Pat, Kate, and Mark charged a total of 162 hours to a certain project. If Pat charged twice as much...

1. Translate the problem requirements: We need to understand that Pat charged twice Kate's hours and 1/3 of Mark's hours, with all three totaling 162 hours. We must find how many more hours Mark charged than Kate.
2. Express all relationships in terms of one variable: Since Pat's time relates to both Kate's and Mark's, we'll use Kate's hours as our base variable and express Pat's and Mark's hours in terms of Kate's.
3. Set up the total equation: Add all three people's hours (expressed in terms of Kate's hours) to equal 162, then solve for Kate's hours.
4. Calculate individual hours and find the difference: Once we know Kate's hours, calculate Mark's hours and find how many more hours Mark charged than Kate.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

• Pat, Kate, and Mark worked on a project together
• Their combined hours total 162
• Pat charged twice as much time as Kate (if Kate worked 10 hours, Pat worked 20 hours)
• Pat charged 1/3 as much time as Mark (if Mark worked 30 hours, Pat worked 10 hours)
• We need to find how many MORE hours Mark charged than Kate

Think of it this way: if Kate is our "baseline" worker, then Pat works double what Kate works, and Mark works so much that Pat's hours are only 1/3 of Mark's hours.

Process Skill: TRANSLATE - Converting the word relationships into mathematical understanding

2. Express all relationships in terms of one variable

Since Pat's time connects to both Kate's and Mark's times, let's use Kate's hours as our starting point. We'll call Kate's hours "K".

If Kate worked K hours, then:
• Pat worked twice as much as Kate, so Pat = \(\mathrm{2K}\) hours
• Pat worked 1/3 as much as Mark, so if Pat = \(\mathrm{2K}\), then Mark = \(\mathrm{3 × (Pat's hours) = 3 × 2K = 6K}\) hours

Let's check this makes sense with a concrete example: If Kate worked 10 hours, then Pat worked 20 hours, and Mark worked 60 hours. Notice that Pat's 20 hours is indeed twice Kate's 10 hours, and Pat's 20 hours is indeed 1/3 of Mark's 60 hours. ✓

So our relationships are:
• Kate: K hours
• Pat: \(\mathrm{2K}\) hours
• Mark: \(\mathrm{6K}\) hours

3. Set up the total equation

Now we can add up all their hours to equal the total of 162:

Kate's hours + Pat's hours + Mark's hours = 162
\(\mathrm{K + 2K + 6K = 162}\)
\(\mathrm{9K = 162}\)
\(\mathrm{K = 162 ÷ 9 = 18}\)

So Kate worked 18 hours.

4. Calculate individual hours and find the difference

Now we can find everyone's actual hours:
• Kate: K = 18 hours
• Pat: \(\mathrm{2K = 2 × 18 = 36}\) hours
• Mark: \(\mathrm{6K = 6 × 18 = 108}\) hours

Let's verify: \(\mathrm{18 + 36 + 108 = 162}\) ✓

The question asks: "How many more hours did Mark charge than Kate?"
Mark's hours - Kate's hours = \(\mathrm{108 - 18 = 90}\) hours

Final Answer

Mark charged 90 more hours to the project than Kate.

The answer is D. 90

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship "Pat charged 1/3 as much time as Mark"
Many students incorrectly think this means Mark = \(\mathrm{Pat/3}\), when it actually means \(\mathrm{Pat = Mark/3}\) (so \(\mathrm{Mark = 3×Pat}\)). This fundamental misunderstanding leads to setting up Mark = \(\mathrm{2K/3}\) instead of Mark = \(\mathrm{6K}\), completely derailing the solution.

2. Confusion about which person to use as the base variable
Students might try to express everything in terms of Pat's hours or Mark's hours instead of Kate's hours. While this approach can work, it often leads to more complex fractions and increases the likelihood of computational errors during execution.

3. Missing the connection between Pat's relationships with both Kate and Mark
Students may treat the two relationships (Pat-Kate and Pat-Mark) as independent constraints rather than recognizing that Pat serves as the linking variable. This can lead to setting up systems with too many variables or incorrectly assuming the relationships are unrelated.

Errors while executing the approach

1. Arithmetic errors when calculating Mark's hours from Pat's relationship
Even if students correctly understand that \(\mathrm{Pat = Mark/3}\), they may make calculation errors when deriving Mark = \(\mathrm{6K}\). For example, incorrectly calculating \(\mathrm{3 × 2K = 5K}\) or \(\mathrm{3 × 2K = 8K}\) due to rushed mental math.

2. Errors in combining like terms in the total equation
When setting up \(\mathrm{K + 2K + 6K = 162}\), students might incorrectly add the coefficients (getting \(\mathrm{8K = 162}\) or \(\mathrm{10K = 162}\) instead of \(\mathrm{9K = 162}\)), leading to wrong values for all individual hours.

3. Division errors when solving for K
Students may incorrectly calculate 162 ÷ 9, potentially getting 16 or 20 instead of 18, which cascades into wrong values for all three people's hours.

Errors while selecting the answer

1. Selecting Mark's total hours instead of the difference
After correctly calculating Mark = 108 hours, students might select answer choice E (108) because they focus on Mark's hours rather than carefully reading that the question asks for "how many MORE hours" Mark charged than Kate.

2. Calculating Kate's hours minus Mark's hours instead of Mark minus Kate
Students might compute \(\mathrm{18 - 108 = -90}\) and then either select a wrong answer or get confused by the negative result, not realizing they should calculate \(\mathrm{108 - 18 = 90}\).