At what rate per cent will a sum of money double itself in `6` years ?

Answer

To find the rate at which a sum of money will double itself in 6 years, we can follow these steps:

### Step 1: Understand the problem
We need to determine the rate of interest (R) that allows a principal amount (P) to double in 6 years.

### Step 2: Define the principal and amount
Let the principal amount be \( P \). If the amount doubles, then the total amount after 6 years will be:
\[ \text{Amount} = 2P \]

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

### Step 4: Use the Simple Interest formula
The formula for Simple Interest is also given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) is the Simple Interest,
- \( P \) is the principal,
- \( R \) is the rate of interest,
- \( T \) is the time in years.

### Step 5: Substitute known values
In our case, we know:
- \( SI = P \)
- \( T = 6 \) years

Substituting these values into the Simple Interest formula:
\[ P = \frac{P \times R \times 6}{100} \]

### Step 6: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 1 = \frac{R \times 6}{100} \]

### Step 7: Solve for R
Now, we can solve for \( R \):
\[ R \times 6 = 100 \]
\[ R = \frac{100}{6} \]
\[ R = \frac{50}{3} \]

### Step 8: Convert to percentage
To express \( R \) as a percentage:
\[ R \approx 16.67\% \]
This can also be expressed as:
\[ R = 16 \frac{2}{3}\% \]

### Final Answer
The rate at which a sum of money will double itself in 6 years is \( 16 \frac{2}{3}\% \).

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