The banker's gain of a sum due 4 yr, hence at 12% per annum is ? 72. Find the present worth of that bill.

Answer

To solve the problem of finding the present worth of a bill given the banker's gain, we can follow these steps:

### Step 1: Understand the given information
We are given:
- Banker's Gain (BG) = 72
- Rate of interest (R) = 12% per annum
- Time (T) = 4 years

### Step 2: Calculate the True Discount (TD)
The formula for True Discount is:
\[ TD = \frac{BG \times 100}{R \times T} \]

Substituting the values:
\[ TD = \frac{72 \times 100}{12 \times 4} \]

### Step 3: Simplify the calculation
Calculating the denominator:
- \( R \times T = 12 \times 4 = 48 \)

Now substituting back into the True Discount formula:
\[ TD = \frac{7200}{48} \]

### Step 4: Perform the division
Calculating:
\[ TD = 150 \]

### Step 5: Calculate the Present Worth (PW)
The formula for Present Worth is:
\[ PW = \frac{TD^2}{BG} \]

Substituting the values:
\[ PW = \frac{150^2}{72} \]

### Step 6: Calculate \(150^2\)
Calculating:
\[ 150^2 = 22500 \]

### Step 7: Divide by Banker's Gain
Now, substituting back:
\[ PW = \frac{22500}{72} \]

### Step 8: Perform the division
Calculating:
\[ PW \approx 312.5 \]

### Final Answer
The present worth of the bill is approximately **312.5**.

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