At r% per annum a sum doubles after 20 years . The rate of interest per annum is :

Options

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

Answer

To solve the problem of finding the rate of interest per annum when a sum doubles after 20 years, we can follow these steps:

### Step 1: Understand the Problem
We know that the principal amount (P) doubles after a certain time period (T). If we let the principal amount be \( x \), then after 20 years, the amount (A) will be \( 2x \).

### Step 2: Identify the Given Values
- Principal (P) = \( x \)
- Amount (A) = \( 2x \)
- Time (T) = 20 years

### Step 3: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = A - P \]
Substituting the values we have:
\[ SI = 2x - x = x \]

### Step 4: Use the Simple Interest Formula
The formula for Simple Interest can also be expressed as:
\[ SI = \frac{P \times r \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal
- \( r \) = Rate of interest per annum
- \( T \) = Time in years

### Step 5: Substitute the Known Values
Now we can substitute the known values into the formula:
\[ x = \frac{x \times r \times 20}{100} \]

### Step 6: Simplify the Equation
We can simplify this equation. First, we can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \frac{r \times 20}{100} \]

### Step 7: Solve for \( r \)
Now, we can solve for \( r \):
\[ 100 = 20r \]
\[ r = \frac{100}{20} = 5 \]

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

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