At `r%` per annum a sum become 2 times after 20 years. The rate of interest per annum is :

Options

A. `4%`
B. `5%`
C. `8%`
D. `10%`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We need to find the rate of interest per annum (r%) at which a sum of money becomes double after 20 years.

### Step 2: Define the Variables
Let the principal amount be \( P \). According to the problem, the amount after 20 years becomes \( 2P \).

### Step 3: Use the Formula for Simple Interest
The formula for the amount \( A \) in terms of principal \( P \), rate \( r \), and time \( t \) in simple interest is:
\[ A = P + \text{SI} \]
Where Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
Thus, we can write:
\[ A = P + \frac{P \times r \times t}{100} \]

### Step 4: Substitute the Known Values
In our case, the amount \( A = 2P \) and the time \( t = 20 \) years. Substituting these values into the equation gives:
\[ 2P = P + \frac{P \times r \times 20}{100} \]

### Step 5: Simplify the Equation
We can simplify the equation by subtracting \( P \) from both sides:
\[ 2P - P = \frac{P \times r \times 20}{100} \]
This simplifies to:
\[ P = \frac{P \times r \times 20}{100} \]

### Step 6: Cancel Out \( P \)
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ 1 = \frac{r \times 20}{100} \]

### Step 7: Solve for \( r \)
Now, we can solve for \( r \):
\[ 1 = \frac{20r}{100} \]
Multiplying both sides by 100 gives:
\[ 100 = 20r \]
Dividing both sides by 20 results in:
\[ r = \frac{100}{20} = 5 \]

### Step 8: Conclusion
Thus, the rate of interest per annum \( r \) is \( 5\% \).

### Final Answer
The rate of interest per annum is **5%**.

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