If a sum of money doubles itself in 8 years . Then the interest rate in percentage is

Options

A. `8(1)/(2) %`
B. 0.1
C. `10(1)/(2)%`
D. `12(1)/(2)%`

Answer

To find the interest rate when a sum of money doubles itself in 8 years using simple interest, we can follow these steps:

### Step 1: Understand the Concept
When a sum of money doubles, it means that the amount at the end of the period is twice the principal.

### Step 2: Define Variables
Let:
- Principal (P) = x (an arbitrary amount)
- Amount (A) = 2x (since the money doubles)
- Time (T) = 8 years
- Rate of Interest (R) = r%

### Step 3: Use the Simple Interest Formula
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]

### Step 4: Calculate Simple Interest
Since the amount doubles, we can express the relationship as:
\[ A = P + SI \]
Substituting the values we have:
\[ 2x = x + SI \]
This implies:
\[ SI = 2x - x = x \]

### Step 5: Substitute SI in the Formula
Now, we can substitute SI in the simple interest formula:
\[ x = \frac{x \times r \times 8}{100} \]

### Step 6: Simplify the Equation
To eliminate x from both sides (assuming x ≠ 0):
\[ 1 = \frac{r \times 8}{100} \]

### Step 7: Solve for r
Now, multiply both sides by 100:
\[ 100 = r \times 8 \]
Now, divide both sides by 8 to find r:
\[ r = \frac{100}{8} = 12.5 \]

### Conclusion
The interest rate is:
\[ r = 12.5\% \]