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
OAB
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Alan and Barney
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j_shreyans
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Hi j_shreyans,
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
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
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Matt@VeritasPrep
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Here's a succinct answer.
Suppose Barney is paid $n per hour on weekdays (and Sundays, I suppose). Then Alan is paid $1.5n per hour, and Barney is paid $2n per hour on Saturday. Each man worked h hours last week. Since Alan is paid $1.5n per hour, he made $1.5n * h, or 1.5nh dollars. Barney must therefore have also made 1.5nh dollars.
Now notice that the average of n and 2n is 1.5n. Since Barney's weekday rate and Saturday rate are weighted EQUALLY in his overall rate, we know that Barney worked HALF of his hours last week, or h/2 hours, on Saturday.
I:
We know that Barney worked half his hours, or h/2 hours, on Saturday.
Since Alan is paid a flat rate every day, he could've worked on any day of the week: all his hours could've been on Saturday, or all his hours could've been on Saturday. This means Alan could've worked anywhere from 0 to h hours on Saturday. We don't know if this is greater than h/2, so it is NOT necessarily true.
II:
This is true, as we saw above.
III:
This is identical to I. If Alan worked all his hours on Saturday, he made more than Barney did. If he worked none of his hours on Saturday, he made less. Hence this is NOT necessarily true.
We're done!
Suppose Barney is paid $n per hour on weekdays (and Sundays, I suppose). Then Alan is paid $1.5n per hour, and Barney is paid $2n per hour on Saturday. Each man worked h hours last week. Since Alan is paid $1.5n per hour, he made $1.5n * h, or 1.5nh dollars. Barney must therefore have also made 1.5nh dollars.
Now notice that the average of n and 2n is 1.5n. Since Barney's weekday rate and Saturday rate are weighted EQUALLY in his overall rate, we know that Barney worked HALF of his hours last week, or h/2 hours, on Saturday.
I:
We know that Barney worked half his hours, or h/2 hours, on Saturday.
Since Alan is paid a flat rate every day, he could've worked on any day of the week: all his hours could've been on Saturday, or all his hours could've been on Saturday. This means Alan could've worked anywhere from 0 to h hours on Saturday. We don't know if this is greater than h/2, so it is NOT necessarily true.
II:
This is true, as we saw above.
III:
This is identical to I. If Alan worked all his hours on Saturday, he made more than Barney did. If he worked none of his hours on Saturday, he made less. Hence this is NOT necessarily true.
We're done!
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Let Barney's wage Monday through Friday = $2 per hour.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.
I only
II only
I and II only
I and III only
II and III only
Barney's Saturday wage = 2*2 = $4 per hour.
Alan's wage = (1.5)*2 = $3 per hour.
Alan's wage is HALFWAY between Barney's two wages.
Thus, for Barney to work the same amount of time as Alan and to earn the same AVERAGE wage as Alan, Barney MUST work for HALF his time at $2 per hour (Monday through Friday) and for HALF his time at $4 per hour (on Saturday), so that his AVERAGE wage = $3 per hour.
Thus, II must be true: Barney must work at least 1 hour on Saturday.
Eliminate any answer choice that does not include II.
Eliminate A and D.
To evaluate I and III, plug in for the number of hours worked.
Try to show that I and III do NOT have to be true.
Let the time worked = 2 hours.
Amount earned by Alan = 2*3 = $6.
Barney works for 1 hour Monday through Friday and for 1 hour on Saturday -- for a total of 2 hours -- earning 2+4 = $6.
Same number of hours for each, same total earnings for each.
I. Alan worked fewer hours Monday through Friday than did Barney.
Since it's possible that Alan worked for 2 hours Monday through Friday while Barney worked for only 1 hour Monday through Friday, eliminate any answer choice that includes I.
Eliminate C.
III. Barney made more money on Saturday than did Alan.
Since it's possible that Alan worked for 2 hours on Saturday -- earning $6 -- while Barney worked for only 1 hour on Saturday -- earning $4 -- eliminate any answer choice that includes III.
Eliminate E.
The correct answer is B.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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