Question 1040986: Over a period of 4 consecutive years an employee has received 7.2, 8.6, 6.9, and 9.8% annual pay increases. The ratios, therefore, of each new salary to the previous year's salary are 1.072, 1.086, 1.069, 1.098. Using logarithms, find the geometric mean for these four ratios and then determine the average percent increase for this employee over the 4-year period.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! here's a reference on geometric mean.
https://www.mathsisfun.com/numbers/geometric-mean.html
there are others.
just do a search on:
geometric mean
your problem states:
Over a period of 4 consecutive years an employee has received 7.2, 8.6, 6.9, and 9.8% annual pay increases.
The ratios, therefore, of each new salary to the previous year's salary are 1.072, 1.086, 1.069, 1.098.
Using logarithms, find the geometric mean for these four ratios and then determine the average percent increase for this employee over the 4-year period.
let x equal his starting salary.
in the second year, he is making 1.072 * x.
in the third year, he is making 1.086 * 1.072 * x.
in the fourth year, he is making 1.069 * 1.086 * 1.072 * x.
in the fifth year, he is making 1.098 * 1.069 * 1.086 * 1.072 * x.
simplify that final expression to get:
in the fifth year, he is making 1.36648433 * x.
since there were 4 salary increases, take the fourth root of that to get:
(1.36648433)^(1/4) = 1.081187868.
this is the common ratio for the geometric progression.
your first term is x.
your common ratio is 1.081187868.
your fifth term is equal to x * (1.081187868)^4 = 1.36648433 * x.
subtract 1 from the common ratio of 1.081187868 to get .081187868.
multiply this by 100 to get 8.11878668%.
that is the geometric mean of annual increases of 7.2% followed by 8.6% followed by 6.9% followed by 9.8%.
you did not need logarithms to solve this.
however, you could also have solved it using logarithms as follows:
let r equal the common ratio.
you get r^4 * x = 1.36648433 * x.
divide both sides of this equation by x to get:
r^4 = 1.36648433
take the log of both sides of the equation to get:
log(r^4) = log(1.36648433)
since log(r^4) is the same as 4*log(r), the equation becomes:
4*log(r) = log(1.36648433)
divide both sides of this equation by 4 to get:
log(r) = log(1.36648433)/4
simplify to get log(r) = .033901164.
this is true if and only if 10^.033901164 = r
solve for r to get r = 1.081187868
this is the same answer you got before, only this time you used logs to get it.
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