The half-life of radioactive material is the length of time it takes the substance to
be halved due to the emission of radiation. If the half-life of a radioactive substance is
50 years, what fraction of the original amount will remain after 150 years?

To determine the fraction of the original amount of a radioactive substance that will remain after 150 years, we can use the concept of half-life.

### Step 1: Determine the number of half-lives
Given:
- Half-life of the substance = 50 years
- Total time = 150 years

The number of half-lives that have passed is:
\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{150}{50} = 3
\]

### Step 2: Calculate the remaining fraction
After each half-life, the remaining amount of the substance is halved. Therefore, after \( n \) half-lives, the fraction of the substance remaining is:
\[
\text{Remaining fraction} = \left(\frac{1}{2}\right)^n
\]
For 3 half-lives:
\[
\text{Remaining fraction} = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]

### Final Answer
After 150 years, **\(\frac{1}{8}\)** of the original amount of the radioactive substance will remain.

Would you like further details or have any questions about this calculation?

Here are 5 related questions to explore further:
1. How much of the substance remains after 200 years if the half-life is 50 years?
2. What is the remaining fraction after 100 years if the half-life is 25 years?
3. How would you calculate the remaining amount after a non-integer multiple of the half-life, say 75 years?
4. If the half-life were instead 30 years, what fraction remains after 90 years?
5. What is the relationship between the number of half-lives and the fraction remaining?

**Tip:** Understanding the exponential nature of decay can help in calculating the remaining amount of a substance over any given period.

Understand how to calculate the fraction of a radioactive substance remaining after 150 years with a 50-year half-life. Explore the exponential decay concept and the formula (1/2)^n. Suitable for advanced high school students.

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