Need help on how to work this problem

Question

Suppose a certain radioactive substance has a half-life of 2 years. Find how long it will take for 400 grams of the substance to decay to 25 grams.

Answer 1

400

200 after 2 years

100 after 4 years

50 after 6 years

25 after 8 years

Answer 2

This one is a bit tricky since your unknown is an exponent.

 

You start with 400 and have to multiply it by 1/2 repeatedly (the halflife) until your result gives you 25.

 

Any number times itself is the same as raising that number to an exponent, so 1/2 x 1/2 is the same as 1/2 squared (1/2)².

 

Trying to turn English into an equation, you have 400 halved a certain number of times is equal to 25 which is the same as 400 x (1/2)^H = 25  (with H being your unknown, the number of halflives).  If you are not using logarithms, the easiest way to do that is below:

divide each side of the above equation by 400 to get as close as you can to isolating your unknown:

(400 x (1/2)^H )/ 400 = 25 /400

(1/2)^H = 25/400 

(1/2)^H = 0.0625

in order to find H you multiple 1/2 times itself over and over until the result is 0.0625.

It takes 1/2 x 1/2 x 1/2 x 1/2 (or 4 iterations) to equal 0.0625 so you know that you will need 4 halflives to reduce the original value of 400 down to 25.

Check this against your original information to confirm that it works.

400 x 1/2 x 1/2 x 1/2 x 1/2 = 25

 

since a halflife is 2 years, the total number of years is 8 (4 sets of 2 yrs).

 

Hope this helps :-)