Lessons · Medicine foundations · half-life, and what is left after n of them
Half-life: halve it, and halve it again
After each half-life half of what was there is gone, so after n half-lives the fraction remaining is one half to the power n.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
The question looks like it wants a calculator and an exponential, and it wants four halvings on the back of a page. Knowing that is worth two minutes in an exam and a great deal of calm in a tutorial where somebody asks you to do it out loud.
How to think about it
Turn the time into a count of half-lives first: hours elapsed divided by the half-life. Then halve the starting amount that many times, or use one half to the power of the count, which is the same thing written shorter. The answer never reaches zero; it just gets small.
Worked example
Fraction remaining after n half-lives = (1/2) to the power nThe rule. n is a count, not a time.
Three half-lives: (1/2)³ = 1/8 = 0.125, an eighth leftHalve, halve, halve. Eighty-seven and a half percent has gone.
Starting from 100 mg: 100 × 0.125 = 12.5 mg remainingThe fraction, applied to the starting amount.
A half-life of 6 h and 18 h elapsed: 18 / 6 = 3 half-lives, so the same eighthTime in, count out. The count is what the power uses.
Your turn
A problem gives a starting amount of 80 mg and two half-lives elapsed. Write the line that gives the amount remaining.
80 × (1/2)² = mg
Solve one, graded on the server
The trap
Halving the time instead of the amount, or subtracting half the amount once for every hour rather than once for every half-life. The clock and the count are different numbers and only one of them goes in the power.