Shyam Lal takes a loan of ₹ 10500 at `10%` p.a. compounded annually which is to be repaid in two equal annual instalments. One at the end of one year and the other at the end of the second year. The value of each instalments is

Options

A. 5987
B. 6050
C. 6352.5
D. 5678.5

Answer

To find the value of each installment that Shyam Lal has to pay back for the loan of ₹10,500 at an interest rate of 10% per annum compounded annually, we can follow these steps:

### Step 1: Understand the Loan Amount and Interest Rate
- **Loan Amount (P)**: ₹10,500
- **Rate of Interest (r)**: 10% per annum

### Step 2: Calculate the Amount after the First Year
The amount to be paid back after the first year can be calculated using the formula for compound interest:
\[ A_1 = P(1 + r) \]
Where \( r \) is expressed as a decimal (10% = 0.10).

Calculating:
\[ A_1 = 10,500 \times (1 + 0.10) \]
\[ A_1 = 10,500 \times 1.10 \]
\[ A_1 = 11,550 \]

### Step 3: Calculate the Amount after the Second Year
For the second year, we apply the interest again on the amount after the first year:
\[ A_2 = A_1(1 + r) \]
Calculating:
\[ A_2 = 11,550 \times (1 + 0.10) \]
\[ A_2 = 11,550 \times 1.10 \]
\[ A_2 = 12,705 \]

### Step 4: Set Up the Equation for Equal Installments
Let the amount of each installment be \( x \). Since Shyam Lal pays two installments, one at the end of the first year and the other at the end of the second year, the equation can be set up as:
\[ x + \frac{x}{(1 + r)} = 10,500 \]
Where the second installment is discounted back to the present value.

### Step 5: Substitute the Values
Substituting \( r = 0.10 \):
\[ x + \frac{x}{1.10} = 10,500 \]

### Step 6: Solve for \( x \)
To solve for \( x \), we first find a common denominator:
\[ x + \frac{x}{1.10} = 10,500 \]
\[ \Rightarrow x + 0.9091x = 10,500 \]
\[ \Rightarrow 1.9091x = 10,500 \]
Now, divide both sides by 1.9091:
\[ x = \frac{10,500}{1.9091} \]
Calculating:
\[ x \approx 5,500 \]

### Conclusion
The value of each installment is approximately **₹5,500**.

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