A owes B 1,573, payable `1(1)/(2)` years hence. Also B owes A 1,444.50, payable 6 months hence. If they want to settle the account forthwith, keeping 14% as the rate of interest, then who should pay whom and how much ?

Options

A. A to B, ` 28.50
B. B to A, ` 37.50
C. A to B,` 50
D. B to A, ` 50

Answer

To solve the problem step by step, we need to calculate the present value of the debts that A owes to B and B owes to A, and then determine who should pay whom and how much.

### Step 1: Calculate the present value of A's debt to B

A owes B ₹1,573, payable in \(1 \frac{1}{2}\) years (or \(1.5\) years). We will calculate the present value using the formula:

\[ PV = \frac{FV}{(1 + r)^t} \]

Where:
- \(PV\) = Present Value
- \(FV\) = Future Value (₹1,573)
- \(r\) = Rate of interest (14% or 0.14)
- \(t\) = Time in years (1.5 years)

Substituting the values:

\[ PV_A = \frac{1573}{(1 + 0.14)^{1.5}} \]

Calculating \( (1 + 0.14)^{1.5} \):

\[ (1.14)^{1.5} \approx 1.14 \times 1.14^{0.5} \approx 1.14 \times 1.069 \approx 1.219 \]

Now substituting back:

\[ PV_A = \frac{1573}{1.219} \approx 1290.76 \]

### Step 2: Calculate the present value of B's debt to A

B owes A ₹1,444.50, payable in 6 months (or \(0.5\) years). We will use the same present value formula:

\[ PV_B = \frac{FV}{(1 + r)^t} \]

Where:
- \(FV\) = Future Value (₹1,444.50)
- \(r\) = Rate of interest (14% or 0.14)
- \(t\) = Time in years (0.5 years)

Substituting the values:

\[ PV_B = \frac{1444.50}{(1 + 0.14)^{0.5}} \]

Calculating \( (1 + 0.14)^{0.5} \):

\[ (1.14)^{0.5} \approx 1.069 \]

Now substituting back:

\[ PV_B = \frac{1444.50}{1.069} \approx 1351.73 \]

### Step 3: Determine the net amount to be settled

Now we compare the present values of the debts:

- Present value of A's debt to B: \(PV_A \approx 1290.76\)
- Present value of B's debt to A: \(PV_B \approx 1351.73\)

To settle the accounts, we find the difference:

\[ \text{Net Amount} = PV_B - PV_A = 1351.73 - 1290.76 \approx 60.97 \]

### Conclusion

Since \(PV_B\) is greater than \(PV_A\), B owes A the difference. Therefore, B should pay A approximately ₹60.97.