Solve for salary given averages of other salaries

[Mindscrape]

My wife is working on problems to study for the GMAT, and asks her fellow math nerd (me) to help on some of them. Originally I had an error and wanted to see if any of you could help me find it, but as I was typing I found it myself! Can I still put this up in case someone stumbles on it and it helps them out? The problem is:

The average weekly salary of 12 workers and 3 managers in a factory was $600. A manager whose salary was $720 was replaced with a new manager, then the average salary of the team fell to $580. What is the salary of the new manager?

So basically we start with (from the first sentence)
$$\frac{tw+tm}{15}=600$$
where tw represents total worker salary and tm represents total manager salary. Now the second sentence says
$$tm=m1+m2+720$$
so the first equation is now (having multiplied out the 15 from before) -- also label this eqn1
$$tw+m1+m2+720=9000$$
continuing with info from the second sentence, we get the second equation for the newly decreased average as
$$\frac{tw+m1+m2+m3}{15}=580$$
now simplifying gives -- and labeling this eqn2
$$tw+m1+m2+m3=8700$$
subtract the two equations (eqn1-eqn2)
$$720-m3=300$$
for the grand finale...
$$m3=420$$

 

[RUber]

Nicely done. Since standardized tests often encourage shortcuts, I'll add a supplemental method which cuts through many of the steps.
In general, if you wanted to affect a change of -$20 in the average of 15 salaries, you have to have a total change of
15(-$20)=-$300 to the sum.
This could be done by reducing one salary by $300, or reducing all salaries by $20, or anywhere in between.
Since the only thing you are changing is the salary starting at $720, you can apply the -$300 to that.