Question

A sum of money is borrowed and paid back in two annual instalments of Rs.\ 882 each, allowing 5% compound interest. The sum borrowed was

• A. Rs.\ 1600
• B. Rs.\ 1620
• C. Rs.\ 1640
• D. Rs.\ 1680
• E. Rs.\ 1650

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to find the total principal sum that was initially borrowed.
We are given that the loan is repaid in exactly two equal annual installments.
The value of each annual installment is Rs. 882.
The rate of compound interest applied to the borrowed sum is 5% per annum.
Step 2: Key Formula or Approach:
The present value of a loan paid off in equal installments is the sum of the present values of each individual installment.
The formula for the principal $P$ when paid in two equal installments $I$ at an interest rate $R$% is:
\[ P = \frac{I}{\left(1 + \frac{R}{100}\right)^1} + \frac{I}{\left(1 + \frac{R}{100}\right)^2} \]
This formula discounts each future payment back to its present value at the start of the borrowing period.
Step 3: Detailed Explanation:

We are given the installment amount $I = 882$ and the interest rate $R = 5$%.

First, we determine the discounting factor $\left(1 + \frac{R}{100}\right)$.

\[ 1 + \frac{5}{100} = 1 + \frac{1}{20} = \frac{21}{20} \]

Now, we substitute this fraction into our present value formula.

The present value of the first installment paid after one year is:

\[ PV_1 = \frac{882}{\frac{21}{20}} = 882 \times \frac{20}{21} \]

We simplify this expression by dividing 882 by 21.

Since $882 \div 21 = 42$, the present value of the first installment is $42 \times 20 = 840$.

The present value of the second installment paid after two years is:

\[ PV_2 = \frac{882}{\left(\frac{21}{20}\right)^2} = \frac{882}{\frac{441}{400}} = 882 \times \frac{400}{441} \]

We simplify this expression by dividing 882 by 441.

Since $882 \div 441 = 2$, the present value of the second installment is $2 \times 400 = 800$.

Finally, we calculate the total principal sum borrowed by adding the present values of both installments.

\[ P = PV_1 + PV_2 = 840 + 800 = 1640 \]

The sum borrowed was Rs. 1640.

Step 4: Final Answer:
Therefore, the initial sum of money borrowed was Rs. 1640.