Mahesh borrowed a sum of money and return 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. Also find the total interest charged.

Options

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

Answer

To solve the problem of finding the sum borrowed by Mahesh and the total interest charged, we will use the formula for the present value of an annuity, as he is repaying the loan in equal quarterly installments.

### Step-by-Step Solution:

**Step 1: Understand the terms of the loan.**
- Mahesh repays the loan in 3 equal quarterly installments of 17,576 each.
- The interest rate is 16% per annum, compounded quarterly.

**Hint:** Remember that the quarterly interest rate is the annual rate divided by 4.

**Step 2: Calculate the quarterly interest rate.**
- Annual interest rate = 16%
- Quarterly interest rate = 16% / 4 = 4% = 0.04 (as a decimal)

**Hint:** Convert the percentage to a decimal by dividing by 100.

**Step 3: Use the present value formula for an annuity.**
The present value (PV) of an annuity can be calculated using the formula:
\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \( P \) = payment per period (17,576)
- \( r \) = interest rate per period (0.04)
- \( n \) = total number of payments (3)

**Hint:** Make sure to substitute the correct values into the formula.

**Step 4: Substitute the values into the formula.**
\[ PV = 17576 \times \left( \frac{1 - (1 + 0.04)^{-3}}{0.04} \right) \]

**Step 5: Calculate \( (1 + r)^{-n} \).**
\[ (1 + 0.04)^{-3} = (1.04)^{-3} \approx 0.889 \]

**Hint:** Use a calculator to find the value of \( (1.04)^{-3} \).

**Step 6: Calculate the present value.**
\[ PV = 17576 \times \left( \frac{1 - 0.889}{0.04} \right) \]
\[ PV = 17576 \times \left( \frac{0.111}{0.04} \right) \]
\[ PV = 17576 \times 2.775 = 48888.8 \]

**Step 7: Round the present value to the nearest whole number.**
The sum borrowed is approximately 48,889.

**Hint:** Always round to the nearest whole number when dealing with currency.

**Step 8: Calculate the total amount paid back.**
Total amount paid back = Number of installments × Amount of each installment
\[ Total = 3 \times 17576 = 52728 \]

**Step 9: Calculate the total interest charged.**
Total interest = Total amount paid back - Sum borrowed
\[ Total Interest = 52728 - 48889 = 3839 \]

### Final Answers:
- **Sum Borrowed:** 48,889
- **Total Interest Charged:** 3,839