Problem Solving

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 worked the same total 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
d I and III
e II and III


OA to follow

IMO E.

I. Alan could not have worked fewers on weekdays than did Barney, because Barney earns twice the normal hourly wage on saturdays and they earned the same total in wages.
II. True. If Barney did not work for atleast an hour on saturday, then Alan would have earned more than Barney would.
III. True. Alan = 3/2 B on weekdays. So, B = 2/3 A. To earn the same wage as Alan, Barney must have made more money than Alan on Saturday.

The answer is E and this is my reasoning.

Let Xa = number of hours Alan work M-F and X'a = number of hours Alan works on Saturday
Let Xb = number of hours Barney work M-F and X'b = number of hours Barney works on Saturday
Let A = Alan's M-F rate and his Saturday rate
B = Barney's M-F rate = 2/3 A. Barney's Saturday rate = 2A [These ratio are given]

Xa + X'a = Xb + X'b [They both work the same amount of hours]
Alan's total income = Barney's total income. [They earn the same amount]

So:

A Xa + A X'a = 2/3 A Xb + 2A X'b which can be simplified to: Xa + X'a = 2/3 Xb + 2X'b. We also know that they work integer hours, so the final equation is 3Xa + 3X'a = 2 Xb + 6 X'b. The final equation is comparing the number of hours both Alan and Barney work during the week after taking into account their rates.

3Xa > 2 Xb , so I is out
6 X'b > 3X'a, so III and II are correct.

What's the OA?

IMO E

i also marked E but OA is not E...and the oE given by Manhattan is %#^$

is it D, becasue II said at least so we can not exactly how many hours braney worked.

IMO - B

OA b

OA is B

OE from MGMAT CAT is

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 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.

Can someone now prove it using #s ?

bhumika.k.shah wrote:
Can someone now prove it using #s ?

III is tricky; let's pick numbers to show that it doesn't have to be true.

Let Barney's regular wage be $2/hour. Accordingly:

Alan makes $3/hour; and

Barney makes $4/hour on Saturday.

We need them to work the same total hours and earn the same total money. So, we're allowed to pick:

Barney: 3 hours on Saturday and 3 hours on Monday (3*$4 + 3*$2 = $18 in 6 hours)

Alan: 6 hours on Saturday (6*$3 = $18 in 6 hours)

Same number of hours, same total salary, but Alan earned $18 on Saturday and Barney only earned $12 on Saturday.

* * *

Thinking it through, we can see that for Barney to earn the same amount of money as Alan in the same number of hours, Barney needs to spend half of his total working hours on Saturday - that way his regular wage and his Saturday wage will average out to Alan's regular wage.

In mathematical terms:

# of hours they each work in total = h
# of hours Barney works on Saturday = s
B's regular wage = x
B's bonus wage = 2x
A's wage = 1.5x

We want:

x(h-s) + 2x(s) = 1.5x(h)

x(h-s + 2s) = 1.5xh

x(h+s) = 1.5x(h)

h + s = 1.5h

s = 1.5h - h

s = .5h

So, Barney has to work half his total hours on Saturday. As long as we fulfil that condition, everything else will work out. What days Alan works is irrelevant, so we can stack all or none of his hours on Saturday and play with III being true or false.

Of course, if we do this through logic or picking numbers instead of math, we're done a LOT quicker and we're MUCH happier.

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