Although I don't think that algebraic way is the best for this problem, here you go:

Let Barney's regular hourly wage be \(x\), then his Saturday wage will be \(2x\) and Alan's hourly wage will be \(1.5x\);

Let the # of hours Barney worked Monday through Friday be \(m\) and on Saturday be \(n\) and the # of hours Alan worked Monday through Friday be \(p\) and on Saturday be \(q\);

Given: \(xm+2xn=1.5x(p+q)\) and \(m+n=p+q\).

\(xm+2xn=1.5x(p+q)\) --> \(m+2n=1.5(m+n)\) --> \(m=n\) --> Barney worked the equal # of hours Monday-Friday and on Saturday.

The above directly tells us that II must be true (as Barney worked total non-zero # of hours and he worked an integer # of hours on any given day then he must have been worked at least one hour on Saturday.)

As for I: Alan may have worked ALL his hours Monday through Friday so in this case this statement is not true (p=total>m). Alan also may have worked all his hours on Saturday. Or algebraically: there are any distribution possible between p and q, p=0 and q=total or p=total and q=0 or any other;

The above means that III is also not always true: if Alan worked all his hours on Saturday then he made all his money on Saturday thus he made more money on Saturday than Barney did.

Answer: B (II only).

But the above can also be done with much less algebra:

As Alan and Barney worked the same # of hours and earned the same amount of money, then their hourly average wages must have been the same: (average wage)=(total amount earned)/(# of hours worked). Now, Alan has constant hourly wage which is \(1.5*x\) and Barney's average (\(\frac{xm+2xn}{m+n}\)) to be equal to this he must have been worked the equal # of hours Monday-Friday and on Saturday, so \(m=n\).

Hope it's clear.
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Bunuel , u humble anyone's approach to logical and or mathematical problems. respect

Hi Bunuel,

I believe the best approach is a mixture of conceptual and number picking, but I'm having a hard time getting to make work fast in under 2 minutes. Would you please show us how you deal with this problem in such way?

Much appreciated!

Cheers
J

Statement I. Alan worked fewer hours Monday through Friday than did Barney, well let's see. Maybe Alan did work fewer hours and then worked those missing hours on Saturday, or maybe not, we really can't tell the split of Alan's hours.

Statement II, what we do know is that Barney must have worked at least 1 hour on Saturday, how else then could he compensate for a lower salary if they work the same number of total hours?

Statement III what if Barney worked fewer hours during the week and then worked on Saturdays, then Alan could compensate by working more hours during the week, and maybe just working a smaller number of hours on Saturday (>=1). In that case we leave the possibility open of Barney earning a salary that is lower than Barney's but still making more money than Barney on Saturday. This can be possible because of the higher amount of hours worked by Alan.

Hence B is the correct choice

Is this clear enough?
Cheers
J

Hi Bunuel,

I'm having a hard time figuring out why statement III is wrong when NOT done algebraically.

I approached this problem conceptually: II is correct is rather easy to see. When it comes to III, the statement says that Barney made more money on Saturday than Alan. Correct?

Doesn't that HAVE to be true? What I mean by that is -- if Barney worked AT LEAST 1 hour on saturday, his salary is 2x vs. Alan's which is 1.5x, so doesn't that inherently make III true?

I would go even further and say that Barney would need to make a ton more money on Saturday to compensate for his lack of pay during the week.

What am I missing here?

We got that Barney worked the same number of hours from Monday to Friday and on Saturday. Thus his wage is split into two parts 1 part is for the work done from Monday to Friday and 1.5 parts for the work done on Saturday.

Now, if Alan worked all his hours on Saturday then he made all his money on Saturday thus he made more money on Saturday than Barney did.

Does this make sense?
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Absolutely. Thanks.

Let total no.of hours worked by each of them be = x

For Mon-Fri :
Hourly wage of B(Barney) = p
Thus Hourly wage of A(Alan) = 1.5p
Total hours worked by A = x-y
Total hours worked by B = x-z

On sat :
Hourly wage of B = 2p while that of A remains same ie = p
Hours worked by A = y
Hours worked by B =z

Now given:
(x-z)p + (z)2p = (x)(1.5p)
On solving we get z=x/2

Since hours worked on saturday(x/2) is non zero and an integer; x/2=2,4,6 & so on ............... (i)

Statement 1:
it says x-y < (x/2)
ie x< 2y
we have not obtained any such relation among x & y above. Thus cannot be true always

Statement 2:
(x/2)>=1
From (i) we know that this is correct always

Statement 3:
it says (x/2)2p > (y)(1.5p)
ie x>1.5y
we have not obtained any such relation among x & y above. Thus cannot be true always

Thus only statement 2 is true always. Hence answer is (B)

A small doubt, may be a silly one:
Is it possible that the previous week Allan worked 6 hrs a day Monday through Friday and Barney worked for 9 hrs a day Monday to Friday?
If this scenario is possible then both of them will end up earning the same amount without B having to work on a Saturday.

Hi,

Could someone help to explain this one, what does "Both men work an integer number of hours on any given day" exactly mean?

When I first read it, I just understood "on any given day" as "on everyday", so I interpreted the statement as "Everyday of the week, each of them work an integer number (not necessarily the same day to day) of hours" and then came to choose II immediately. But the explanation from MGMAT said that "As for the other statements (I and III), we cannot tell, because Alan may or may not have worked on Saturday." I think I am misunderstanding something here What is the meaning of on any given day in English?

Thanks for your help
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Hi,
the sentence means that both have worked in integer hours, that is 1 hr or 2 hour or 6 hour etc, the hour will be an integer..
secondly any given day means - any day when they work which means it is possible that they didn't work at all OR if they worked they worked for integer hours..

Hope it helps
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It really helps. Thanks chetan2u
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Took me a hell lot of time to get through this one. I need more questions like this, please someone tell me whats the source.

I solved the problem by picking numbers.I found algebric aproach a little time consuming.

Suppose Alalns hourly wage is A and that of Brian is B. AND THEY BOTH HAVE WORKED FOR n NUMBER OF HOURS ON THE LAST WEEK.

So n*A=n*B where A ≠B

Now lets attack the options.
1.According to this condition: Alan worked fewer hours Monday through Friday than did Barney.

Suppose A=15 and B=10 and n=6

then for A its 15*6=90
Now we have to make 6*10(Brian's wage)=90

That is possible when 3*2B+3*B;Then we can conclude that statement 1 doesn't work

I tried this by picking other numbers and got the same result

So we can eliminate 1.

2.Its definitely true and we don't need any calculation for that.

3.Its not always true because we don't know the number of hours Alan work on Staurday.So we can eliminate this also.

HENCE THE ANSWER IS B

who says alan even paid on weekend? :D I that was confusing for me. I also couldn't catch average wage approach makes the question very easy actually. thank you for the exlanation.

Official Solution:

Because Alan and Barney worked the same total number of hours last week and earned the same total in wages, they must have had the same average hourly wage. Alan’s hourly wage is constant, equal to 1.5 times Barney’s regular wage. Therefore, last week, Barney’s average hourly wage must also have been equal to 1.5 times his regular hourly wage. This is only possible if half of Barney’s working hours were at his regular wage, and the other half of his working hours were at twice his regular wage, i.e., on Saturday. Therefore II is definitely true.

As for the other statements, we cannot tell, because Alan may or may not have worked on Saturday. For example, suppose Barney worked one hour on Monday and one hour on Saturday, for 2 hours total. If Alan worked both of his total of 2 hours on Tuesday, then I is false. If, on the other hand, Alan worked both of his hours on Saturday, then III is false.

The correct answer is B.

Hi All,

This question is more of a logic question than anything else (although you can TEST VALUES to prove what MUST be true)

We're given the hourly pay rates for Alan and Barney:

Alan = 1.5X
Barney = X
Barney (on Saturday) = 2X

We're told that each person work an integer number of hours on any given day and that each work the SAME NUMBER OF HOURS and earned the SAME TOTAL PAY. We're asked which of the 3 Roman Numeral MUST be true (which really means "which of the following is ALWAYS TRUE no matter how many examples you come up with?")

Before we look at the Roman Numerals though, we should take a moment to review the situation. What would have to happen for the two people to work the SAME NUMBER OF HOURS and earn the SAME TOTAL PAY? Alan makes more money per hour than Barney EXCEPT on Saturdays, so Barney MUST have worked some Saturday hours (otherwise the total pay for each would have been different. Keep THAT in mind when working through the 3 Roman Numerals.

I. Alan worked fewer hours Monday-Friday than did Barney.

We know that Barney worked some hours on Saturday, but Alan COULD have worked on ANY day. Thus, this statement isn't necessarily true.

We can prove it by TESTing VALUES

Alan: works 2 hours on Friday = 2(1.5X) = 3X in pay
Barney: works 1 hour on Friday and 1 hour on Saturday = 1(X) + 2(X) = 3X in pay
Same total hours, same total pay
Alan did NOT worker fewer hours Monday-Friday than Barney
#1 is NOT necessarily true.

II. Barney worked at least one hour on Saturday.

We determined this already; this MUST be true.

III. Barney made more money on Saturday than did Alan.

If we use the example from Roman Numeral 1 and shift Alan's work to Saturday, then we can prove that this statement is NOT necessarily true.

Alan: works 2 hours on Saturday = 2(1.5X) = 3X in pay
Barney: works 1 hour on Friday and 1 hour on Saturday = 1(X) + 2(X) = 3X in pay
Alan on Saturday = 3X
Barney on Saturday = 2X
#3 is NOT necessarily true.

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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-- EASY SOLUTION --

I. Not necessarily — They could work equal hours from M. to F. with W{Alan}=7.5x and W{Barney}=5x. To have equal wage and hrs it must be true for Saturday that 2.5x + 1.5xt = 2xt or t=5!

II. Must be true — Otherwise it isn’t possible for them to have equal wage and equal hrs, since from M. to F. we have Wage{Alan} > Wage{Barney}!

III. Not necessarily — Alan could make more on Saturday if Alan didn’t work all the days from M. to F.

Bunuel check

I - Can not be true
II - IS true all the time
III - May or May not be true.