Shyam Lal takes a loan of Rs. 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 solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Loan Details
Shyam Lal takes a loan of Rs. 10,500 at an interest rate of 10% per annum compounded annually. He will repay this loan in two equal annual installments.

### Step 2: Calculate the Amount After One Year
The amount after one year can be calculated using the formula for compound interest:
\[ A = P(1 + r)^n \]
Where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal (initial amount)
- \( r \) = Rate of interest (as a decimal)
- \( n \) = Number of years

For the first year:
- \( P = 10,500 \)
- \( r = 10\% = 0.10 \)
- \( n = 1 \)

Calculating the amount after one year:
\[ A = 10,500 \times (1 + 0.10)^1 \]
\[ A = 10,500 \times 1.10 \]
\[ A = 11,550 \]

### Step 3: Calculate the Amount After Two Years
Now, we need to calculate the amount after the second year using the amount from the first year as the principal for the second year.

For the second year:
- \( P = 11,550 \)
- \( n = 1 \)

Calculating the amount after the second year:
\[ A = 11,550 \times (1 + 0.10)^1 \]
\[ A = 11,550 \times 1.10 \]
\[ A = 12,705 \]

### Step 4: Set Up the Equation for Equal Installments
Let the equal installment amount be \( x \). The total amount paid in two installments must equal the total amount due after two years.

The first installment \( x \) will be paid at the end of the first year, and the second installment \( x \) will be paid at the end of the second year. The first installment will accrue interest for one year, and the second installment will not accrue any interest.

Thus, the equation becomes:
\[ x + \frac{x}{(1 + r)} = 12,705 \]
Where \( r = 0.10 \).

Substituting the value of \( r \):
\[ x + \frac{x}{1.10} = 12,705 \]

### Step 5: Solve for \( x \)
To solve for \( x \), we first find a common denominator:
\[ x + \frac{x}{1.10} = 12,705 \]
\[ \Rightarrow \frac{1.10x + x}{1.10} = 12,705 \]
\[ \Rightarrow \frac{2.10x}{1.10} = 12,705 \]

Now, multiply both sides by 1.10 to eliminate the fraction:
\[ 2.10x = 12,705 \times 1.10 \]
\[ 2.10x = 13,975.5 \]

Now, divide both sides by 2.10:
\[ x = \frac{13,975.5}{2.10} \]
\[ x = 6,650 \]

### Step 6: Final Answer
Thus, each installment is Rs. 6,650.