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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Verb Tenses Simplified

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