A certain sum of money will be doubled in 15 years at the rate of simple interest percent per annum of

Options

A. 25
B. `5 (1)/(2)`
C. 6
D. `6 (2)/(3)`

Answer

To solve the problem of finding the rate of simple interest at which a certain sum of money will double in 15 years, we can follow these steps:

### Step 1: Understand the Problem
We need to find the rate of simple interest (R) that will allow a principal amount (P) to double in 15 years.

### Step 2: Set Up the Equation
When a sum of money doubles, the total amount (A) becomes:
\[ A = 2P \]
The formula for the amount in simple interest is:
\[ A = P + I \]
where \( I \) is the interest earned. The interest can be calculated using:
\[ I = \frac{P \times R \times T}{100} \]
where \( T \) is the time in years.

### Step 3: Substitute the Values
Since we want the amount to double:
\[ 2P = P + \frac{P \times R \times 15}{100} \]

### Step 4: Simplify the Equation
Subtract \( P \) from both sides:
\[ 2P - P = \frac{P \times R \times 15}{100} \]
This simplifies to:
\[ P = \frac{P \times R \times 15}{100} \]

### Step 5: Cancel Out the Principal
Assuming \( P \) is not zero, we can divide both sides by \( P \):
\[ 1 = \frac{R \times 15}{100} \]

### Step 6: Solve for R
To find \( R \), multiply both sides by 100:
\[ 100 = R \times 15 \]
Now divide both sides by 15:
\[ R = \frac{100}{15} \]

### Step 7: Simplify the Rate
Calculating \( \frac{100}{15} \):
\[ R = \frac{20}{3} \]
This can be converted to a mixed fraction:
\[ R = 6 \frac{2}{3} \]

### Conclusion
The rate of simple interest at which the sum will double in 15 years is:
\[ R = 6 \frac{2}{3} \% \]