For each landscaping job that takes more than 4 hours, a certain contractor charges a total of r dollars for...

Understanding the Question

We have a contractor with a specific pricing structure for landscaping jobs that take more than 4 hours:

• First 4 hours: r dollars (where \(\mathrm{r > 100}\))
• Each additional hour (or fraction thereof): \(\mathrm{0.2r}\) dollars

The question asks: Did a particular landscaping job take more than 10 hours?

This is a yes/no question. To be sufficient, a statement must tell us definitively either "Yes, it took more than 10 hours" or "No, it took 10 hours or less."

Key Insight

The charging structure creates a predictable pattern. For any job taking h hours (where \(\mathrm{h > 4}\)), the total charge equals \(\mathrm{r + 0.2r(h - 4)}\). This means if we know both the total charge AND the value of r, we can determine the exact hours worked.

Analyzing Statement 1

Statement 1: The contractor charges a total of \(\mathrm{$288}\) for the job.

What This Tells Us

We know the exact dollar amount charged (\(\mathrm{$288}\)), but we don't know the value of r. This creates a problem because different values of r would result in different numbers of hours worked.

Testing the Logic

Let's think about what would happen if the job took exactly 10 hours:

• Base charge for first 4 hours: r
• Additional 6 hours at 0.2r each: \(\mathrm{6 \times 0.2r = 1.2r}\)
• Total for 10 hours: \(\mathrm{r + 1.2r = 2.2r}\)

For this to equal \(\mathrm{$288}\), we'd need: \(\mathrm{r = $288 \div 2.2 \approx $131}\)

But here's the critical question: What if r has a different value?

• If \(\mathrm{r = $120}\) (which is > 100): The job would need MORE than 10 hours to reach \(\mathrm{$288}\)
• If \(\mathrm{r = $144}\) (which is > 100): The job would need FEWER than 10 hours to reach \(\mathrm{$288}\)

Since different valid values of r lead to opposite conclusions about whether the job exceeded 10 hours, we cannot answer our question definitively.

Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: The contractor charges a total of \(\mathrm{2.4r}\) dollars for the job.

The Power of Proportional Thinking

This time, the charge is expressed as a multiple of r rather than a fixed dollar amount. This changes everything!

Since the base charge for the first 4 hours is r, and the total charge is 2.4r, we can determine:

• Additional charge beyond the first 4 hours: \(\mathrm{2.4r - r = 1.4r}\)

Now here's the key insight: Each additional hour costs exactly 0.2r, so:

• Number of additional hours = \(\mathrm{1.4r \div 0.2r = 7}\) hours
• Total hours worked = \(\mathrm{4 + 7 = 11}\) hours

Why This Works

Notice that r cancels out in our calculation (\(\mathrm{1.4r \div 0.2r = 7}\)). This means no matter what specific value r takes (as long as r > 100), the job took exactly 11 hours.

Since \(\mathrm{11 > 10}\), we can definitively answer "Yes" to our question.

Statement 2 is sufficient.

[STOP - Sufficient!] This eliminates choices C and E.

The Answer: B

Statement 2 alone tells us the job took exactly 11 hours, which definitively answers our question with "Yes." Statement 1 alone leaves us uncertain because different values of r lead to different conclusions about the hours worked.

Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."