This is a weighted average question.
We have two types of employees, M and D; we want to know what % of the employees are Ds.
Statements (1) and (2) aren't sufficient separately, since each one doesn't give enough information about the big picture.
However, when we combine the statements, we can use the weighted average formula to figure out the weights of each group.
Overall Average = (weight group 1)(avg group 1) + (weight group 2)(avg group 2) + ...
If we let the overall average of this group = x, we know that:
x = M%(x-5000) + (100 - M)%(x+15000)
x = M%x - M%(5000) + 100%x - M%x + 100(15000) - M%(15000)
x = 100% - 50M -150M + 1500000
200M = 1500000
M = 7500
M% = 75
Now, that might seem like a lot of complicated math. The good news is that, since this is a DS question, we didn't actually have to do it - as soon as we decide that the equation is solvable for M, we're done.
If we understood the concepts, we could have just avoided the math entirely (which is why it's good to understand concepts!). Let's picture a number line:
(x-5000) ............... x ........................................... (x+15000)
As we can see, x is considerable closer to the average of M than to the average of D. The overall average is drawn closer to the group with the greatest weight. Since we know exactly where x is in relation to both groups, it MUST be possible to determine the weight of each group.
In this case, x is 25% of the way from group M to group D, so group M must have 75% weight and group D must have 25% weight.
GMAT Prep #8 please help
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Stuart@KaplanGMAT
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Stuart, I found this problem quite tough as well so trying to understand this concept. One doubt I have in the formula as well as the diagram+explanation above is : In both these, is there any assumption that salary of directors and manager is the same. The 1:3 split of the number line above could also be because of managers (x-5000) group having a greater salary as averages are affected both by salary and number of memebers in that group. Apologize if this is some weird thought...Stuart Kovinsky wrote: If we understood the concepts, we could have just avoided the math entirely (which is why it's good to understand concepts!). Let's picture a number line:
(x-5000) ............... x ........................................... (x+15000)
As we can see, x is considerable closer to the average of M than to the average of D. The overall average is drawn closer to the group with the greatest weight. Since we know exactly where x is in relation to both groups, it MUST be possible to determine the weight of each group.
In this case, x is 25% of the way from group M to group D, so group M must have 75% weight and group D must have 25% weight.
Thx,
Ash
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codesnooker
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The answer in fact is 'C' but there could be a different approach to solve this problem.
Solution:
Lets assume,
Total Number of employees on the task force = N
Total Number of Manager among these employees = M
Total Number of Directors among these employees = D = (N-M) ----> Equation #1
Salary of Employee #1 = S1
Salary of Employee #2 = S2
--------------------------------------------------
--------------------------------------------------
Salary of Employee #m = Sm
--------------------------------------------------
--------------------------------------------------
Salary of Employee #n = Sn
Also lets assume that first M employees are manager and rest are Directors.
Now Average Salary of All Employees will be = A
A = (S1 + S2 + ---- + Sn) / N -----> Equation 2
According to condition #1
Av. salary of Managers = Av. Salary of All employees - 5000
i.e. (S1 + S2 + ---- + Sm) / M = A - 5000
Move denominator (M) of LHS to the RHS (right hand side) of the equation.
i.e. (S1 + S2 + ---- + Sm) = (A - 5000).(M) --------> Equation #3
According to condition #2
Av. salary of Directors = Av. Salary of All employees + 15000
i.e.
(Sm+1 + Sm+2 + ---- + Sn) / (N - M)= A + 15000
Similarly, Move (N - M) on the RHS (right hand side) of the equation.
i.e. (Sm+1 + Sm+2 + ---- + Sn) = (A + 15000).(N - M) --------> Equation #4
(why here Sm+1 ? --> Because we have consider first 'm' employees are managers and after that counters are for directors i.e. rest are directors)
Now add equation #3 and equation #4
(S1 + S2 + ----- + Sm) + (Sm+1 + Sm+2 + ----- + Sn) = (A + 15000).(N - M) + (A - 5000).(M) -----------> equation #5
Now multiply and divide LHS by N
==> LHS = N.A ---> I guess its easy to understood by it is reduced to A.
Therefore equation #5 will reduce to
N.A = N.A + 15000N - MA - 15000M + MA - 5000M
MA and NA will cancel out in the above equation and we left with
0 = 15000N - 20000M
=> 4M = 3N
=> M = (3/4)N = 75%
from equation #1,
=> D = (N-M) = (1/4)N = 25%
Solution:
Lets assume,
Total Number of employees on the task force = N
Total Number of Manager among these employees = M
Total Number of Directors among these employees = D = (N-M) ----> Equation #1
Salary of Employee #1 = S1
Salary of Employee #2 = S2
--------------------------------------------------
--------------------------------------------------
Salary of Employee #m = Sm
--------------------------------------------------
--------------------------------------------------
Salary of Employee #n = Sn
Also lets assume that first M employees are manager and rest are Directors.
Now Average Salary of All Employees will be = A
A = (S1 + S2 + ---- + Sn) / N -----> Equation 2
According to condition #1
Av. salary of Managers = Av. Salary of All employees - 5000
i.e. (S1 + S2 + ---- + Sm) / M = A - 5000
Move denominator (M) of LHS to the RHS (right hand side) of the equation.
i.e. (S1 + S2 + ---- + Sm) = (A - 5000).(M) --------> Equation #3
According to condition #2
Av. salary of Directors = Av. Salary of All employees + 15000
i.e.
(Sm+1 + Sm+2 + ---- + Sn) / (N - M)= A + 15000
Similarly, Move (N - M) on the RHS (right hand side) of the equation.
i.e. (Sm+1 + Sm+2 + ---- + Sn) = (A + 15000).(N - M) --------> Equation #4
(why here Sm+1 ? --> Because we have consider first 'm' employees are managers and after that counters are for directors i.e. rest are directors)
Now add equation #3 and equation #4
(S1 + S2 + ----- + Sm) + (Sm+1 + Sm+2 + ----- + Sn) = (A + 15000).(N - M) + (A - 5000).(M) -----------> equation #5
Now multiply and divide LHS by N
==> LHS = N.A ---> I guess its easy to understood by it is reduced to A.
Therefore equation #5 will reduce to
N.A = N.A + 15000N - MA - 15000M + MA - 5000M
MA and NA will cancel out in the above equation and we left with
0 = 15000N - 20000M
=> 4M = 3N
=> M = (3/4)N = 75%
from equation #1,
=> D = (N-M) = (1/4)N = 25%
So, there is no need to assume any one's salary.ash g wrote:In both these, is there any assumption that salary of directors and manager is the same.
Last edited by codesnooker on Tue Mar 11, 2008 3:41 am, edited 1 time in total.
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ikant
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Well.. i think that one is doing a DS problem one need not try and calculate the final answer.
The concept mentioned is the key.
Avg. salary of emp = T
Avg. salary of managers = M
Avg. salary of directors = D
Relations:: M + 5000 = T
D = T + 15000
Hence the difference in manager and director is 20000.
Average salary lies at 5000 from manager and 15000 from director. Hence, the ratio of them will be 1: 3.
Solved.
Now considering the assumption thing please make a note that we are talking average salaries and not individual salaries. Hence, there is no assumption made.
The concept mentioned is the key.
Avg. salary of emp = T
Avg. salary of managers = M
Avg. salary of directors = D
Relations:: M + 5000 = T
D = T + 15000
Hence the difference in manager and director is 20000.
Average salary lies at 5000 from manager and 15000 from director. Hence, the ratio of them will be 1: 3.
Solved.
Now considering the assumption thing please make a note that we are talking average salaries and not individual salaries. Hence, there is no assumption made.
"To do is to be"
I agree with the long solution. I am just trying to understand the shortcut mentioned by Stuart. 1:3 split represents average salaries. Question asks about percentage of number of directors. Can weighted average formula/diagram concept mentioned by Stuart be applied.ash g wrote:Stuart, I found this problem quite tough as well so trying to understand this concept. One doubt I have in the formula as well as the diagram+explanation above is : In both these, is there any assumption that salary of directors and manager is the same. The 1:3 split of the number line above could also be because of managers (x-5000) group having a greater salary as averages are affected both by salary and number of memebers in that group. Apologize if this is some weird thought...Stuart Kovinsky wrote: If we understood the concepts, we could have just avoided the math entirely (which is why it's good to understand concepts!). Let's picture a number line:
(x-5000) ............... x ........................................... (x+15000)
As we can see, x is considerable closer to the average of M than to the average of D. The overall average is drawn closer to the group with the greatest weight. Since we know exactly where x is in relation to both groups, it MUST be possible to determine the weight of each group.
In this case, x is 25% of the way from group M to group D, so group M must have 75% weight and group D must have 25% weight.
Thx,
Ash
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Stuart@KaplanGMAT
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The 1:3 ratio is the weight of each group and relates to the number of members of the groups, not the amount of the salaries.ash g wrote:I agree with the long solution. I am just trying to understand the shortcut mentioned by Stuart. 1:3 split represents average salaries. Question asks about percentage of number of directors. Can weighted average formula/diagram concept mentioned by Stuart be applied.
Looking back at the weighted average formula:
Overall Average = (weight group 1)(avg group 1) + (weight group 2)(avg group 2) + ...
We solved for the weight of M, not the average salary of M. Similarly, using the number line method, we're seeing how much impact the M salaries have on the average vs how much impact the D salaries have. The impact (weight) is a function of the ratio of members in each group.
If the overall average were right in the middle, then each group would have 50% weight and the ratio of M:D would be 1:1. If the overall average slides toward one side, then that side must have greater pull, i.e. more weight.
For problem solving, we need to know exactly how to calculate the weights. For data sufficiency, we merely need to recognize that if we know exactly where the overall average is in relation to the group averages, it must be possible to figure out exactly what weight each group has.
Last edited by Stuart@KaplanGMAT on Tue Nov 25, 2008 1:10 pm, edited 1 time in total.
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musicdaemon
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ikant wrote:Well.. i think that one is doing a DS problem one need not try and calculate the final answer.
The concept mentioned is the key.
Avg. salary of emp = T
Avg. salary of managers = M
Avg. salary of directors = D
Relations:: M + 5000 = T
D = T + 15000
Hence the difference in manager and director is 20000.
Average salary lies at 5000 from manager and 15000 from director. Hence, the ratio of them will be 1: 3.
Solved.
Now considering the assumption thing please make a note that we are talking average salaries and not individual salaries. Hence, there is no assumption made.
I think that this is the best approach..... short & tidy.... and Simple
Let the Game begin
