John and Mary were each paid x dollars in advance to do a certain job together. John worked on the...

1. Translate the problem requirements: Both John and Mary received x dollars initially. John worked 10 hours, Mary worked 8 hours (2 less than John). Mary gave John y dollars so their hourly wages became equal. We need to find John's initial payment (x) in terms of y.
2. Set up the final payment scenario: After the transfer, John has \((\mathrm{x} + \mathrm{y})\) dollars and Mary has \((\mathrm{x} - \mathrm{y})\) dollars, and their hourly wages are equal.
3. Apply the equal hourly wage condition: Since hourly wage equals total payment divided by hours worked, we can set up the equation that John's hourly wage equals Mary's hourly wage.
4. Solve for the initial payment: Use the equality of hourly wages to create an equation and solve for x in terms of y.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know in plain English:

• John and Mary each got paid the same amount upfront: x dollars each
• John worked for 10 hours on the job
• Mary worked for 8 hours (which is 2 hours less than John's 10 hours)
• After the work was done, Mary gave John y dollars from her payment
• This transfer made their hourly wages equal
• We need to find what x (John's initial payment) equals in terms of y

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up the final payment scenario

After Mary gives John y dollars, let's see what each person has:

• John's final payment: He started with x dollars and received y dollars from Mary, so he ends up with \((\mathrm{x} + \mathrm{y})\) dollars
• Mary's final payment: She started with x dollars and gave away y dollars to John, so she ends up with \((\mathrm{x} - \mathrm{y})\) dollars

The key insight is that after this transfer, both John and Mary have the same hourly wage, even though they worked different numbers of hours.

3. Apply the equal hourly wage condition

Now we use the fact that hourly wage equals total payment divided by hours worked:

• John's hourly wage after transfer = \((\mathrm{x} + \mathrm{y})\) dollars ÷ 10 hours
• Mary's hourly wage after transfer = \((\mathrm{x} - \mathrm{y})\) dollars ÷ 8 hours

Since these hourly wages are equal, we can set them equal to each other:

\(\frac{\mathrm{x} + \mathrm{y}}{10} = \frac{\mathrm{x} - \mathrm{y}}{8}\)

Process Skill: INFER - Recognizing that equal hourly wages means we can set up an equation

4. Solve for the initial payment

Now we solve this equation step by step:

\(\frac{\mathrm{x} + \mathrm{y}}{10} = \frac{\mathrm{x} - \mathrm{y}}{8}\)

Cross multiply to eliminate fractions:
\(8(\mathrm{x} + \mathrm{y}) = 10(\mathrm{x} - \mathrm{y})\)

Expand both sides:
\(8\mathrm{x} + 8\mathrm{y} = 10\mathrm{x} - 10\mathrm{y}\)

Move all terms with x to one side and all terms with y to the other:
\(8\mathrm{y} + 10\mathrm{y} = 10\mathrm{x} - 8\mathrm{x}\)
\(18\mathrm{y} = 2\mathrm{x}\)

Solve for x:
\(\mathrm{x} = \frac{18\mathrm{y}}{2} = 9\mathrm{y}\)

Process Skill: MANIPULATE - Using algebraic techniques to isolate the variable we want

5. Final Answer

John was paid \(9\mathrm{y}\) dollars in advance.

Let's verify this makes sense: If John got \(9\mathrm{y}\) dollars initially and Mary gave him \(\mathrm{y}\) dollars, John ends up with \(10\mathrm{y}\) dollars for 10 hours of work, giving him an hourly wage of \(\mathrm{y}\) dollars per hour. Mary ends up with \(8\mathrm{y}\) dollars for 8 hours of work, also giving her an hourly wage of \(\mathrm{y}\) dollars per hour. The hourly wages are indeed equal!

The answer is (E) \(9\mathrm{y}\).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "same hourly wage" condition
Students might incorrectly think that John and Mary should receive the same total payment after the transfer, rather than the same hourly wage. This leads to setting up the wrong equation: \((\mathrm{x} + \mathrm{y}) = (\mathrm{x} - \mathrm{y})\) instead of \(\frac{\mathrm{x} + \mathrm{y}}{10} = \frac{\mathrm{x} - \mathrm{y}}{8}\).

Faltering Point 2: Confusion about the direction of money transfer
Students may get confused about who gives money to whom. The problem states "Mary gave John y dollars," but students might set up the equation assuming John gave money to Mary, leading to John having \((\mathrm{x} - \mathrm{y})\) and Mary having \((\mathrm{x} + \mathrm{y})\).

Faltering Point 3: Misunderstanding Mary's work hours
Students might misread "Mary worked 2 hours less than John" and incorrectly calculate Mary's hours as 12 hours instead of 8 hours, leading to the wrong equation setup.

Errors while executing the approach

Faltering Point 1: Cross-multiplication errors
When solving \(\frac{\mathrm{x} + \mathrm{y}}{10} = \frac{\mathrm{x} - \mathrm{y}}{8}\), students often make mistakes in cross-multiplication, writing \(10(\mathrm{x} + \mathrm{y}) = 8(\mathrm{x} - \mathrm{y})\) instead of the correct \(8(\mathrm{x} + \mathrm{y}) = 10(\mathrm{x} - \mathrm{y})\).

Faltering Point 2: Sign errors during algebraic manipulation
When expanding \(8(\mathrm{x} + \mathrm{y}) = 10(\mathrm{x} - \mathrm{y})\) to get \(8\mathrm{x} + 8\mathrm{y} = 10\mathrm{x} - 10\mathrm{y}\), students frequently make sign errors when moving terms across the equals sign, such as writing \(8\mathrm{y} - 10\mathrm{y} = 10\mathrm{x} - 8\mathrm{x}\) instead of \(8\mathrm{y} + 10\mathrm{y} = 10\mathrm{x} - 8\mathrm{x}\).

Faltering Point 3: Arithmetic errors in final calculation
Students may correctly get to \(18\mathrm{y} = 2\mathrm{x}\) but then make arithmetic mistakes when dividing, such as calculating \(\mathrm{x} = \frac{18\mathrm{y}}{2} = 8\mathrm{y}\) instead of \(9\mathrm{y}\).

Errors while selecting the answer

Faltering Point 1: Answering for Mary's payment instead of John's
Students might solve correctly but then select the answer that represents Mary's initial payment rather than John's initial payment, especially if they calculated both values during their work.

Faltering Point 2: Selecting the transfer amount instead of initial payment
Students may confuse what the question is asking for and select an answer choice that represents the transfer amount \((\mathrm{y})\) or some multiple of it that appeared in their calculations, rather than the initial payment amount \((9\mathrm{y})\).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for y

Let's set \(\mathrm{y} = 1\) dollar. This will make our calculations cleaner and easier to follow.

Step 2: Set up the scenario with concrete numbers

• John worked 10 hours
• Mary worked 8 hours (2 less than John)
• Both initially received x dollars
• Mary gave John 1 dollar (our chosen value for y)
• After the transfer: John has \((\mathrm{x} + 1)\) dollars, Mary has \((\mathrm{x} - 1)\) dollars

Step 3: Apply the equal hourly wage condition

Since their hourly wages are equal after the transfer:
John's hourly wage = Mary's hourly wage
\(\frac{\mathrm{x} + 1}{10} = \frac{\mathrm{x} - 1}{8}\)

Step 4: Solve for x

Cross multiply: \(8(\mathrm{x} + 1) = 10(\mathrm{x} - 1)\)
\(8\mathrm{x} + 8 = 10\mathrm{x} - 10\)
\(8 + 10 = 10\mathrm{x} - 8\mathrm{x}\)
\(18 = 2\mathrm{x}\)
\(\mathrm{x} = 9\)

Step 5: Express the result in terms of y

Since we chose \(\mathrm{y} = 1\) and found \(\mathrm{x} = 9\), we can see that \(\mathrm{x} = 9\mathrm{y}\).
We can verify this relationship holds generally: when \(\mathrm{y} = 1\), \(\mathrm{x} = 9(1) = 9\).

Step 6: Verification

With \(\mathrm{x} = 9\) and \(\mathrm{y} = 1\):

• John's final payment: \(9 + 1 = 10\) dollars for 10 hours = $1/hour
• Mary's final payment: \(9 - 1 = 8\) dollars for 8 hours = $1/hour ✓

Therefore, John was paid \(9\mathrm{y}\) dollars in advance.