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For each landscaping job that takes more than 4 hours, a certain contractor charges a total of \(\mathrm{r}\) dollars for the first 4 hours plus \(0.2 \times \mathrm{r}\) dollars for each additional hour or fraction of an hour, where \(\mathrm{r} > 100\). Did a particular landscaping job take more than 10 hours?
We have a contractor with a specific pricing structure for landscaping jobs that take more than 4 hours:
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."
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.
Statement 1: The contractor charges a total of \(\mathrm{$288}\) for the job.
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.
Let's think about what would happen if the job took exactly 10 hours:
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?
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.
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.
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:
Now here's the key insight: Each additional hour costs exactly 0.2r, so:
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.
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."