A Manhattan GMAT Problem-John and Mary

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pkw209: Master | Next Rank: 500 Posts; Posts: 266; Joined: Mon Oct 19, 2009 9:46 pm; Thanked: 8 times; GMAT Score:690

by pkw209 » Wed Feb 03, 2010 5:28 pm

A brief explanation would be appreciated. thanks.

Question-

John and Mary were each paid x dollars in advance to do a certain job together. John worked on the job for 10 hours and Mary worked 2 hours less than John. If Mary gave John 4 dollars of her payment so that they would have received the same hourly wage, what was the dollar amount in terms of y that John was paid in advance?

Answer to come...

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Source: Beat The GMAT — Problem Solving | Wed Feb 03, 2010 5:28 pm

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Wed Feb 03, 2010 6:15 pm

pkw209 wrote:A brief explanation would be appreciated. thanks.

Question-

John and Mary were each paid x dollars in advance to do a certain job together. John worked on the job for 10 hours and Mary worked 2 hours less than John. If Mary gave John y dollars of her payment so that they would have received the same hourly wage, what was the dollar amount in terms of y that John was paid in advance?

Answer to come...

I think that's an official problem, not an MGMAT problem, and in the original version of the question, where you've written '4 dollars', it ought to read 'y dollars' -- otherwise the question doesn't make much sense, since there is no y anywhere! I've noted the correction in red above.

So, after Mary gives John y dollars, Mary has x-y dollars, and John has x+y dollars. John works for 10 hours, so his hourly wage is (x+y)/10 (just divide the total money earned by the number of hours worked), and Mary works for 8 hours, so her hourly wage is (x-y)/8. These hourly wages are equal:

(x+y)/10 = (x-y)/8
8x + 8y = 10x - 10y
18y = 2x
9y = x

Since x is the amount John was paid in advance, the answer is 9y.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Wed Feb 03, 2010 10:38 pm

pkw209 wrote:A brief explanation would be appreciated. thanks.

Question-

John and Mary were each paid x dollars in advance to do a certain job together. John worked on the job for 10 hours and Mary worked 2 hours less than John. If Mary gave John 4 dollars of her payment so that they would have received the same hourly wage, what was the dollar amount in terms of y that John was paid in advance?

Answer to come...

I've noticed that you've posted a number of questions without providing answer choices; doing so cripples our ability to discuss strategic approaches to the questions.

In future, please post the answer choices as well, since on a high percentage of high level questions, strategic approaches are superior to pure math.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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bluebird42: Junior | Next Rank: 30 Posts; Posts: 12; Joined: Thu Apr 28, 2011 8:17 am

by bluebird42 » Tue May 24, 2011 4:56 pm

Here are the multiple choice answers:
A. 4y
B. 5y
C. 6y
D. 8y
E. 9y

I too had difficulties with this problem, specifically setting up the correct equations. There is a lot of talk about strategically guessing on the GMAT. With regards to this question, what answers could you eliminate, and how would you go about guessing the right answer?

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed May 25, 2011 3:08 am

bluebird42 wrote:Here are the multiple choice answers:
A. 4y
B. 5y
C. 6y
D. 8y
E. 9y

I too had difficulties with this problem, specifically setting up the correct equations. There is a lot of talk about strategically guessing on the GMAT. With regards to this question, what answers could you eliminate, and how would you go about guessing the right answer?

We can plug in a value for the hourly rate.

Let hourly rate = $1.
Since John works 10 hours, he earns 10*1 = $10.
Since Mary works 10-2=8 hours, she earns 8*1 = $8.
Thus, the total advance payment to John and Mary = 10+8 = $18.
Since each receives x dollars in advance, x = 18/2 = $9.

John is given x=9 in advance but must earn $10.
Mary is given x=9 in advance but must earn $8.
Thus, Mary gives John y=1.

The question asks for the amount that John is paid in advance: $9.
This is our target.
Now we plug y=1 into the answers to see which yields our target of 9.

Only answer choice E works:
9y = 9*1 = 9.

The correct answer is E.

Plugging in is an efficient way to solve that requires no algebraic insight.

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bluebird42: Junior | Next Rank: 30 Posts; Posts: 12; Joined: Thu Apr 28, 2011 8:17 am

by bluebird42 » Wed May 25, 2011 6:04 am

Thanks Mitch, that was very helpful!!

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TOPGMAT: Senior | Next Rank: 100 Posts; Posts: 86; Joined: Wed Aug 25, 2010 11:00 am; Thanked: 1 times

by TOPGMAT » Fri Dec 02, 2011 9:52 am

Mitch.. You are the best!!!

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samru130: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Fri Mar 19, 2010 9:22 am

by samru130 » Mon Jun 25, 2012 11:12 am

My approach would be,

total money given=2x.
Total Time spent=18hrs.
Hourly Rate: 2x=18h || let h=hourly rate

h= 1/9x

Now, x+y = 10/9x
x-y = 8/9x

x= 9y.

Hope it helps someone.

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HAmjad: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Thu Jan 07, 2016 9:15 pm

by HAmjad » Sat Jan 30, 2016 9:15 pm

GMATGuruNY wrote:
bluebird42 wrote:Here are the multiple choice answers:
A. 4y
B. 5y
C. 6y
D. 8y
E. 9y

I too had difficulties with this problem, specifically setting up the correct equations. There is a lot of talk about strategically guessing on the GMAT. With regards to this question, what answers could you eliminate, and how would you go about guessing the right answer?
We can plug in a value for the hourly rate.

Let hourly rate = $1.
Since John works 10 hours, he earns 10*1 = $10.
Since Mary works 10-2=8 hours, she earns 8*1 = $8.
Thus, the total advance payment to John and Mary = 10+8 = $18.
Since each receives x dollars in advance, x = 18/2 = $9.

John is given x=9 in advance but must earn $10.
Mary is given x=9 in advance but must earn $8.
Thus, Mary gives John y=1.

The question asks for the amount that John is paid in advance: $9.
This is our target.
Now we plug y=1 into the answers to see which yields our target of 9.

Only answer choice E works:
9y = 9*1 = 9.

The correct answer is E.

Plugging in is an efficient way to solve that requires no algebraic insight.

I plugged in the answers this way:

Advance payment to both: $80
So Mary= $40 hourly wage: $4/hr
John= $40 hourly wage: $4/hr

But Mary gets 40/8= $5/hr and John gets 40/10= $4/hr
since Mary works 2 hours less so she gives 5*2= $10 to John (y)

The advance payment that John got was $40. So shouldn't the answer be A? 4y = 4(10) = $40.

Please explain how am I doing it wrong?

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MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 amFollowed by: 140 members; Location: https://martymurraycoaching.com/; Thanked: 955 times; GMAT Score:800

by MartyMurray » Sun Jan 31, 2016 4:55 am

HAmjad wrote:I plugged in the answers this way:

Advance payment to both: $80
So Mary= $40 hourly wage: $4/hr
John= $40 hourly wage: $4/hr

Where did you get these hourly wages? This implies that they each worked for 10 hours, when actually Mary worked for only 8.

But Mary gets 40/8= $5/hr and John gets 40/10= $4/hr
since Mary works 2 hours less so she gives 5*2= $10 to John (y)

The advance payment that John got was $40. So shouldn't the answer be A? 4y = 4(10) = $40.

Please explain how am I doing it wrong?

Look at what you arrived at.

Mary: 40 - 10 = 30 30/8 = 3.75 per hour

John: 40 + 10 = 50 50/10 = 5.00 per hour

The error you made is that rather than doing something based on making Mary's and John's proportions of the total payments match the hours each worked, you first used for Mary a rate, $5/hour, that was not going to hold for either Mary or John once she gave some money to John, and then, using that rate, gave payment for two hours to John.

So John worked for 10/18 = 5/9 of the hours, but ended up with 5/8 of the money.

Even if instead of using Mary's initial rate, you had used John's initial rate, $4/hour, or a rate based on the total amount of money paid, your method still would not have worked, because the way to get them to parity is not to give John two hours' worth of payments, but to transfer to him an amount that makes the proportions paid match the proportions of the hours worked.

Look at Mitch's example above. They each end up making $1 per hour, but Mary does not give John $2. She only has to give him $1 to make the proportions match.

The amount that Mary gives John has to in some way involve calculating what proportion of the total pay John should get and then figuring out how much Mary has to give him in order for him to get that proportion.

Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Sun Jan 31, 2016 7:16 pm

I actually think this is easiest algebraically.

Total money = $2x

John worked 10 hours, Mary worked 8, so John should get (10/18), or (5/9) of the money, and Mary should get the rest.

SHOULD GET:
John = 2x * (5/9) = 10x/9
Mary = 2x * (4/9) = 8x/9

ACTUALLY GOT:
John = x
Mary = x

So Mary should give John (x/9), or 1/9 of her money, to even things out. Hence y = x/9, and x = 9y.

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