The present worth of a sum due sometimes hence is Rs. 576 and the banker's gain is Rs. 1. The true discount is-----

Options

A. Rs. 16
B. Rs. 18
C. Rs. 24
D. Rs. 32

Answer

To solve the problem step by step, we will use the given information and the formula for true discount.

### Step 1: Identify the given values
- Present Worth (PW) = Rs. 576
- Banker's Gain (BG) = Rs. 1

### Step 2: Understand the relationship between True Discount (TD), Present Worth (PW), and Banker's Discount (BD)
The formula for True Discount (TD) is:
\[ TD = \sqrt{PW \times BD} \]
Where:
- PW = Present Worth
- BD = Banker's Discount

### Step 3: Relate Banker's Gain (BG) to Banker's Discount (BD)
Banker's Gain (BG) is defined as:
\[ BG = BD - TD \]
From the problem, we know:
\[ BG = 1 \]
So we can express Banker's Discount (BD) as:
\[ BD = TD + 1 \]

### Step 4: Substitute the expression for BD into the True Discount formula
Now we can substitute \( BD \) in the True Discount formula:
\[ TD = \sqrt{PW \times (TD + 1)} \]

### Step 5: Substitute the value of Present Worth (PW)
Now, substituting the value of PW:
\[ TD = \sqrt{576 \times (TD + 1)} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ TD^2 = 576 \times (TD + 1) \]

### Step 7: Rearranging the equation
This expands to:
\[ TD^2 = 576 \times TD + 576 \]
Rearranging gives:
\[ TD^2 - 576 \times TD - 576 = 0 \]

### Step 8: Solve the quadratic equation
We can use the quadratic formula \( TD = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -576, c = -576 \):
- Calculate the discriminant:
\[ b^2 - 4ac = (-576)^2 - 4 \times 1 \times (-576) \]
\[ = 331776 + 2304 = 334080 \]
- Now apply the quadratic formula:
\[ TD = \frac{576 \pm \sqrt{334080}}{2} \]

### Step 9: Calculate the square root and find TD
Calculating \( \sqrt{334080} \) gives approximately 578.
Thus:
\[ TD = \frac{576 \pm 578}{2} \]
Calculating the two possible values:
1. \( TD = \frac{1154}{2} = 577 \) (not valid, as it exceeds PW)
2. \( TD = \frac{-2}{2} = -1 \) (not valid)

### Step 10: Find the True Discount using the earlier method
From earlier steps, we can also directly calculate True Discount using:
\[ TD = \sqrt{576 \times 1} = \sqrt{576} = 24 \]

### Final Answer
The True Discount is Rs. 24.

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