Step by step
The question
Question: A substance has a half-life of 8 days.
How much of 50 mg is left after 20 days?
1 Count the half-lives
Every 8 days, half is left.
Half means times 1/2.
every 8 days: × 12
How many half-lives in 20 days?
20 divided by 8: 2.5.
half-lives = 208 = 2.5
2 Halve it that many times
A = the start times (1/2) to the power 2.5.
A = 50 × (12)2.5
The power is not whole.
With a calculator, (1/2) to the power 2.5 is about 0.176777.
(12)2.5 ≈ 0.176777
Times 50.
A ≈ 50 × 0.176777
That is about 8.84 mg, to two decimal places.
A ≈ 8.84 mg
3 Check
Let's check if it makes sense.
2 half-lives leave 12.5 mg.
3 half-lives leave 6.25 mg.
2 half-lives: 12.5 mg, 3 half-lives: 6.25 mg
2.5 is between 2 and 3, and our answer is between them too.
It works!
6.25 < 8.84 < 12.5
Let's look back.
First: count the half-lives.
Then: halve it that many times.
Last: check.
Answer
A ≈ 8.84 mg