Last year, the five employees of company X took an average

1) Three employees had a 50% increase in thier number of vacation days and two employees has a 50% decrease
2) Three employees had 10 more vacation days each , and two employees has 5 fewer vacation days each.


given 5 employees 16 days average so 80 days leave last year

stmt 1 doesn't tell for which employess 50% increase and for which employess 50% decrease in vacation days

so not sufficient

stmt 2
if 3 employees had 10 more vacation days each then total vacation days is increased 80+3*10=>110

and if 2 employee vacation days decreased by 5 days each then total vacation days will come down to 110-2*5 =>100 days

so average vac. days for emp. this year will be 100/5 => 20 days

so ans is B

I evaluated statement 1 intuitively:
the 50% increase could be referring to those with the most vacation days, so the average would increase
likewise, it could refer to those with the least vacation days, in which case the average would decrease
no way to tell, so 1 = insufficient

statement 2:
5x16 = 80 (total)
take away 3x10 and add 2x5: 100
100/5 = 20 sufficient

Clearly B...

We have Sum of all vacations taken by 5 employess = 16x5=80
S1. gives % increases of 3 employees and % decrease of 2 employees... This isn't sufficient. INSUFF...
S2: This can gives us the average as we need to add 10 and subtract 5 from 80 and then divide by 5... SUFF

Hi . .Can you pls explain the variables x and y and how you arrived at the equations ?

(1) Three employees had a 50% increase in their number of vacation days and two employees has a 50% decrease.

Say x is the total # vacations days taken last year by the three employees mentioned and y is the total # of days taken last year by two employees mentioned.

Now, since we are told that "last year the five employees of company X took an average of 16 vacation days each", then (total # of vacation days)/(# of employees)=(x+y)/5=16.

Next, the first statement says that "the three employees had a 50% increase in their number of vacation days", so those three had 1.5x vacation days this year, and the other two had 0.5y vacation days this year. We need the new average for this year, so the value of (1.5x+0.5y)/5.

The same way for the second statement.

Hope it's clear.

A non algebraic approach:

Total number of vacationing days => \(16 * 5 = 80 days\)

A) Insufficient:
Let the employees be segregated as \(a--b--16--x--y\) (where x & y have taken more than 16 days of vacationing last year and a & b have taken less than 16 days of vacationing last year). Nowhere does it mention the three/two are < 16 days guy (ie a/b) or > 16 days guy (ie x/y). It only mentions that 3 employees had a 50% incre. and 2 employees 50% decrease.

B) Sufficient:
Three guys have taken more than 10 days of last years' average, each (each is important here!). Therefore, these three have taken \(16*3 (avg.) + 10*3 (this years' incre.) =\) \(78 days\) of vacationing. Similarly, the remaining two have taken \(22 days\) of vacationing \([ie 2*(16-5)=11].\)

Avg. days of vacationing this year = \(\frac{78+22}{5}\)=\(20 days.\)

Ans: B

Regards,
Pratik

Hello Sir,

Option B never mentions that the information is for this year, this might as well be of the last year. In the case the we dont have any information for the this year and so B is not the answer. thoughts?

Hi Bunuel, this very reasoning made me select option E. The GMAT uses such vague language sometimes it becomes really hard to try to understand what the question is truly asking. So, sometimes you get the right answer and sometimes you don't. What to do in such situations?

Since we are talking about averages; it is quite safe to assume :-
There can be a great range in the elements of the set
or
There can be no difference in the elements of set.

Statement 1)Three employees had a 50% increase in thier number of vacation days and two employees has a 50% decrease
Insufficient :- We don't know which employees increased or decreased their vacations percent.
Case 1) The employees that took more holidays last year increased their holidays by 50%
Case 2) The employee who took less holidays last year increased their holidays by 50 %

Cannot reach a definite value

(2) Three employees had 10 more vacation days each , and two employees has 5 fewer vacation days each.
Sufficient
Last year (a)+(b)+(c)+(d)+(e)=80

this year (a+10)+(b+10)+(c+10)+(d-5)+(e-5)?

==> a+b+c+d+e+30-10
==> a+b+c+d+e+20
==>80+20
==> 100
average number of vacation per employee=100/5= 20 holidays per employee

B is the answer

simple B

no calculation also required.

1) its clearly infsuff we don't know each employee vacation days .

2)avg=16 for each employee

10+10+10-5-5=20

20/5=4 increase avg by 4 from 16.Easy

Statement 1

In this statement we can use a smaller example- not a perfect example though possibly a practical one for understanding the problem/statement

A set of 5 employees has an average of 6 days

5 employees [4,6,4,6,10]

In this circumstance it actually matters which group of 3 employees has a 50% increase and 50% decrease- this makes the average number of days variable

[6,9,6] + [3,5] = 28/5 average days

whereas

[15, 12,8]+[2,3]= 40/5 average days

Insufficient

Statement 2

We can actually solve for the number of additional days by using the formula

3 employees (10 days)-2 employees (5 days) = 20 additional days

80 days last year + 20 additional = 100 days taken this year

100/5 = 20 average days

Sufficient

Total number of vacation days last year = (number of employees)(average number of days per employee) = 5*16 = 80.
To determine the average number of vacations days for the 5 employees this year, we need to know the TOTAL NUMBER of vacation days this year.
Question stem, rephrased:
What was the total number of vacation days this year?

Statement 1: Three employees had a 50% increase in their number of vacation days, and two employees had a 50% decrease
Case 1: The 3 red employees took a total of 60 days last year, and the 2 blue employees took a total of 20 days last year, for a total of 80 vacation days last year
Total this year for the 3 red employees = 60 + (50% of 60) = 60 + 30 = 90.
Total this year for the 2 blue employees = 20 - (50% of 20) = 20 - 10 = 10.
Total this year for all 5 employees = 90 + 10 = 100.

Case 2: The 3 red employees took a total of 20 days last year, and the 2 blue employees took a total of 60 days last year, for a total of 80 vacation days last year
Total this year for the 3 red employees = 20 + (50% of 20) = 20 + 10 = 30.
Total this year for the 2 blue employees = 60 - (50% of 60) = 60 - 30 = 30.
Total this year for all 5 employees = 30 + 30 = 60.

Since the total this year can be different values, INSUFFICIENT.

Statement 2: Three employees had 10 more vacation days each, and two employees had 5 fewer vacation days each.
Increase this year for the 3 red employees = 3*10 = 30.
Decrease this year for the 2 blue employees = 2*5 = 10.
Since last year's total = 80 days, we get:
Total this year for all 5 employees = 80 + 30 - 10 = 100.
SUFFICIENT.

"Last year, the five employees of company X took an average of 16 vacation days each"

How do we know that this doesn't imply that each employee had an average of 16 days ie 16 + 16 + 16 + 16 + 16 /5 = 16?
From the solutions presented, the interpretation is that the average of the vacations days is 16 ie a + b + c +d + e/ 5 = 16

This method described above is a very clear explanation. Most people would think of this as a weighted averages problem because it seems so. However, it is a basic question of average confusing you with its language. All the other methods will complicate it for you. Use this one prescribed above for best clarity on the problem.

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