A sum becomes 6 fold at 5% per annum. At what rate, the sum becomes 12 fold?

Options

A. 0.1
B. 0.12
C. 0.09
D. 0.11

Answer

To solve the problem step by step, we will use the concept of simple interest and the relationship between the folds of the sum and the rate of interest.

### Step 1: Understand the problem
We know that a certain sum of money becomes 6 times (6 fold) at an interest rate of 5% per annum. We need to find out the rate at which the same sum becomes 12 times (12 fold).

### Step 2: Define the variables
Let:
- \( m = 12 \) (the fold we want to achieve)
- \( n = 6 \) (the fold we already know)
- \( r_1 = 5\% \) (the rate of interest for the 6 fold)

### Step 3: Use the formula for rate of interest
The formula to find the rate of interest when comparing different folds is given by:
\[ r_2 = \frac{m - 1}{n - 1} \times r_1 \]
Where:
- \( r_2 \) is the rate of interest we want to find.

### Step 4: Substitute the values into the formula
Substituting the values into the formula:
\[ r_2 = \frac{12 - 1}{6 - 1} \times 5 \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ r_2 = \frac{11}{5} \times 5 \]

### Step 6: Calculate the rate of interest
Now, we can cancel out the 5 in the numerator and denominator:
\[ r_2 = 11\% \]

### Step 7: Conclusion
Thus, the rate at which the sum becomes 12 fold is **11% per annum**.

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