The Half Life of anything is the amount of time that the substance's total amount is halved. A common real world example of half life is the decay of radioactive substances.

Formula for Half Life

y = 100(½)^{x}

x represents the interval between half lives

y = the current amount

Example

If a radioactive substance has a half life of 1 day, and there is 100 grams of this substance

to find the amount in 1 day

current amount= (starting amount)(½)^{(days passed)}

current amount= 100(½)¹ = 50 grams

to find the amount in 2 days

current amount = (100)(½)^{2} = 25 grams

to find the amount in 5 days

current amount = (100)(½)^{5} = 3.125 grams

Example Two (Interval of 10 minutes)

If the the half life of substance X is 10 minutes and substance X starts out at a total of 100 grams then

at t =0, the current amount =100 grams

t = 10 mins, current amount =50 grams

t = 20 mins, there is 25 grams

t = 30 mins, 12.5 grams

and so on and so forth.

It should be noted that the total substance will never actually become zero since half of anything, no matter how small and insignificant a number, is still not ZERO. Therefore, the half life of radioactive and other substances approaches zero, getting closer and closer to zero but never actually reaching it. See the
interactive demo just below for a real time example of how half of a number never actually reaches zero.

Problem 1
A certain substance has a half life of 1 hour, if you start with 1000 grams of the substance, how much will you have in 1 hour?

in this case the interval between half lives is 1 hour so

current amount = 1,000(½)^{1} = 500

How much will there be in 3 hours? and in 9 hours?

Answer

in 3 hours

current amount = 1,000(½)^{3} = 125

In 9 hours

current amount = 1,000(½)^{9} = 1.95

Radioactive Half Life

Problem 2)Technetium-99m is a radioactive substance used to diagnose brain, thyroid liver and kidney diseases. This radioactive substance has a half life of 6 hours. If there are 200 mgs of this technetium-99m, how much will there be in
6 hours?

in this case the interval between half lives is 6 hours so

current amount = 200(½)^{1} = 100

How much will there be in

12 hours?

1 day?

Answer

The challenging part to this problem is figuring out how many half lives occur. If the half life is 6 hour hours, then in 12 hours there 2 half lives occurred (12/6 = 2) and in 1 day which is 24 hours 4 half lives happened (24/6 = 4) .