Step by step
The question
Question: A substance has a half-life of 5 years.
How much of 80 g is left after 12 years?
1 Count the half-lives
Every 5 years, half is left.
Half means times 1/2.
every 5 years: × 12
How many half-lives in 12 years?
12 divided by 5: 2.4.
half-lives = 125 = 2.4
2 Halve it that many times
A = the start times (1/2) to the power 2.4.
A = 80 × (12)2.4
The power is not whole.
With a calculator, (1/2) to the power 2.4 is about 0.189465.
(12)2.4 ≈ 0.189465
Times 80.
A ≈ 80 × 0.189465
That is about 15.16 g, to two decimal places.
A ≈ 15.16 g
3 Check
Let's check if it makes sense.
2 half-lives leave 20 g.
3 half-lives leave 10 g.
2 half-lives: 20 g, 3 half-lives: 10 g
2.4 is between 2 and 3, and our answer is between them too.
It works!
10 < 15.16 < 20
Let's look back.
First: count the half-lives.
Then: halve it that many times.
Last: check.
Answer
A ≈ 15.16 g