Exponential Decay Problem

Question

The yearly inflation rate tells the percentage by which prices increase. In 1990 an individual retired on a fixed income of per year. Assuming that the inflation rate remains constant at 9% determine how long it will take in years (rounded to the nearest hundredth) for the retirement income to deflate to half its 1990 value. (Note: To say that retirement income has deflated to half its 1990 value means that prices have doubled.)

Answer 1

To do this as a decay problem, would take:

 

Easiest way would be (1)/(1 + 0.09)^t

 

Try t (time) in years, to get close to the denominator of 2.0

 

If do:  (1 + 0.09)⁸, we get 1.99 (about 2)

 

So, doubling, which leaves to ½ of spending power, is done in about 8 years, or 1998.

 

There are other, perhaps more official ways to do this.  BUT, this seems like an easy way to get the answer, in terms of whole years, even for a math novice.

 

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To check our work, we could employ the 'Rule of 72', in which we divide out the interest, to get the number of years; 

 

So, 72 ÷ 9 = 8    [Checks Out]