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Shawna and Jia worked together to paint a house. Combined they worked

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Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 18 Aug 2018, 02:45
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Question Stats:

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Shawna and Jia worked together to paint a house. Combined they worked for a total of y hours. Before the job started, Shawna paid t dollars to purchase paint and other supplies for the job. When the job was completed, Shawna was given a total of d dollars to pay for the work as well as reimburse her for the supplies. If Shawna worked for x more hours than Jia, how much money should Shawna give to Jia such that Shawna and Jia are each paid the same hourly rate for their work?

A) \(\frac{(d-t)(y-x)}{2y}\)

B) \(\frac{(d-t)}{y}\)

C) \(\frac{(d-x)}{y} - t\)

D) \(\frac{(dx - t)}{2}\)

E) \(\frac{(d-t)(y+x)}{2y}\)

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Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 18 Aug 2018, 02:50
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dabaobao wrote:
Shawna and Jia worked together to paint a house. Combined they worked for a total of y hours. Before the job started, Shawna paid t dollars to purchase paint and other supplies for the job. When the job was completed, Shawna was given a total of d dollars to pay for the work as well as reimburse her for the supplies. If Shawna worked for x more hours than Jia, how much money should Shawna give to Jia such that Shawna and Jia are each paid the same hourly rate for their work?

A) \(\frac{(d-t)(y-x)}{2y}\)

B) \(\frac{(d-t)}{y}\)

C) \(\frac{(d-x)}{y} - t\)

D) \(\frac{(dx - t)}{2}\)

E) \(\frac{(d-t)(y+x)}{2y}\)


Official Solution (Credit: Manhattan Prep)



Step 1: Glance Read Jot

This problem contains a lot of words that you need to turn into equations so this problem is about Algebraic Translations. Also, note that there are variables in the answer choices. Jot down the variables and what you are solving for.

d = total pay
t = cost of supplies
y = total hours
x = additional hours worked by Shawna

Money to Jia? → Equal hourly wage



Step 2: Reflect Organize

When there are variables in the answers, choosing Smart Numbers is often a good strategy. The other possibility is to work the translations. In this case, the algebra is likely to get messy given the number of variables, so smart numbers may be the better choice. Both strategies are provided below.



Step 3: Work

Smart Numbers



Start by picking numbers for the variables in the problem. Since you will have to divide the dollar amounts by hours to calculate hourly wages, make the dollar values multiples of 10 and set the total hours equal to 10 to keep the division easy.

d = total pay = $100
t = cost of supplies = $20
y = total hours = 10
x = additional hours worked by Shawna = 2

Now, calculate the target values: the amount of money Shawna should give Jia to make their hourly wages equal.

The amount of money for total wages is equal to the total pay minus the supply costs:
$100 – $20 = $80

The hourly wage is the amount for wages divided by the total hours worked:
$80/10 = $8

Calculate the number of hours Jia worked. The two women worked a total of 10 hours and Shawna worked 2 more hours than Jia. Therefore, Shawna worked 6 hours and Jia worked 4 hours.

If Jia worked 4 hours and the hourly wage is $8, then she is owed $32 ($8 × 4).

$32 is your target value; this is the value you will get when you plug your smart numbers for the other variables into the correct answer.

Test Choice (A).

(d − t)(y + x)/2y=(100 − 20)(10 − 2)/2(10)=(80)(8)/20=4(8)=32

If you are short on time, you may choose to stop once you find a match for your target. Otherwise, test the remaining answers to ensure that no other answer matches the target.

Test Choice (B).

(d − t)/y=(100 − 20)/10=8


Test Choice (C).

(d − x)/y−t=((100 − 2)/10)−20≈−10


Test choice (D).

(dx − t)/2=(100(2) − 20)/2=90


Test choice (E).

(d − t)(y + x)/2y=(100 − 20)(10 + 2)/(2(10))=(80)(12)/20=48



Only choice (A) matches the target value.



Algebra



The goal on this problem is to calculate how much money Jia should receive so that the two women earn the same hourly wage. Multiply the number of hours Jia worked times the hourly wage.

Jia’s Pay = Jia’s Hours Worked × Hourly Wage

Consider how to calculate those two things separately.

Jia’s Hours Worked
Two pieces of information are provided about the hours: The total hours worked were y and Shawna worked x more hours. Create equations for J, the hours Jia worked, and S, the hours Shawna worked.

J + S = y
J + x = S

Since the answers are in terms of x and y (not S), solve for J in terms of these two variables.

J + S = y Substitute in for S.
J + J + x = y Subtract x from both sides.
2J = y ­­ – x Divide both sides by 2.
J=(y − x)/2

Hourly Wage
To calculate the hourly wage, subtract the money that serves as reimbursement for the supplies from the total pay, then divide by the total number of hours worked by both women.

Hourly wage = (d − t)/y

Finally, multiply Jia’s hours by the hourly wage to get the total amount of money owed to Jia.

(y − x)/2×(d − t)/y=(y − x)(d − t)/(2y)



The correct answer is (A).
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 05:59
I almost cried while attempting this question. So many variables so many conditions. It took me five minutes on the test and I still ended up getting it wrong.
Can such kind of questions come in the exam?
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 11:02
Micky1005 wrote:
I almost cried while attempting this question. So many variables so many conditions. It took me five minutes on the test and I still ended up getting it wrong.
Can such kind of questions come in the exam?

Hi Micky1005 ,

In my opinion, the question is absolute GMAT-like... but the official solution is NOT!

Please follow my reasoning (in the post below) after reading the question stem carefully (more than once) for (say) approximately one minute.

I explain: the first reading must be fast - say 20 seconds - for the brain to know which "drawers to open". The second reading takes double-time, to start "the structure"!

Important: this is not time-wasting... it is time-investment!

(This advice follows the GMATH method.)

If you have any doubts, please feel free to ask me about it.

Regards and success in your studies!
Fabio.
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Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 11:08
dabaobao wrote:
Shawna and Jia worked EACH ONE ALONE to paint a house. Combined they worked for a total of y hours. Before the job started, Shawna paid t dollars to purchase paint and other supplies for the job. When the job was completed, Shawna was given a total of d dollars to pay for the work as well as reimburse her for the supplies. If Shawna worked for x more hours than Jia, how much money should Shawna give to Jia such that Shawna and Jia are each paid the same hourly rate for their work?

A) \(\frac{(d-t)(y-x)}{2y}\)

B) \(\frac{(d-t)}{y}\)

C) \(\frac{(d-x)}{y} - t\)

D) \(\frac{(dx - t)}{2}\)

E) \(\frac{(d-t)(y+x)}{2y}\)

\(?\,\,\,:\,\,\,{\text{Jia}}\,\,{\text{fair}}\,\,{\text{payment}}\,{\text{for}}\,\,{\text{her}}\,\,{\text{work}}\)

The careful reading suggested (in the post above) is enough to understand the following:

1. The $t paid by Shawna is reimbursed, therefore what is paid for the work is simply $ (d-t) and THIS is the amount that must be divided.

Conclusion: alternative choices (C) and (D) are not good candidates.

2. If we imagine y=3 (3h for Shawna and Jia to work, each one alone) and x=1 (so that Jia works 1h and Shawna works 2h), we are sure
$(d-t) must be divided in three equal parts, and Jia deserves one third of it.

Conclusion: when we explore the particular case x=1 and y=3 , our FOCUS, in this case the TARGET expression, is (d-t)/3 (in dollars).

Checking (A), (B) and (E), there is only one survivor (check that), hence this choice (A) is the correct choice!


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 11:42
1
Total hours worked = y hours
Shawn worked "x" more than Jia. Therefore , Jia worked \(\frac{(y-x)}{2}\) hours

Shawn paid "t" for supplies. He received "d" as total payment + reimbursement for supplies.
Let k be their hourly rate which is same for both.

d = t + ky

=>\(\frac{(d-t)}{ky} = 1\) (wage for 1 hour)

=> \(\frac{(d-t)}{ky} * \frac{(y-x)k}{2}\) = Jia's wage

=> \(\frac{(d-t)(y-x)}{2y}\)
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Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 14:46
dabaobao wrote:
Shawna and Jia worked EACH ONE ALONE to paint a house. Combined they worked for a total of y hours. Before the job started, Shawna paid t dollars to purchase paint and other supplies for the job. When the job was completed, Shawna was given a total of d dollars to pay for the work as well as reimburse her for the supplies. If Shawna worked for x more hours than Jia, how much money should Shawna give to Jia such that Shawna and Jia are each paid the same hourly rate for their work?

Very nice approach, pandeyashwin !

Without exploring a particular case (as I did previously), let´s show the "GMATH´s way", using UNITS CONTROL, one of the most powerful tools of our method!

\(?\,\,\,\,:\,\,\,\,{\rm{Jia}}\,\,{\rm{fair}}\,\,\$ \,\,{\rm{payment}}\,{\rm{for}}\,\,{\rm{her}}\,\,{\rm{work}}\,\,\,\,\,\,\,\,\left[ {\,\$ \,c\,\,\, = \,\,\,{\rm{common}}\,\,{\rm{hourly}}\,\,\,{\rm{rate}}\,} \right]\,\,\,\)

\(y\,\,{\rm{h}}\,\,\,\left\{ \matrix{
\,{\rm{Shawna}}\,\,:\,\,\left( {{y \over 2} + {x \over 2}} \right)\,\,{\rm{h}} \hfill \cr
\,{\rm{Jia}}\,\,:\,\,\left( {{y \over 2} - {x \over 2}} \right)\,\,{\rm{h}} \hfill \cr} \right.\,\,\,\,\,\,\,\left[ {\,{\rm{Sum}}\,\,y\,\,,\,\,\,x\,\,{\rm{difference}}\,\,{\rm{,}}\,\,{\rm{Shawna}}\,\,{\rm{more}}\,\,{\rm{time}}\,} \right]\)

\(\left( {{{y + x} \over 2}} \right)\,\,{\rm{h}}\,\, \cdot \,\,\left( {{{\,\$ \,\,c\,} \over {1\,\,{\rm{h}}}}} \right)\,\,\,\,\, + \,\,\left( {{{y - x} \over 2}} \right)\,\,{\rm{h}}\,\, \cdot \,\,\left( {{{\,\$ \,\,c\,} \over {1\,\,{\rm{h}}}}} \right)\,\,\,\, = \,\,\,\,\$ \,\,\left( {d - t} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,yc = d - t\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,c = {{d - t} \over y}\,\,\,\,\,\left[ {\rm{$}} \right]\,\)

\(?\,\,\, = \,\,\,\,\left( {{{y - x} \over 2}} \right)\,\,h\,\,\, \cdot \,\,\,\left( {{{\,\$ \,\,\left( {d - t} \right)\,} \over {y\,\,\,{\rm{h}}}}} \right)\,\,\,\,\, = \,\,\,\,{{\,\left( {y - x} \right)\left( {d - t} \right)\,} \over {2y}}\,\,\,\,\,\,\left[ {\rm{\$ }} \right]\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 22:58
"Shawna and Jia worked together to paint a house. Combined they worked for a total of y hours".

If shawna and jia worked together, how come shawna worked for more number of hours than jia?

Why are we not using this:
s = time taken by shawna to do the work
j = time taken by jia to do the work

1/s + 1/j = 1/y ?

I understood the solution given by Manhattan, but if the question is worded in the above way, should we not follow this approach?
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 22 Oct 2018, 23:27
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PriyankaPalit7 wrote:
"Shawna and Jia worked together to paint a house. Combined they worked for a total of y hours".

If shawna and jia worked together, how come shawna worked for more number of hours than jia?

Why are we not using this:
s = time taken by shawna to do the work
j = time taken by jia to do the work

1/s + 1/j = 1/y ?

I understood the solution given by Manhattan, but if the question is worded in the above way, should we not follow this approach?


PriyankaPalit7,

When we say " They worked together or simultaneously for y hours" => Shawna worked for y hrs and Jia also worked for y hrs.

On the other hand, if we say "Combined they worked for a total of y hours" => Shawna worked for a hrs (say) and Jia worked for b hrs (say), then a+b = y hrs.

i.e.
Combined they worked for a total of y hours \(\neq\) they worked together for y hours.

Hope this helps.

Thanks.
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 23 Oct 2018, 04:51
Hi PriyankaPalit7 !

"Shawna and Jia worked together to paint a house." was something that bothered me, too.

Please have a look at my posts above, in which I changed (slightly) the question stem to avoid this confusion.

Regards and success in your studies!
Fabio.
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 23 Oct 2018, 18:21
dabaobao wrote:
Shawna and Jia worked together to paint a house. Combined they worked for a total of y hours. Before the job started, Shawna paid t dollars to purchase paint and other supplies for the job. When the job was completed, Shawna was given a total of d dollars to pay for the work as well as reimburse her for the supplies. If Shawna worked for x more hours than Jia, how much money should Shawna give to Jia such that Shawna and Jia are each paid the same hourly rate for their work?

A) \(\frac{(d-t)(y-x)}{2y}\)

B) \(\frac{(d-t)}{y}\)

C) \(\frac{(d-x)}{y} - t\)

D) \(\frac{(dx - t)}{2}\)

E) \(\frac{(d-t)(y+x)}{2y}\)


We are given that both Shawn and Jia worked for y hours. Since Shawna worked for x more hours than Jia, she worked (y + x)/2 hours and Jia worked (y - x)/2 hours. (Notice that (y + x)/2 + (y - x)/2 = 2y/2 = y and (y + x)/2 - (y - x)/2 = 2x/2 = x.)

Since Shawna paid t dollars for paint and supplies and was paid d dollars for the job and reimbursement, she received (d - t) dollars for the time she and Jia worked. Since they worked a total of y hours, the hourly wage should be
(d - t)/y dollars per hour. Since Jia worked (y - x)/2, then he should receive

(d - t)/y * (y - x)/2 = (d - t)(y - x)/(2y) dollars

Answer: A
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Re: Shawna and Jia worked together to paint a house. Combined they worked  [#permalink]

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New post 24 Oct 2018, 16:04
"Shawna was given a total of d dollars to pay for the work".. i messed up. i thought for Shawna's work :(
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Re: Shawna and Jia worked together to paint a house. Combined they worked &nbs [#permalink] 24 Oct 2018, 16:04
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