Pumima borrowed a sum of money and returned it in three equal quarterly instalments of ₹ 17576 each. Find the sum borrowed, if the rate of interest charged was `16%` per annum compounded quarterly. Find also the total interest charged:

Options

A. 46900 and 4700
B. 48775 and 3953
C. 68320 and 1200
D. none of these

Answer

To solve the problem, we need to find the sum borrowed by Fundima, given that she returned it in three equal quarterly installments of ₹ 17,576 each, with an interest rate of 16% per annum compounded quarterly.

### Step-by-Step Solution:

1. **Identify the Quarterly Interest Rate:**
The annual interest rate is 16%. Since the interest is compounded quarterly, we need to divide this rate by 4 (the number of quarters in a year).
\[ \text{Quarterly Interest Rate} = \frac{16\%}{4} = 4\% = \frac{4}{100} = 0.04 \]

2. **Determine the Present Value of the Installments:**
The formula for the present value (PV) of an annuity (which in this case are the quarterly installments) is given by:
\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
where:
- \( P \) = amount of each installment = ₹ 17,576
- \( r \) = quarterly interest rate = 0.04
- \( n \) = number of installments = 3

Plugging in the values:
\[ PV = 17576 \times \left( \frac{1 - (1 + 0.04)^{-3}}{0.04} \right) \]

3. **Calculate \( (1 + r)^{-n} \):**
\[ (1 + 0.04)^{-3} = (1.04)^{-3} \approx \frac{1}{1.124864} \approx 0.887 \]
Therefore:
\[ 1 - (1 + 0.04)^{-3} \approx 1 - 0.887 = 0.113 \]

4. **Calculate the Present Value:**
Now substituting back into the present value formula:
\[ PV = 17576 \times \left( \frac{0.113}{0.04} \right) \approx 17576 \times 2.825 = 49766.8 \]

5. **Round the Present Value:**
The sum borrowed is approximately ₹ 49,767 (rounding to the nearest whole number).

6. **Calculate Total Amount Paid:**
The total amount paid back by Fundima is:
\[ \text{Total Amount Paid} = 3 \times 17576 = 52728 \]

7. **Calculate Total Interest Charged:**
The total interest charged can be calculated as:
\[ \text{Total Interest} = \text{Total Amount Paid} - \text{Sum Borrowed} \]
\[ \text{Total Interest} = 52728 - 49767 = 2951 \]

### Final Answers:
- **Sum Borrowed:** ₹ 49,767
- **Total Interest Charged:** ₹ 2,951