x = 10,500 \)

7. **Final Answer:**
The three installments are:
- First installment: ₹ 7,000
- Second installment: ₹ 14,000
- Third installment: ₹ 10,500