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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 [#permalink]

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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,
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Re: MGMAT CAT1 Question 11 [#permalink]

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mmcooley33 wrote:
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

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,


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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Re: MGMAT CAT1 Question 11 [#permalink]

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New post 25 May 2013, 05:10
Bunuel wrote:
mmcooley33 wrote:
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

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,


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.


Bunuel , u humble anyone's approach to logical and or mathematical problems. respect :)
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Re: MGMAT CAT1 Question 11 [#permalink]

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New post 14 Jan 2014, 11:24
Bunuel wrote:
mmcooley33 wrote:
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

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,


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.


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
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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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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?
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Re: MGMAT CAT1 Question 11 [#permalink]

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New post 10 May 2014, 12:04
Bunuel wrote:
mmcooley33 wrote:
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

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,


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.


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?
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Re: MGMAT CAT1 Question 11 [#permalink]

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New post 11 May 2014, 05:27
Expert's post
russ9 wrote:
Bunuel wrote:
mmcooley33 wrote:
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

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,


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.


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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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: MGMAT CAT1 Question 11 [#permalink]

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New post 15 May 2014, 16:35
Bunuel wrote:
russ9 wrote:
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?


Absolutely. Thanks.
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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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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)
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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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New post 27 Aug 2015, 12:13
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.
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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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New post 27 Apr 2016, 08:30
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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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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New post 27 Apr 2016, 09:11
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thuyduong91vnu wrote:
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 :)



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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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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New post 27 Apr 2016, 09:21
It really helps. Thanks chetan2u :-D
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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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New post 08 Jun 2016, 22:52
mmcooley33 wrote:
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,


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.
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Re: Alan’s regular hourly wage is 1.5 times Barney’s regular [#permalink]

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New post 11 Jun 2016, 07:04
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
Re: Alan’s regular hourly wage is 1.5 times Barney’s regular   [#permalink] 11 Jun 2016, 07:04
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