A owes B Rs 2600, payable `2(1)/(2)` yr hence. Also B owes A Rs 3180, Payable 6 months hence. If they want to settle the account forthwith, keeping 12% as the rate of simple interest, then who should pay whom and how much?

Options

A. Rs 1500
B. Rs 2000
C. Rs 1000
D. Rs 3000

Answer

To solve the problem, we need to find out how much A owes B and how much B owes A, and then determine who should pay whom and how much, considering the simple interest.

### Step 1: Calculate the Present Worth of A's Debt to B

A owes B Rs 2600, payable in \(2\frac{1}{2}\) years (which is \(2.5\) years). The rate of interest is 12%.

Using the formula for present worth:
\[ \text{Present Worth} = \frac{\text{Amount}}{(1 + \text{Rate} \times \text{Time})} \]
Where:
- Amount = Rs 2600
- Rate = 12% = 0.12
- Time = \(2.5\) years

Calculating:
\[ \text{Present Worth of A's Debt} = \frac{2600}{1 + (0.12 \times 2.5)} = \frac{2600}{1 + 0.3} = \frac{2600}{1.3} \]
\[ = \frac{2600 \times 10}{13} = \frac{26000}{13} \approx 2000 \]

### Step 2: Calculate the Present Worth of B's Debt to A

B owes A Rs 3180, payable in 6 months (which is \(0.5\) years). The rate of interest is also 12%.

Using the same formula:
\[ \text{Present Worth of B's Debt} = \frac{3180}{1 + (0.12 \times 0.5)} = \frac{3180}{1 + 0.06} = \frac{3180}{1.06} \]
Calculating:
\[ = \frac{3180 \times 100}{106} \approx \frac{318000}{106} \approx 3000 \]

### Step 3: Compare the Present Worths

- Present Worth of A's Debt to B = Rs 2000
- Present Worth of B's Debt to A = Rs 3000

### Step 4: Determine Who Pays Whom and How Much

Since B's present worth (Rs 3000) is greater than A's present worth (Rs 2000), B owes A more than A owes B.

To settle the accounts:
\[ \text{Net Amount} = \text{Present Worth of B's Debt} - \text{Present Worth of A's Debt} = 3000 - 2000 = 1000 \]

Thus, B should pay A Rs 1000 to settle the accounts.

### Final Answer:
B should pay A Rs 1000.

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