A has to pay Rs. 220 to B after 1 years. B asks A to pay Rs. 110 in cash and defer the payment of Rs. 110 for 2 years. A agrees to it, Counting the rate of interest at `10%` per annum in this new mode of payment-

Options

A. There is no gain or loss to any one
B. A gains Rs. `7cdot34`
C. A loses Rs. `7cdot34`
D. A loses Rs. `1cdot66`

Answer

To solve the problem step-by-step, we need to determine how much A should actually pay B in the new payment arrangement.

### Step 1: Calculate the Present Worth of Rs. 220 due in 1 year
A has to pay Rs. 220 after 1 year. To find the present worth (PW) of this amount, we use the formula:

\[ PW = \frac{FV}{(1 + r)^n} \]

Where:
- \(FV\) = Future Value (Rs. 220)
- \(r\) = Rate of interest (10% or 0.10)
- \(n\) = Number of years (1)

Substituting the values:

\[ PW = \frac{220}{(1 + 0.10)^1} = \frac{220}{1.10} = 200 \]

### Step 2: Calculate the Present Worth of Rs. 110 deferred for 2 years
B asks A to pay Rs. 110 in cash and defer the payment of another Rs. 110 for 2 years. We need to calculate the present worth of the Rs. 110 that is deferred for 2 years.

Using the same formula:

\[ PW = \frac{FV}{(1 + r)^n} \]

Where:
- \(FV\) = Future Value (Rs. 110)
- \(r\) = Rate of interest (10% or 0.10)
- \(n\) = Number of years (2)

Substituting the values:

\[ PW = \frac{110}{(1 + 0.10)^2} = \frac{110}{(1.10)^2} = \frac{110}{1.21} \approx 90.91 \]

### Step 3: Calculate the total present worth of the payments
Now, we need to find the total present worth of the payments A is making:

- Cash payment: Rs. 110
- Deferred payment present worth: Rs. 90.91

Total present worth:

\[ Total PW = 110 + 90.91 = 200.91 \]

### Step 4: Determine the difference from the original present worth
Now we compare the total present worth of the new payment arrangement to the original present worth of Rs. 220:

\[ Difference = 200 - 200.91 = -0.91 \]

This means A would be paying slightly more than the original present worth of Rs. 220.

### Conclusion
A should pay Rs. 200.91 in total under the new payment arrangement.

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