In a certain time, a sum becomes 4 times at the rate of 5% per annum. At what rate of simple interest, the same sum becomes 8 times in the same duration?

Options

A. `12(2)/(3)%`
B. `11(3)/(5)%`
C. `11(2)/(3)%`
D. `12(3)/(5)%`

Answer

To solve the problem step by step, we will use the concept of simple interest and the relationship between the principal, interest, and time.

### Step 1: Understand the problem
We know that a certain sum of money becomes 4 times itself at a rate of 5% per annum in a certain time period. We need to find the rate at which the same sum becomes 8 times itself in the same time period.

### Step 2: Set up the equations
Let the principal amount be \( P \) and the time period be \( T \).

From the problem, we know:
- When the sum becomes 4 times, the total amount \( A_1 = 4P \).
- The interest earned in this case is \( I_1 = A_1 - P = 4P - P = 3P \).

Using the formula for simple interest:
\[ I = P \times r \times T \]
where \( r \) is the rate of interest in decimal form.

For the first case:
\[ 3P = P \times 0.05 \times T \]

### Step 3: Simplify the equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = 0.05 \times T \]
Now, solve for \( T \):
\[ T = \frac{3}{0.05} = 60 \text{ years} \]

### Step 4: Analyze the second case
Now, we need to find the rate \( r_2 \) for the case when the sum becomes 8 times itself.

In this case:
- The total amount \( A_2 = 8P \).
- The interest earned is \( I_2 = A_2 - P = 8P - P = 7P \).

Using the simple interest formula again:
\[ 7P = P \times r_2 \times T \]

### Step 5: Simplify the second equation
Dividing both sides by \( P \):
\[ 7 = r_2 \times T \]
Substituting \( T = 60 \):
\[ 7 = r_2 \times 60 \]
Now, solve for \( r_2 \):
\[ r_2 = \frac{7}{60} \]

### Step 6: Convert to percentage
To convert \( r_2 \) into a percentage, multiply by 100:
\[ r_2 = \frac{7}{60} \times 100 = \frac{700}{60} \approx 11.67\% \]

### Final Answer
The rate of simple interest at which the same sum becomes 8 times in the same duration is approximately **11.67%**.

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