Half Life Example Problems- Worked Examples for Mastery

Half Life Example Problems: No More Guessing

Half-life problems trip up most students. The math looks intimidating, but once you see the pattern, it's simple arithmetic with a decay formula. These worked examples will get you there.

What Half Life Actually Means

Half-life is the time it takes for half of a radioactive sample to decay. If you start with 100 grams and the half-life is 10 years, you have 50 grams left after 10 years. That's it. No hidden tricks.

The formula you need:

N = N₀ × (½)^t/T

• N = remaining amount
• N₀ = starting amount
• t = elapsed time
• T = half-life period

Worked Example 1: Basic Half Life Calculation

Problem: You have 80 grams of Carbon-14. Its half-life is 5730 years. How much remains after 17,190 years?

Step 1: Figure out how many half-lives passed.

17,190 ÷ 5,730 = 3 half-lives

Step 2: Cut in half three times.

80 → 40 → 20 → 10 grams

Answer: 10 grams remain.

Worked Example 2: Finding the Half Life

Problem: A 200 gram sample decays to 25 grams in 60 days. What is the half-life?

Step 1: Determine how many half-lives reduced 200 to 25.

200 → 100 → 50 → 25

That's 3 half-lives

Step 2: Divide total time by number of half-lives.

60 days ÷ 3 = 20 days per half-life

Worked Example 3: Using the Formula Directly

Problem: A sample has 150 atoms. The half-life is 5 minutes. How many atoms remain after 20 minutes?

Using the formula:

N = 150 × (½)^20/5

N = 150 × (½)⁴

N = 150 × 1/16

N = 9.375 atoms

Round to 9 atoms. You can't have partial atoms in a real sample, but for math problems, use the calculated value.

Worked Example 4: Time Required to Reach a Target

Problem: How long will it take for a 1000 Bq sample to decay to 125 Bq? Half-life is 3 years.

Step 1: Find the ratio.

125 ÷ 1000 = 0.125 = 1/8

Step 2: Determine how many half-lives to reach 1/8.

1/2 → 1/4 → 1/8

That's 3 half-lives

Step 3: Calculate total time.

3 × 3 years = 9 years

Quick Reference Table

Common Mistakes Students Make

Mistake 1: Forgetting to divide time by half-life to find the exponent. The exponent in (½)^t/T is the number of half-lives, not the time itself.

Mistake 2: Using the wrong starting amount. Some problems give intermediate values. Always identify N₀ clearly.

Mistake 3: Rounding too early. Keep full decimal values until your final answer.

Mistake 4: Confusing decay constant (λ) with half-life. They are related, but different calculations. Stick with half-life formulas unless the problem specifically asks for λ.

Getting Started: Your Approach

When you see a half-life problem:

1. Identify what you know: N₀, N, T, or t. Write them down.
2. Identify what you need: Which variable is missing?
3. Choose your method: Count half-lives for simple problems. Use the formula for complex ones.
4. Check your units: Time and half-life must match. Convert if needed.
5. Verify the answer: Does 3 half-lives mean 1/8 remaining? Quick sanity check.

The Honest Take

Half-life problems are pattern recognition. You either see how many times you halve the starting amount, or you plug numbers into the formula. Both approaches work. Pick whichever clicks faster for you.

Most exam questions give you the same structure: starting amount, half-life, elapsed time, find remaining amount. Once you practice 10 problems, you'll stop thinking about it. It becomes automatic.

Don't memorize every variation. Learn the one formula, practice the halving pattern, and you can solve any half-life problem that comes at you.