in what time a certain sum will become 3 1/4 time at 5%?

To determine the time it takes for a certain sum to become 3 1/4 times its original value at a 5% interest rate, you would need to use the compound interest formula. The formula is:

A = P(1 + r/n)^(nt)

Where:
A = Final amount
P = Principal (initial amount)
r = Interest rate (in decimal form)
n = Number of times interest is compounded per time period
t = Number of time periods

In this case, since you want to find the time it takes for the sum to become 3 1/4 times the original value (3.25), the equation becomes:

3.25 = P(1 + 0.05/n)^(nt)

Now, let's solve for time (t).

Step 1: Simplify the equation
Divide both sides of the equation by P:
3.25 / P = (1 + 0.05/n)^(nt)

Step 2: Take the natural logarithm of both sides
ln(3.25 / P) = ln[(1 + 0.05/n)^(nt)]

Step 3: Apply the power rule of logarithms to bring down the exponent
ln(3.25 / P) = (nt) * ln(1 + 0.05/n)

Step 4: Solve for t
Divide both sides by ln(1 + 0.05/n):
t = [ln(3.25 / P)] / [ln(1 + 0.05/n)]

Once you have the principal amount (P) and the value of n (number of times compounded per time period), you can substitute those values into the equation to calculate the time (t) it would take for the sum to become 3 1/4 times its original value at 5% interest.