Solution To: Better? Or Worse? That Is The Question.

Here is the original problem:

Is a 10% raise followed by a 10% cut in pay better, worse, or the same as a 10% cut followed by a 10% raise?

Does either of them yield the original salary?

Explain your answer. (Or better yet, prove you answer!)

 

Solution:

 

Both scenarios yeild the same salary in the end, though neighter of them yeild the original salary.

 

Proof:

 

Now you could just test out a couple of different salaries and come to the solution but that by no means proves that that solution would hold up for every given salary.  So here is a simple proof for you:

 

 

Let  x = any given salary such that x>0 and x is a real number.

 

First scenario:  Given x (our original salary), find x after a 10% raise followed by a 10% cut.

 

x     Given (our original salary)

1.1x  Receiving a 10% raise means we are receiving 110% of our original salary.  110% as a decimal
         is 1.1     Multiply x by 1.1 to get our salary after the raise.

(1.1x)(.9)  Receiving a 10% cut means we are receiving 90% of our salary.  90% as a decimal is .9 
                 Multiply our new salary by .9 to fund our salary after the pay cut.

.99x  Combine like terms and we find that after a 10% raise and then a 10% cut we are receiving 99%
         of our original salary.

 

Second scenario:  Given x, find x after a 10% cut followed by a 10% raise.

 

x

.9x

(.9x)(1.1)

.99x  Again, after a 10% cut followed by a 10% raise, we are receiving 99% of our original salary.

 

Therefore, neither yeild the original salary, they both yeild the same salary, and either way you are getting the shaft.

 

Hopefully none of my math professors read this...that is one sloppy proof.

 

See you next week with another problem.

 

Sincerely,
The Math Freak