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.
In beta. This lesson was written for Hone and has not yet been checked by a physician. Practice material, not professional advice. What that means.
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.