At his regular hourly rate, Don had estimated the labor cost of a repair job as $336 and he was...

1. Translate the problem requirements: Don estimated a job would cost \(\$336\) based on his regular rate. The actual job took 4 hours longer than estimated, so his effective hourly rate was \(\$2\) less, but he still received \(\$336\). We need to find his original time estimate.
2. Set up the rate-time relationship: Establish equations connecting estimated time, regular rate, actual time, and actual rate using the fact that rate × time = total payment in both scenarios.
3. Create the constraint equation: Since the payment amount stayed the same despite the time and rate changes, set up an equation that reflects this constraint.
4. Solve for the estimated time: Substitute known relationships to eliminate variables and solve the resulting quadratic equation for the original time estimate.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English:

Don runs a repair business and quotes jobs based on his regular hourly rate. For this particular job:

• He estimated it would take a certain number of hours at his regular rate
• His quote was \(\$336\), and that's exactly what he got paid
• The job actually took 4 hours longer than he thought it would
• Because he took longer but got the same total payment, his effective hourly rate was \(\$2\) less than his regular rate

We need to find: How many hours did Don originally estimate the job would take?

Let's define our key terms:

• Let \(\mathrm{t}\) = estimated time (hours) - this is what we want to find
• Let \(\mathrm{r}\) = regular hourly rate (dollars per hour)
• Actual time = \(\mathrm{t} + 4\) hours
• Actual rate = \(\mathrm{r} - 2\) dollars per hour
• Total payment = \(\$336\) (same in both cases)

Process Skill: TRANSLATE - Converting the story into mathematical relationships

2. Set up the rate-time relationship

The fundamental relationship here is simple: Rate × Time = Total Payment

Let's think about this in two scenarios:

Scenario 1 (Original Estimate):
Don thought: "At my regular rate of r dollars per hour, this job will take t hours"
So: \(\mathrm{r} \times \mathrm{t} = \$336\)

Scenario 2 (What Actually Happened):
Reality: "The job took (t + 4) hours, so my effective rate was (r - 2) dollars per hour"
So: \((\mathrm{r} - 2) \times (\mathrm{t} + 4) = \$336\)

Now we have two equations that both equal \(\$336\):

• \(\mathrm{r} \times \mathrm{t} = 336\)
• \((\mathrm{r} - 2) \times (\mathrm{t} + 4) = 336\)

3. Create the constraint equation

Since both expressions equal \(\$336\), we can set them equal to each other:

\(\mathrm{r} \times \mathrm{t} = (\mathrm{r} - 2) \times (\mathrm{t} + 4)\)

From the first equation, we know that \(\mathrm{r} = \frac{336}{\mathrm{t}}\)

Let's substitute this into our constraint equation:
\(\frac{336}{\mathrm{t}} \times \mathrm{t} = \left(\frac{336}{\mathrm{t}} - 2\right) \times (\mathrm{t} + 4)\)

The left side simplifies to: \(336\)

For the right side, let's expand step by step:
\(\left(\frac{336}{\mathrm{t}} - 2\right) \times (\mathrm{t} + 4)\)
\(= \frac{336}{\mathrm{t}} \times (\mathrm{t} + 4) - 2 \times (\mathrm{t} + 4)\)
\(= \frac{336(\mathrm{t} + 4)}{\mathrm{t}} - 2(\mathrm{t} + 4)\)
\(= 336 + \frac{1344}{\mathrm{t}} - 2\mathrm{t} - 8\)
\(= 328 + \frac{1344}{\mathrm{t}} - 2\mathrm{t}\)

So our equation becomes:
\(336 = 328 + \frac{1344}{\mathrm{t}} - 2\mathrm{t}\)

4. Solve for the estimated time

Let's rearrange our equation:
\(336 = 328 + \frac{1344}{\mathrm{t}} - 2\mathrm{t}\)
\(336 - 328 = \frac{1344}{\mathrm{t}} - 2\mathrm{t}\)
\(8 = \frac{1344}{\mathrm{t}} - 2\mathrm{t}\)

To eliminate the fraction, multiply everything by t:
\(8\mathrm{t} = 1344 - 2\mathrm{t}^2\)

Rearrange to standard form:
\(2\mathrm{t}^2 + 8\mathrm{t} - 1344 = 0\)

Divide by 2 to simplify:
\(\mathrm{t}^2 + 4\mathrm{t} - 672 = 0\)

Now we need to factor this quadratic. We're looking for two numbers that multiply to -672 and add to 4.

Let's think about factors of 672: \(672 = 2^4 \times 3 \times 7 = 16 \times 42\)
Since we need a difference of 4 (because one factor is positive and one negative), we try:
\(28 \times 24 = 672\), and \(28 - 24 = 4\) ✓

So: \(\mathrm{t}^2 + 4\mathrm{t} - 672 = (\mathrm{t} + 28)(\mathrm{t} - 24) = 0\)

This gives us: \(\mathrm{t} = -28\) or \(\mathrm{t} = 24\)

Since time cannot be negative, \(\mathrm{t} = 24\) hours.

Process Skill: MANIPULATE - Using algebraic techniques to solve the quadratic equation

5. Final Answer

Don estimated the job would take 24 hours.

Let's verify: If the estimated time was 24 hours at \(\$336\) total, his regular rate was \(\$336 \div 24 = \$14\) per hour.
The actual job took \(24 + 4 = 28\) hours at a rate of \(\$14 - \$2 = \$12\) per hour.
Check: \(28 \times \$12 = \$336\) ✓

The answer is (B) 24.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the payment structure: Students often assume Don gets paid at his actual hourly rate multiplied by actual hours worked, rather than understanding he gets paid a fixed amount (\(\$336\)) regardless of how long the job actually takes. This leads to setting up incorrect equations like \((\mathrm{r}-2) \times (\mathrm{t}+4)\) = some different amount instead of \(\$336\).

2. Confusing what needs to be found: Students may set up equations to solve for Don's regular hourly rate or the actual time taken, rather than focusing on the estimated time. This confusion stems from not clearly identifying that 't = estimated time' is the target variable.

3. Incorrectly relating the rate reduction: Students might think the \(\$2\) reduction applies to the total payment rather than the hourly rate, leading them to set up equations like \(\mathrm{r} \times \mathrm{t} = 336\) and \(\mathrm{r} \times (\mathrm{t}+4) = 334\), missing the key insight that the hourly rate decreases while total payment stays the same.

Errors while executing the approach

1. Algebraic manipulation errors when expanding: When expanding \(\left(\frac{336}{\mathrm{t}} - 2\right) \times (\mathrm{t} + 4)\), students commonly make errors like forgetting to distribute properly or making sign errors. For example, they might get \(336 + \frac{1344}{\mathrm{t}} + 2\mathrm{t} + 8\) instead of \(336 + \frac{1344}{\mathrm{t}} - 2\mathrm{t} - 8\).

2. Factoring the quadratic incorrectly: Students often struggle to factor \(\mathrm{t}^2 + 4\mathrm{t} - 672 = 0\) because 672 is a large number. They may attempt to use the quadratic formula incorrectly or miss that \(672 = 28 \times 24\), leading to computational errors or giving up on the problem entirely.

3. Arithmetic errors when clearing fractions: When multiplying the equation \(8 = \frac{1344}{\mathrm{t}} - 2\mathrm{t}\) by 't' to eliminate fractions, students frequently make sign errors or forget to multiply all terms, resulting in incorrect quadratic equations like \(8\mathrm{t} = 1344 + 2\mathrm{t}^2\) instead of \(8\mathrm{t} = 1344 - 2\mathrm{t}^2\).

Errors while selecting the answer

1. Accepting the negative solution: After solving the quadratic \((\mathrm{t} + 28)(\mathrm{t} - 24) = 0\), students might not recognize that \(\mathrm{t} = -28\) is impossible since time cannot be negative, and either select a wrong answer or get confused about which solution to use.

2. Confusing estimated vs. actual time: Even after correctly solving for \(\mathrm{t} = 24\), students may mistakenly think this represents the actual time taken and add 4 hours, selecting 28 as their final answer instead of recognizing that 24 hours is the estimated time the question asks for.