Given:
- The sum of money trebles itself in 15 years 6 months.

To find:
- In how many years would it double itself?

Solution:

To solve this problem, we can use the concept of compound interest.

Step 1: Understanding the given information
- When a sum of money trebles itself, it means that the amount becomes three times its original value.
- So, if the original amount is A, then the new amount after 15 years 6 months will be 3A.

Step 2: Understanding the formula for compound interest
- The formula for compound interest is given by: A = P(1 + r/n)^(nt)
- Where A is the final amount, P is the principal amount (original amount), r is the interest rate, n is the number of times interest is compounded per year, and t is the time period in years.

Step 3: Applying the formula
- In this case, we want to find the time period it takes for the amount to double itself.
- Let's assume the original amount is A and the new amount after doubling is 2A.
- Using the compound interest formula, we can write: 2A = A(1 + r/n)^(nt)

Step 4: Simplifying the equation
- Dividing both sides of the equation by A, we get: 2 = (1 + r/n)^(nt)
- Taking the natural logarithm (ln) of both sides, we get: ln(2) = ln((1 + r/n)^(nt))
- Using the property of logarithms, we can write: nt * ln(1 + r/n) = ln(2)
- Rearranging the equation, we get: t = ln(2) / (n * ln(1 + r/n))

Step 5: Applying the given information
- We know that the amount trebles itself in 15 years 6 months, which is equivalent to 15.5 years.
- Using this value in the equation, we get: 15.5 = ln(2) / (n * ln(1 + r/n))

Step 6: Calculating the time period to double the amount
- We need to find the value of t when the amount becomes double, which is equivalent to 2.
- Substituting this value in the equation, we get: 2 = ln(2) / (n * ln(1 + r/n))
- Rearranging the equation, we get: n * ln(1 + r/n) = ln(2) / 2

Step 7: Calculating the time period
- To calculate the time period, we need to know the values of n and r.
- Since these values are not given, we cannot determine the exact time period.
- However, we can compare the options given and choose the closest value based on the calculations.

Option B: 7 years 9 months
- Comparing the options, option B is the closest value to the calculated time period.
- Therefore, the answer is option B) 7 years 9 months.

Conclusion:
- In this problem, we