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

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).

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

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

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.
_________________

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

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.
_________________

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.
_________________

"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.

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.
_________________

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
_________________

"Shawna was given a total of d dollars to pay for the work".. i messed up. i thought for Shawna's work

We know that Shawna worked extra X hours, So: Shawna = Jia + X Hours
It's given that Shawna's time + Jia's time = Y Hours and if we replace Shawna we get:

Jia + Jia + X = Y => 2Jia + X =Y => Jia's Time = (Y - X)/2

Shawna was paid D$ = Both of their pays + reimbursement for buying supplies => Both of their pays + T$
So their hourly pay was: (D$ (total amount) - T$ (supplies))/ Y (time for finishing their work)

To get Jia's payment we must multiply her working hours with their hourly rate:
(Y - X)/2 * (D$ - T$)/Y => (Y - X)(D-T)/2Y - A.

Question:
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?

Solution: Shortest (1-liner) approach:

Total time for which they worked = y hours
Shawna worked for x hours more than Jia

If x = y, it would imply that Shawna worked for the entire time. Thus, payment for Jia would be $0.

Working with the options: We plugin x = y in each:

Option A: This reduces to $0 - possibly correct
Option B: This does not reduce to $0 - incorrect
Option C: This does not reduce to $0 - incorrect
Option D: This does not reduce to $0 - incorrect
Option B: This does not reduce to $0 - incorrect

Thus, Option A is correct



Normal Approach:

Total time for which they worked = y hours
Shawna worked for x hours more than Jia

=> Time for which Shawna worked = (y + x)/2 hours and Time for which Jia worked = (y - x)/2 hours

Amount paid to Shawna = $d
Amount spent by Shawna on tools = $t

=> Amount that goes towards payment for total y hours of work = $(d - t)

=> Amount paid per hour (to either Shawna or Jia) = $[(d - t) ÷ y]

=> Amount that goes towards Jia's payment for (y - x)/2 hours of work

= $[(d - t) ÷ y] * (y - x)/2
= $[(d - t)/y * (y - x)/2]
= \($[\frac{(d - t)(y - x)}{2y}]\)

Answer A
_________________

Combined they worked for a total of y hours.
Shawna worked for x more hours than Jia.
Let y=2 and x=2.
Since a total of 2 hours are worked -- and Shawna works 2 more hours than Jia -- Shawna works 2 hours, while Jia works 0 hours.
In other words:
Shawna on her own completes the entire 2-hour job.

How much money should Shawna give to Jia?
Since Shawna works the entire job, the amount that should be given to Jia = 0.
When y=2 and x=2, the values of d and t become IRRELEVANT.
No matter what values are chosen for d and t, the correct answer MUST yield a value of 0 when y=2 and x=2.

Only A is guaranteed to yield a value of 0 for any possible combination of values assigned to d and t:
\(\frac{(d-t)(y-x)}{2y} = \frac{(d-t)(2-2)}{2y} = 0\)


_________________

I too struggled with the question at the first glance but after thinking about it a little bit i realized it is a simple equation formation question.

The average wage for both J & S is equal so overall average wage is (d-t)/y --- (1)

Now total hours, j + s = y but j + x + j = y so 2j + x = y
Thus j = (y-x)/2

Jia' average wage = Jia wage / (y-x)/2 --- (2)

From the question stem we know (1) = (2)

S0, (d-t)/y = Jia wage / (y-x)/2
Jia wage = (d-t)(y-x) / 2y

Hope it helps..

Given: 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.

Asked: 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?

Let the hours Jia worked be z.
Shawna worked for = (z+x) hours

Combined they worked for total of y hours.
y = z + (z+x) = 2z + x
z = (y-x)/2

Value of their combined work = (d- t) dollars
Total hours worked = y = 2z + x hours
Hourly rate = (d-t)/y dollars/hour

The money Shawna should give to Jia \(= \frac{(d-t)}{y} * z = \frac{(d-t)}{y }* \frac{(y-x)}{2} = \frac{(d-t)(y-x)}{2y}\)

IMO A

Been doing more practice with the timer, this one took me around 9 minutes to solve and even then I accidentially solved for Shawna's wages as opposed to Jia's wages.

Feels fairly impossible at my current skill level to do this question in 2 minutes or anywhere close to that so I still have a long ways to go to practice.