At what rate per cent per annum simple interest will a sum double itself in `10` years?

Options

A. `8%`
B. `10%`
C. `12%`
D. `12.1/2%`

Answer

To find the rate per cent per annum at which a sum will double itself in 10 years using simple interest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the rate of interest (r) at which a principal amount (P) will double in 10 years. If we denote the principal amount as \( P \), then the amount after 10 years will be \( 2P \).

2. **Set Up the Equation**: The formula for the amount (A) in terms of principal (P) and simple interest (SI) is:
\[ A = P + SI \]
Since the amount doubles, we have:
\[ 2P = P + SI \]

3. **Calculate Simple Interest (SI)**: Rearranging the equation gives us:
\[ SI = 2P - P = P \]
So, the simple interest earned in 10 years is equal to the principal amount \( P \).

4. **Use the Simple Interest Formula**: The formula for simple interest is:
\[ SI = \frac{P \times r \times t}{100} \]
Here, \( SI = P \), \( t = 10 \) years, and \( P \) is the principal.

5. **Substitute the Values**: Plugging in the values we have:
\[ P = \frac{P \times r \times 10}{100} \]

6. **Simplify the Equation**: To eliminate \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 1 = \frac{r \times 10}{100} \]

7. **Solve for r**: Rearranging gives:
\[ r \times 10 = 100 \]
Dividing both sides by 10:
\[ r = 10 \]

8. **Conclusion**: The rate of interest \( r \) is 10%.

### Final Answer:
The rate per cent per annum at which a sum will double itself in 10 years is **10%**.

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