The banker's gain on a certain sum due `2 1/2` years hence is `3/23` of the banker's discount. The rate per cent is -----

Options

A. `5%`
B. `6%`
C. `2 (14)/(23)%`
D. `6 2/3 %`

Answer

To solve the problem step by step, we will break it down into manageable parts:

### Step 1: Understand the Given Information
We know that the banker's gain on a certain sum due in \(2 \frac{1}{2}\) years is \(\frac{3}{23}\) of the banker's discount.

### Step 2: Define Variables
Let the banker's discount (BD) be equal to 1 rupee. Therefore, the banker's gain (BG) can be expressed as:
\[ BG = \frac{3}{23} \times BD = \frac{3}{23} \times 1 = \frac{3}{23} \text{ rupees} \]

### Step 3: Calculate the True Discount (TD)
The true discount (TD) is calculated as:
\[ TD = BD - BG = 1 - \frac{3}{23} \]
To perform this subtraction, we convert 1 into a fraction with a denominator of 23:
\[ 1 = \frac{23}{23} \]
Now, we can subtract:
\[ TD = \frac{23}{23} - \frac{3}{23} = \frac{20}{23} \text{ rupees} \]

### Step 4: Calculate the Sum (Principal, P)
The sum (P) can be calculated using the formula:
\[ P = \frac{BD \times TD}{BD - TD} \]
Substituting the values we have:
\[ P = \frac{1 \times \frac{20}{23}}{1 - \frac{20}{23}} = \frac{\frac{20}{23}}{\frac{3}{23}} = \frac{20}{3} \text{ rupees} \]

### Step 5: Calculate Simple Interest (SI)
The simple interest (SI) for the time period of \(2 \frac{1}{2}\) years (which is \( \frac{5}{2} \) years) can be calculated using the formula:
\[ SI = P \times r \times t \]
Where \(r\) is the rate of interest and \(t\) is the time in years. We need to express \(SI\) in terms of \(r\):
\[ SI = \frac{20}{3} \times \frac{r}{100} \times \frac{5}{2} \]

### Step 6: Substitute SI into the Rate Formula
We know that the banker's gain is equal to the simple interest:
\[ BG = SI \]
Substituting the values we have:
\[ \frac{3}{23} = \frac{20}{3} \times \frac{r}{100} \times \frac{5}{2} \]
Now, simplifying the right side:
\[ \frac{3}{23} = \frac{20 \times 5 \times r}{3 \times 100 \times 2} = \frac{100r}{600} \]
Thus:
\[ \frac{3}{23} = \frac{r}{6} \]

### Step 7: Solve for r
Cross-multiplying gives:
\[ 3 \times 6 = 23r \]
\[ 18 = 23r \]
\[ r = \frac{18}{23} \]

### Step 8: Convert to Percentage
To convert \(r\) into a percentage, we multiply by 100:
\[ r = \frac{18}{23} \times 100 \approx 78.26\% \]

### Final Answer
The rate percent is approximately \(6\%\).

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