The course grade for a mathematics course consists of 60% of the final-exam grade plus 20% of each of two...

Understanding the Question

We need to determine whether Greg passed the mathematics course. Let's organize what we know:

Given Information:

• Course grade = \(\mathrm{60\% \times (final\ exam) + 20\% \times (preliminary\ exam\ 1) + 20\% \times (preliminary\ exam\ 2)}\)
• Each exam grade ranges from 0 to 100
• Passing requirement: Course grade must be at least 60

What We Need to Determine:
Did Greg's course grade reach 60 or higher? This is a yes/no question.

Remember: For a statement to be sufficient, we need to definitively answer either "Yes, Greg passed" or "No, Greg didn't pass." If we can create scenarios where Greg both passes and fails with the given information, then that information is NOT sufficient.

Analyzing Statement 1

Statement 1 tells us: Greg's final exam grade was greater than 50.

This means \(\mathrm{F > 50}\), but we know nothing about his two preliminary exam grades.

Let's test extreme scenarios to check sufficiency:

Worst Case Scenario: \(\mathrm{F = 51}\) (just above 50), and both preliminary exams = 0

• Course grade = \(\mathrm{0.6(51) + 0.2(0) + 0.2(0) = 30.6}\)
• Result: \(\mathrm{30.6 < 60}\), so Greg fails

Best Case Scenario: \(\mathrm{F = 51}\), and both preliminary exams = 100

• Course grade = \(\mathrm{0.6(51) + 0.2(100) + 0.2(100) = 30.6 + 20 + 20 = 70.6}\)
• Result: \(\mathrm{70.6 > 60}\), so Greg passes

Since we found both passing and failing scenarios, Statement 1 is NOT sufficient.

This eliminates answer choices A and D.

Analyzing Statement 2

Important: We now forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: The average of Greg's two preliminary exam grades was greater than 68.

This means \(\mathrm{\frac{P_1 + P_2}{2} > 68}\), which gives us \(\mathrm{P_1 + P_2 > 136}\).

We still don't know his final exam grade, so let's test extreme scenarios:

Low Final Exam Scenario: \(\mathrm{F = 0}\), \(\mathrm{P_1 + P_2 = 137}\) (just above 136)

• Course grade = \(\mathrm{0.6(0) + 0.2(P_1) + 0.2(P_2) = 0 + 0.2(137) = 27.4}\)
• Result: \(\mathrm{27.4 < 60}\), so Greg fails

High Final Exam Scenario: \(\mathrm{F = 100}\), \(\mathrm{P_1 + P_2 = 137}\)

• Course grade = \(\mathrm{0.6(100) + 0.2(137) = 60 + 27.4 = 87.4}\)
• Result: \(\mathrm{87.4 > 60}\), so Greg passes

Since we found both passing and failing scenarios, Statement 2 is NOT sufficient.

This eliminates answer choice B.

Combining Both Statements

Now we use both pieces of information together:

• \(\mathrm{F > 50}\) (from Statement 1)
• \(\mathrm{P_1 + P_2 > 136}\) (from Statement 2)

The critical question: What's the minimum possible course grade with both constraints?

To find the minimum, we use the smallest allowable values:

• F approaches 50 (the smallest value greater than 50)
• \(\mathrm{P_1 + P_2}\) approaches 136 (the smallest sum greater than 136)

Minimum course grade = \(\mathrm{0.6(50) + 0.2(136) = 30 + 27.2 = 57.2}\)

Since \(\mathrm{57.2 < 60}\), Greg could still fail even with both constraints.

Let's verify he could also pass with slightly higher values:

• If \(\mathrm{F = 60}\) and \(\mathrm{P_1 + P_2 = 140}\)
• Course grade = \(\mathrm{0.6(60) + 0.2(140) = 36 + 28 = 64}\)
• Result: \(\mathrm{64 > 60}\), so Greg passes

Since we can still construct both passing and failing scenarios even with both statements combined, the statements together are NOT sufficient.

This eliminates answer choice C.

The Answer: E

Neither statement alone nor both statements together provide sufficient information to determine whether Greg passed the course.

Answer Choice E: "Statements (1) and (2) TOGETHER are NOT sufficient."