The present worth of a sum due sometime hence is ? 400 and the true discount is ? 80. Find the banker's gain.

Answer

To find the banker's gain, we can follow these steps:

### Step 1: Understand the given values
We are given:
- Present Worth (PW) = ₹400
- True Discount (TD) = ₹80

### Step 2: Use the formula for Banker's Gain
The formula for Banker's Gain (BG) is:
\[ \text{Banker's Gain} = \text{True Discount} - \text{Interest on Present Worth} \]

### Step 3: Calculate the Interest on Present Worth
To find the interest on the present worth, we can use the relationship between True Discount and Present Worth. The True Discount can also be expressed as:
\[ \text{True Discount} = \text{Present Worth} \times \frac{r}{100} \times t \]
Where \( r \) is the rate of interest and \( t \) is the time period.

However, we can also derive the Banker's Gain directly from the given values:
\[ \text{Banker's Gain} = \text{True Discount} \times \frac{\text{Present Worth}}{\text{Present Worth} - \text{True Discount}} \]

### Step 4: Substitute the values into the formula
Now substituting the values:
- Present Worth = ₹400
- True Discount = ₹80

\[ \text{Banker's Gain} = 80 \times \frac{400}{400 - 80} \]
\[ = 80 \times \frac{400}{320} \]
\[ = 80 \times 1.25 \]
\[ = 100 \]

### Step 5: Conclusion
Thus, the Banker's Gain is ₹100.

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