Alan’s regular hourly wage is 1.5 times Barney’s regular

Question

Alan’s regular hourly wage is 1.5 times Barney’s regular hourly wage, but Barney gets paid at twice his regular wage for any hours he works on Saturday. Both men work an integer number of hours on any given day. If Alan and Barney each worked for the same total non-zero number of hours last week, and earned the same total in wages, which of the following must be true?

I. Alan worked fewer hours Monday through Friday than did Barney. 
II. Barney worked at least one hour on Saturday. 
III. Barney made more money on Saturday than did Alan.

A.I only
B. II only
C. I and II only
D. I and III only
E. II and III only

I would just like to see the equation written out, so I can visualize the problem. I am having trouble conceptualizing the problem. I understand the a=1.5b but I would like to see how the second part is written when they have equal hours and equal pay. Thanks,

Bunuel · Accepted Answer

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.

yezz · Answer

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

russ9 · Answer

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?

Bunuel · Answer

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?

shankyGMAT · Answer

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)

thuyduong91vnu · Answer

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

What is the meaning of on any given day in English? Thanks for your help

chetan2u · Answer

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

sahilvijay · Answer

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.

EMPOWERgmatRichC · Answer

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:  B

GMAT assassins aren't born, they're made, 
Rich

ajarpit60 · Answer

Alan’s regular hourly wage is 1.5 times Barney’s regular hourly wage, but Barney gets paid at twice his regular wage for any hours he works on Saturday. Both men work an integer number of hours on any given day. If Alan and Barney each worked for the same total non-zero number of hours last week, and earned the same total in wages, which of the following must be true?

I. Alan worked fewer hours Monday through Friday than did Barney. 
II. Barney worked at least one hour on Saturday. 
III. Barney made more money on Saturday than did Alan.

A.I only
B. II only
C. I and II only
D. I and III only
E. II and III only

In such complex problems, first think whether you can convert it in simple mathematical equation and assign simple values to variables. 
 Step 1: Assign variables to wages 
Let, Barney’s regular hourly wage=a1=10 (assume), Alan’s regular hourly wage=1.5a1=15
Alan’s Saturday wage=15, Barney’s Saturday wage=2*10=20
 Step 2: Assign variables to no of hours worked 
Alan weekday hours= p, Alan Saturday hours=r
Barney weekday hours= q, Barney Saturday hours=s
 Step3: Make equations 
They worked total equal no of hours:  ….. Equation 1 
Wages equation: Total wage equal :  :  …  equation 2 
 Using equation 1 in equation 2  :   :  
Clearly, II is valid. Can’t say anything about I and II.
 Ans: Option B

GMATGuruNY · Answer

Let Barney's hourly wage Monday through Friday = $2 per hour, implying that Barney's wage on Saturday = 2*2 = $4 per hour and that Alan's hourly wage = (1.5)(2) = $3 per hour.

For Barney to work the same number of hours as Alan and earn the same amount of money, Barney's AVERAGE hourly wage must be equal to Alan's hourly wage ($3 per hour).
Implication:
Since $3 is HALFWAY between Barney's two wages -- $2 and $4 -- Barney must work half of his total hours on WEEKDAYS (earning $2 per hour) and the other half on SATURDAY (earning $4 per hour), with the result that his AVERAGE hourly wage = $3 per hour.

Let the time for each worker = 2 hours, implying that Barney works half the time (1 hour) Monday through Friday and the other half (1 hour) on Saturday.
Alan can work his 2 hours any day of the week.
Total income for each worker = (average hourly wage)(number of hours) = 3*2 = $6.

I. Alan worked fewer hours Monday through Friday than did Barney.  
Whereas Alan can work all of his 2 hours Monday through Friday, Barney works only 1 weekday hour.
Thus, Statement I does not have to be true.
Eliminate A, C and D.

III: Barney made more money on Saturday than did Alan.  
Whereas Barney's 1 hour of work on Saturday earns him only $4, Alan's TOTAL income of $6 can be earned on Saturday.
Thus, Statement III does not have to be true.
Eliminate E.

B

Mathematical proof for Barney: 
Let w = Barney's weekday hours and s = Barney's Saturday hours.
Since Barney has a weekday rate of $2 per hour and a Saturday rate of $4 per hour, we get:
Total wages = 2w+4s

Since Barney must earn the same amount as Alan -- $6 -- we get:
2w+4s = 6
 w+2s = 3

Since Barney must work the same number of hours as Alan, we get:
 w+s = 2

Subtracting the blue equation from the red equation, we get:
(w+2s) - (w+s) = 3-2
s=1, implying that w=1.
Implication:
To earn the same amount as Alan, Barney must work ONE WEEKDAY HOUR AND ONE SATURDAY HOUR.

vaibhav1221 · Answer

I think approaching this question more logically (like a CR question) than mathematically (like a quant question) would be way easier.

We know that A and B earned the same amount and worked the same number of hours and that A earns more than B does but B gets paid more on Saturdays.
For A and B to receive an equal amount by putting an equal number of hours, B must've worked at least a few hours on Saturday.

I. Alan worked fewer hours Monday through Friday than did Barney. 
Alan must've worked more hours from Monday to Friday because for the condition to be true, B must've worked some hours on Saturday. So, this is out. This can never be the case, however, the opposite of this could be true.

II. Barney worked at least one hour on Saturday. 
Definitely true as explained above.

III. Barney made more money on Saturday than did Alan. 
Not always true. It is possible that A worked only on Saturdays as he gets paid equally for all days.
(Mathematical example - A: 2 hours on Saturday; B: 1 hour during the weekdays and 1 hour on Saturday. Both get the same amount for the same number of hours)

bumpbot · Answer

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Alan’s regular hourly wage is 1.5 times Barney’s regular

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Alan’s regular hourly wage is 1.5 times Barney’s regular

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