Each employee on a certain task force is either a manager or a director. What percent of the employees on the task force are directors.

A) The average salary of the managers on the task force is 5,000 less than the average salary of all employees on the task force.

B) The average salary of the directors on the task force is 15,000 greater than the average salary of all employees on the task force.

Answer is C.

Is there a quick to determine the sufficiency without getting long algebraic equations?

Complicated. If you understand really well how weighted averages work, you can do something here without using algebra. Otherwise, this is probably a good one on which to make an educated guess and move on.

I can't just average the two averages, right? The point of a weighted average is to know how much weight to give these two individual groups, the managers and the directors.

Statement 1 tells me how the managers' salaries relate to the all employee average but, since I know how weighted averages work (that is, I know that I need to know something about both groups), this isn't enough info.

Statement 2:
Again, I need to know something about both groups; not enough info.

Statements 1 and 2 together:
Now I really have to understand how weighted averages work. The manager average is 5000 less than the combined average. The director average is 15000 greater than the combined average. The difference between the manager average and the director average is 20000. If there were an equal number of managers and directors, they would each be 10000 off of the combined average - that would be a 50/50 weighting. But the combined average is closer to the manager average, so there are more managers than directors. And I can actually use the above three numbers to know how much: the difference is 20000, and the combined average is three-quarters of the way towards the manager average. Sketch it out:
manager avg ----- combined average ----- ----- ----- director average (each dash represents 1000)

So 3/4 of the employees ar managers and 1/4 are directors. Sufficient.

Here are the steps I used to confirm C makes sense:
1) The difference between managers's average and group average is 5,000
2) The corresponding figure for directors is 15,000.
3) The ratio between the differences is 1:3. Since the average is a weighted average, the corresponding ratio for the number of managers to number of directors will be 3:1.
4) Do a quick check to make sure this is right. Assume the average is 10, and use 5 for manager's average and 25 for directors. Assume 3 managers and 1 directors. Hence, 3x5 + 1x25 = 15+25 = 40. Divide by number of employees 40/(3+1) = 10.

Hi instructors,

Is it possible to solve this problem using algebra as well, if so how can one go about doing so?

Thanks

In addition to the question above could define by what you mean by weighted average and when to use it.
Thanks

can you pls confirm borocho's method is correct?

stacey, i'm not sure where the 10,000 comes from.

1) M_sal = All_sal - 5000
2) D_sal = All_sal + 15000

3) We know that All_sal = M_sal*M + D_sal*D / M+D

Substitute 1 and 2 in 3.

yeah, you can do that; the ratio of the 'weights' is the inverse of the ratio of the distances between the weighted average and the individual data points.
(there is absolutely no way to express this truth in a way that's actually easy to read; it's very easy to illustrate in a couple minutes of a live session.)

well, sure, of course it's possible, although the process is somewhat formidable.

here's how you do it:

let M be the number of managers, and D the number of directors; also, let S stand for the average salary.
also, express everything in thousands (so that you don't have to waste time writing ",000" on the end of every single number in the problem)
statement (1) says that the managers' salary is S - 5, and statement (2) says that the directors' salary is S + 15.
so
the sum of everybody's salaries is M(S - 5) + D(S + 15).
but
the sum of everybody's salaries is also given by the average formula: Sum = Average x Number, or (M + D)S.

therefore,
(M + D)S = M(S - 5) + D(S + 15)
MS + DS = MS - 5M + DS + 15D
0 = -5M + 15D
5M = 15D
M = 3D
the ratio of M to D is 3:1
sufficient