At what rate per cent per annum simple interest will a sum double itself in 8 years ?

Answer

To find the rate per cent per annum simple interest at which a sum will double itself in 8 years, we can follow these steps:

### Step 1: Understand the Problem
We need to find the rate of interest (r) when a principal amount (P) doubles in 8 years.

### Step 2: Define the Variables
- Let the principal amount be \( P \).
- After 8 years, the amount will be \( 2P \) (since it doubles).
- The time period \( T \) is given as 8 years.

### Step 3: Calculate Simple Interest
The formula for simple interest (SI) is:
\[ SI = \text{Amount} - \text{Principal} \]
Substituting the values:
\[ SI = 2P - P = P \]

### Step 4: Use the Simple Interest Formula
The formula for simple interest can also be expressed as:
\[ SI = \frac{P \times r \times T}{100} \]
Substituting the known values:
\[ P = \frac{P \times r \times 8}{100} \]

### Step 5: Simplify the Equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 1 = \frac{r \times 8}{100} \]

### Step 6: Solve for \( r \)
Rearranging the equation gives:
\[ r \times 8 = 100 \]
\[ r = \frac{100}{8} \]
\[ r = 12.5 \]

### Step 7: Conclusion
The rate of interest is:
\[ \text{Rate of Interest} = 12.5\% \]

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