Step by step
The question
Question: A substance has a half-life of 3 years.
How much of 90 g is left after 10 years?
1 Count the half-lives
Every 3 years, half is left.
Half means times 1/2.
every 3 years: × 12
How many half-lives in 10 years?
10 divided by 3: 3.333333.
half-lives = 103 = 3.3333
2 Halve it that many times
A = the start times (1/2) to the power 3.333333.
A = 90 × (12)3.333333
The power is not whole.
With a calculator, (1/2) to the power 3.333333 is about 0.0992126.
(12)3.333333 ≈ 0.0992126
Times 90.
A ≈ 90 × 0.0992126
That is about 8.93 g, to two decimal places.
A ≈ 8.93 g
3 Check
Let's check if it makes sense.
3 half-lives leave 11.25 g.
4 half-lives leave 5.625 g.
3 half-lives: 11.25 g, 4 half-lives: 5.625 g
3.333333 is between 3 and 4, and our answer is between them too.
It works!
5.625 < 8.93 < 11.25
Let's look back.
First: count the half-lives.
Then: halve it that many times.
Last: check.
Answer
A ≈ 8.93 g