If rate of interest and time on a certain bill are numerically equal and true discount is 81 times of banker's gain, find the rate of interest.

Options

A. `2(9)/(13)%`
B. `1(2)/(9)%`
C. `1(7)/(9)%`
D. `1(1)/(9)%`

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the Given Information
We know that:
- The rate of interest (R) is numerically equal to the time (T).
- The true discount (Td) is 81 times the banker's gain (Bg).

### Step 2: Set Up the Relationship
Since R = T, we can denote both as R. Therefore, we have:
- Td = 81 * Bg

### Step 3: Use the Formula for True Discount and Banker's Gain
We know the relationship between true discount and banker's gain:
\[ Td = \frac{Bg \times 100}{R \times T} \]

Since R = T, we can substitute T with R:
\[ Td = \frac{Bg \times 100}{R \times R} \]
\[ Td = \frac{Bg \times 100}{R^2} \]

### Step 4: Substitute True Discount in Terms of Banker's Gain
From the information given, we can also write:
\[ Td = 81 \times Bg \]

### Step 5: Equate the Two Expressions for True Discount
Now we can equate the two expressions for Td:
\[ 81 \times Bg = \frac{Bg \times 100}{R^2} \]

### Step 6: Simplify the Equation
Assuming Bg is not zero, we can divide both sides by Bg:
\[ 81 = \frac{100}{R^2} \]

### Step 7: Rearrange to Solve for R
Now, rearranging the equation gives:
\[ R^2 = \frac{100}{81} \]

### Step 8: Take the Square Root
To find R, we take the square root of both sides:
\[ R = \sqrt{\frac{100}{81}} \]
\[ R = \frac{10}{9} \]

### Step 9: Convert to Percentage
To express R as a percentage, we can convert it:
\[ R = \frac{10}{9} \times 100 = 111.11\% \]

### Final Answer
Thus, the rate of interest is approximately **11.11%**.

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