Roy was paid at a fixed hourly rate for the 38 hours that he worked in a certain week and...

1. Translate the problem requirements: Roy worked 38 hours in week 1 and 30 hours in week 2 at different hourly rates, but earned the same total pay both weeks. We need to find the percent change in his hourly rate from week 1 to week 2.
2. Set up the equal pay relationship: Since total weekly pay was identical, we can establish that \(38 \times \mathrm{rate_1} = 30 \times \mathrm{rate_2}\), creating a direct relationship between the two rates.
3. Find the rate ratio using inverse proportionality: When hours decrease from 38 to 30 while total pay remains constant, the hourly rate must increase proportionally. The rate ratio equals the inverse of the hours ratio.
4. Calculate percent change from the ratio: Convert the rate ratio into a percent change by subtracting 1 and expressing as a percentage, then match to the closest answer choice.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what happened in plain English. Roy worked two different weeks with different hours but earned the exact same total pay both weeks. This is the key insight we need to work with.

In Week 1: Roy worked 38 hours at some hourly rate (let's call it \(\mathrm{rate_1}\))
In Week 2: Roy worked 30 hours at a different hourly rate (let's call it \(\mathrm{rate_2}\))

The important constraint is that his total weekly pay was identical for both weeks. This means:
Total pay Week 1 = Total pay Week 2

We need to find the percent change in his hourly rate from Week 1 to Week 2.

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

2. Set up the equal pay relationship

Since we know that total pay equals hours worked times hourly rate, and the total pay was the same both weeks, we can write this relationship in everyday terms first:

"38 hours times the first week's rate equals 30 hours times the second week's rate"

Now we can express this mathematically:
\(38 \times \mathrm{rate_1} = 30 \times \mathrm{rate_2}\)

This equation captures the essential constraint of the problem - equal total pay despite different hours worked.

3. Find the rate ratio using inverse proportionality

Here's the key insight: when the total pay stays the same but the hours worked decrease, the hourly rate must increase to compensate. This is an inverse relationship.

Think about it this way: Roy worked fewer hours (30 instead of 38) but earned the same total amount. So he must have been paid more per hour in the second week.

From our equation: \(38 \times \mathrm{rate_1} = 30 \times \mathrm{rate_2}\)

We can solve for the relationship between \(\mathrm{rate_2}\) and \(\mathrm{rate_1}\):
\(\mathrm{rate_2} = \frac{38}{30} \times \mathrm{rate_1}\)
\(\mathrm{rate_2} = \frac{19}{15} \times \mathrm{rate_1}\)

This means \(\mathrm{rate_2}\) is \(\frac{19}{15}\) times as large as \(\mathrm{rate_1}\).

Process Skill: INFER - Recognizing the inverse relationship between hours and rate when total pay is constant

4. Calculate percent change from the ratio

Now we need to convert this ratio into a percent change. The percent change formula in plain English is:

"How much bigger (or smaller) is the new value compared to the original value, expressed as a percentage?"

Percent change = \(\frac{\mathrm{New\ Value} - \mathrm{Original\ Value}}{\mathrm{Original\ Value}} \times 100\%\)

Substituting our rates:
Percent change = \(\frac{\mathrm{rate_2} - \mathrm{rate_1}}{\mathrm{rate_1}} \times 100\%\)
Percent change = \(\frac{\frac{19}{15} \times \mathrm{rate_1} - \mathrm{rate_1}}{\mathrm{rate_1}} \times 100\%\)
Percent change = \(\left(\frac{19}{15} - 1\right) \times 100\%\)
Percent change = \(\left(\frac{19}{15} - \frac{15}{15}\right) \times 100\%\)
Percent change = \(\frac{4}{15} \times 100\% = \frac{400}{15}\% = 26.67\%\)

This rounds to approximately 27%.

Since \(\mathrm{rate_2}\) is larger than \(\mathrm{rate_1}\), this represents an increase.

5. Final Answer

Roy's hourly pay rate increased by approximately 27% from the first week to the following week.

The answer is (E) 27% increase.

We can verify this makes sense: Roy worked about 21% fewer hours (from 38 to 30), so to maintain the same total pay, his hourly rate needed to increase by about 27%. The inverse relationship between hours and rate when total pay is constant explains this result perfectly.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint of equal total pay

Students may focus on the different hours worked (38 vs 30) without properly recognizing that the total weekly pay was identical for both weeks. This leads them to set up incorrect equations or attempt to solve for individual rates without establishing the crucial relationship that \(38 \times \mathrm{rate_1} = 30 \times \mathrm{rate_2}\).

2. Confusion about which direction the rate should change

Students might incorrectly assume that since Roy worked fewer hours in the second week, his hourly rate must have decreased as well. They fail to recognize the inverse relationship: when total pay stays constant but hours decrease, the hourly rate must increase to compensate.

Errors while executing the approach

1. Arithmetic errors when simplifying the fraction 38/30

Students may incorrectly simplify \(\frac{38}{30}\) or make calculation errors when converting to \(\frac{19}{15}\). Some might also struggle with the fraction arithmetic when calculating \(\left(\frac{19}{15} - 1\right)\) or converting \(\frac{4}{15}\) to a percentage.

2. Setting up the percent change formula incorrectly

Students often confuse which rate should be the "original" versus "new" value in the percent change formula. Since we're looking for the change from first week to second week, \(\mathrm{rate_1}\) is original and \(\mathrm{rate_2}\) is new, but students might reverse this relationship.

Errors while selecting the answer

1. Choosing "decrease" instead of "increase"

Even after calculating the correct magnitude (around 27%), students might select answer choice (A) "27% decrease" instead of (E) "27% increase" due to confusion about the direction of change or misinterpreting their calculated result.

2. Selecting the wrong percentage due to approximation confusion

Students might calculate 26.67% correctly but then select answer choice (B) "21% decrease" or (D) "21% increase" because they're unsure whether 26.67% should round to 21% or 27%, not recognizing that 27% is the much closer approximation.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient total weekly pay
Since we need the same total pay for both weeks, let's choose \$1140 as Roy's weekly pay for both weeks. This number works well because it's divisible by both 38 and 30, giving us clean hourly rates.

Step 2: Calculate the hourly rate for Week 1
Week 1: Roy worked 38 hours for \$1140
Hourly rate₁ = \(\\$1140 \div 38 = \\$30\) per hour

Step 3: Calculate the hourly rate for Week 2
Week 2: Roy worked 30 hours for \$1140
Hourly rate₂ = \(\\$1140 \div 30 = \\$38\) per hour

Step 4: Calculate the percent change
Percent change = \(\frac{\mathrm{New\ rate} - \mathrm{Old\ rate}}{\mathrm{Old\ rate}} \times 100\%\)
Percent change = \(\frac{\\$38 - \\$30}{\\$30} \times 100\%\)
Percent change = \(\frac{\\$8}{\\$30} \times 100\% = \frac{8}{30} \times 100\% = 26.67\%\)

Step 5: Match to closest answer choice
26.67% rounds to approximately 27%, which corresponds to a 27% increase.

Answer: (E) 27% increase

Note: The smart number \$1140 was strategically chosen because it's the LCM of 38 and 30, ensuring both hourly rates come out as whole numbers, making calculations cleaner and reducing arithmetic errors.