Each employee of a certain task force is either a manager or a directo

Question 1:

Okay, let us assume the number of managers are M and the number of directors are D. So the total number of employees = M +D

Average Salary of Manager = x
Average Salary of Director = y

So, we have: Total salary of all employees = Mx + Dy
Average salary of employee = Total salary/Total employees = \(\frac{Mx+Dy}{M+D}\)

Statement 1:

Average salary of manager = Average salary of employee - 5000

\(x = \frac{Mx+Dy}{M+D} - 5000\)

Cross multiplying to the other side we get:

\(x(M+D) = Mx + Dy - 5000\)

Cancelling Mx on both sides and rearranging like terms together we get:

\(D(y-x) = 5000\) - (1)

However, this doesn't say anything specific to us. So we move on to the second statement.

Statement 1:

Average salary of director = Average salary of employee + 15000

\(y = \frac{Mx+Dy}{M+D} + 15000\)

Cross multiplying to the other side we get:

\(y(M+D) = Mx + Dy + 15000\)

Cancelling Mx on both sides and rearranging like terms together we get:

\(M(y-x) = 15000\) - (2)

This doesn't say anything by itself either. But when we put the results of the two statements [(1) and (2)] together, we get:

\(D(y-x) = 5000\)
\(M(y-x) = 15000\)

Dividing statement 1 by 2 we get

\(\frac{D}{M} = \frac{5000}{15000} = \frac{1}{3}\)

M = 3D

Total = M+D = 4D

From here, we can say percentage of directors = \(\frac{D}{4D}*100 = 25%\).

Hence answer is C.

(1) : Tells us nothing about how many directors or managers
(2) : Again tells us nothing about how many

(1+2) : Say average salary is x and there be m fraction of managers and hence (1-m) directors

m(x-5000) + (1-m)(x+15000) = x
mx - 5000m + x + 15000 -mx - 15000m = x
15000 - 20000m = 0
m = (3/4)
Hence fraction of directors = (1/4)

Sufficient !
Answer is (c)

Bunuel - is my minimalist solution of just identifying 3 equations satisfactory?

Or do you recommend fully solving it out to see if I net a result?

Generally you can stop solving a DS question at the point you realize a statement is sufficient to get the answer. Note that for this question we don't need to solve for unknowns, we need to get the ratio of directors to total employees. I don't know what 3 equations are you talking about but the statements together are indeed sufficient to get the desired ratio and one can get this even not calculating its exact value.

One has to understand the concept of weighted averages pretty well to understand my solution.

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 how the managers' salaries relate to employee average but there is no information about how the directors' salaries relate to the employee average.
Insufficient
Eliminate A & D

Statement 2 tells how the directors' salaries relate to employee average but there is no information about how the managers' salaries relate to the employee average.
Insufficient
Eliminate B

Statements 1 and 2 together:
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. 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.

Hence, 3/4 of the employees are managers & 1/4 are directors.
Sufficient.

Hence C.

This can be solved in allegations method.
The ratio of the components is inverse of the ratio of their differences from the average.

#Managers:#Directors = Difference between salary between combined avg and directors : Difference between salary between combined avg and managers

M:D = 15000:500 = 3:1

D:M = 1:3

D:(M+D) = 1: (1+3) = 1:4

this way you can solve under 30 secs.

This is a question involving weighted average.
Having two quantities \(Q_1\) and \(Q_2\) with averages \(a_1\) and \(a_2\) respectively, if the combined average is \(a\), and let assume that \(a_1>a>a_2,\) then we can write:

\(\frac{a_1Q_1+a_2Q_2}{Q_1+Q_2}=a\) from which \(a_1Q_1+a_2Q_2=aQ_1+aQ_2\) or \((a_1-a)Q_1=(a-a_2)Q_2,\) which means that the distances from the combined average are inversely proportional to the quantities.
This equality we can also be written as \(\frac{a_1-a}{a-a_2}=\frac{Q_2}{Q_1}.\)

To answer the question it is enough to know the ratio between the two types of employees.
In our case we have a certain number of managers \(Q_1\) and a certain number of directors \(Q_2.\)
From the above, if we know the two differences between the combined average (average salary of all employees) and each type of average, then in fact we have the ratio between \(Q_1\) and \(Q_2.\)
Sufficient

Answer C

Let T be the total average salary.
M be the no of Managers
D be no of Directors.

1st Statement: Aveg salary of Managers is T-5000.
2nd Statement: Avg salary of Directors is T+15000

1st things first. None of the statements provide sufficient information when taken one at a time. So the answer is either C or E
Combine two statements and you get one equation:

(T-5000)*M + (T+15000)*D = T*(M+D)
(Average Salary of M * no of Managers + Average salary of D* no of directors = Total Average Salary T * (M+D)

solving this we get M=3D

So ratio of Directors = D/(D+M) = D/(D + 3D) = 1/4 = 0.25

Hence 25%. So the answer is C

DJ

This is actually a direct application of weighted averages. If you recall the scale method (explained here: https://www.youtube.com/watch?v=_GOAU7moZ2Q ), this is what the number line will look like

AVG-5000 _____________ AVG _________________________AVG+15000

Number of managers/Number of directors = [(AVG+15000) - AVG]/[AVG - (AVG - 5000)] = 3/1

Percentage of directors = 1/(1+3) = 1/4 = 25%

Hi Karishma, I also tried the same method for this but the ratio of 3:1 s the ration of average salary. How can we say that it would be the ratio of Managers and Directors?
Thanks

How would you find the average salary of both - Managers + Directors

Average salary of group = (Avg sal of Managers * No of managers + Avg sal of Directors * No of Directors) / (No of Managers + No of Directors)

So note that the weights in weighted average formula are 'no of managers' and 'no of directors'.

The formula w1/w2 = (A2 - Aavg)/(Aavg - A1) is just an arrangement of the above formula.

w1 and w2 are the weights. A2 is the average salary of Directors, Aavg is the average salary of the group and A1 is the average salary of Managers.

Stmt. 1. The average (Arithmetic Mean) salary of the mangers on the task force is $5,000 less than the average salary of all employees on the task force.
This means Aavg - A1 = 5,000

Stmt 2. The average (Arithmetic Mean) salary of the directors on the task force is $15,000 greater than the average salary of all employees on the task force.
This means A2 - Aavg = 15,000

Q stem: What percent of the employees on the task force are directors?
So we need some values for number of directors/Managers to get to a definite answer.

St(1):The average salary for manager is $5,000 less than the total average salary
This does not tell us of the number of Managers/Directors. (Insufficient) Eliminate A and D.

St(2):The average salary for directors is $15,000 more than the total average salary.
This does not tell us of the number of Managers/Directors. (Insufficient) Eliminate B.

Lets combine the information. Use concept of Weighted Average to use the information.
Let there be m managers and d directors. Let the average salary be y
y-5000 y y+5000

Ratio of m/d = (y+15000)-y / y - (y-5000) =3:1

Directors represent 1 part and hence it is 1/(1+3)*100 =25%.Eliminate E.
(option c)
D.S
GMAT SME

This is a fascinating DS / weighted average question.

Many students who do not test examples or successfully complete the (complicated!) algebra will use theory to arrive at Choice E, because the difference between the average salary of this task force and the average salary of managers and directors does not appear to be enough to deduce the percentage of directors — which is in fact true for both conditions 1 and 2 individually (cross off A, D, and B).

However, when we combine the conditions in considering Choice C vs. Choice E, we realize that because both conditions are tied to the average salary of an employee, we in fact cannot change the ratio of managers to directors without changing the average salary, thus "breaking" the conditions. Hence, Choice C is correct.

Keep in mind that if the question asked for the number of directors, not the percentage of directors, then Choice E would in fact be correct, because all we can deduce is the correct ratio of managers to directors in the task force — not the actual number of directors (or managers, for that matter).

One final note: we cannot always infer a percentage from a ratio. We can only do so in cases where we know that the ratio includes all the elements in the set, which is why the question specifies that the task force consists of only managers and directors.

I guess everyone could be familiar with the question below:
In a grocery store, each apple sold for 10 and each banana sold for 20. Zoe bought some apples and bananas, and the average money she paid for each fruit was 12.
What was the ratio of the apple she bought to the ratio of the banana she bought?

And this director & manager question is just a inverted vertion of apple & banana question.

Bunuel

Is this question similar to a question type like mixing/mixture ? [what is the ratio of 1st and 2nd juice if we mix 30% juice concentration and 40% concentration and the final mixture is 37%]

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Statement 1 gives you the total deficit from average
Statement 2 gives you the total surplus from average

deficit is equal to surplus since the average doesn't change

number of managers * 5000 = number of directors * 15000 --> it is possible to find the ratio between the number of directors and managers --> it is possible to find the ratio between any one of them and total which is basically percentage --> answer is C.

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