I am working through a problem in a study guide that asks:

**"The half-life of a certain isotope is 5.5 years. If there were 20 grams of one such isotope left after 22 years, what was the original weight?"**

To set this up, there are 4 halvings (22 years/5.5 years) = t. The rate, r, is 0.5 (for half-life). The final mass, y, is 20g. Using y = a(1 - r)

^{t}:

20 = a(1 - 0.5)

^{4}; 20 = a(0.5)

^{4}; 20 = a(0.0625)

**a = 320 grams original weight**

Straightforward enough. To understand this process better, I tried using the equation A(t) = A

*e*

^{-kt}

where

A(t) = 0.5A

t = 5.5

solve for k:

0.5A = A

*e*

^{-k(5.5)}

0.5 =

*e*

^{-k(5.5)}

ln(0.5) = -k(5.5)

-0.693 = -k(5.5)

k = 0.126

Returning to the earlier equation and using t = 22 with this calculated per-year rate, I tried:

20 = a(1 - 0.126)

^{22}; 20 = a(0.874)

^{22}; 20 = a(0.0517) ;

**a = 386.85.**Similarly,

20 = 320(1 - 0.126)

^{t}; 0.0625 = (1- 0.126)

^{t}; 0.0625 = (0.874)

^{t}; log

_{0.874}(0.0625) = 20.58 years.

Why don't these two methods match up? What assumption or math error am I making that I shouldn't?

Thanks!