Answer is 8 month here is how :
**in 1st four months half of substance will decay that is 1/2 is decayed
**and after 2nd four month 1/2 again gone means
1/2 +1/2=3/4
so, 8 months for 3/4 to gone that is (2 half lives)………
Hope this will help you

Introduction:
A radioactive substance undergoes decay over time, and its half-life is the time it takes for half of the substance to decay. In this case, we have a radioactive substance with a half-life of four months. We need to determine the time it takes for three-fourths of the substance to decay.

Understanding the half-life:
Before we proceed to solve the problem, let's first understand the concept of a half-life. The half-life of a radioactive substance is the time it takes for half of the initial amount to decay. After one half-life, only half of the original substance remains. After two half-lives, only one-fourth of the original substance remains, and so on.

Finding the number of half-lives:
To determine the time it takes for three-fourths of the substance to decay, we need to find the number of half-lives required. Since we know the half-life is four months, we can calculate the number of half-lives as follows:

Number of half-lives = (time elapsed) / (half-life)

In this case, we want to find the time it takes for three-fourths of the substance to decay, so the time elapsed would be the total time required. Let's denote the total time required as "t."

Number of half-lives = t / 4 months

Calculating three-fourths decay:
Since we want to find the time it takes for three-fourths of the substance to decay, we can set up the following equation:

(1/2)^(number of half-lives) = 3/4

Taking the logarithm of both sides allows us to solve for the number of half-lives:

log[(1/2)^(number of half-lives)] = log(3/4)

Using the logarithmic property, we can bring down the exponent:

(number of half-lives) * log(1/2) = log(3/4)

Now, we can solve for the number of half-lives:

number of half-lives = log(3/4) / log(1/2)

Calculating the total time:
Finally, we can substitute the number of half-lives into the equation we derived earlier to find the total time required:

t = (number of half-lives) * 4 months

Now, we can calculate the total time required for three-fourths of the substance to decay.

Conclusion:
To determine the time it takes for three-fourths of a radioactive substance with a half-life of four months to decay, we need to calculate the number of half-lives required. By setting up the appropriate equations and using logarithms, we can find the number of half-lives and then calculate the total time required.