On a recent test taken by 21 students, each student's test score was between 0 and 100, inclusive. Was the...

Understanding the Question

We have 21 students who took a test, with each score between 0 and 100 inclusive. We need to determine: Was the average score greater than 75?

This is a yes/no question. We'll have sufficient information if we can definitively answer YES or definitively answer NO.

What We Need to Know

For the average to exceed 75, the sum of all 21 scores must exceed \(75 \times 21 = 1,575\). So we're really asking: Is the sum > 1,575?

Analyzing Statement 1

Statement 1: The median of the 21 test scores was 80.

With 21 scores (odd number), the median is the 11th score when arranged in ascending order. This tells us:

• 10 scores are ≤ 80
• The 11th score = 80
• 10 scores are ≥ 80

Let's test different scenarios to see if we can get different answers:

Scenario 1 - Minimum possible average:

• First 10 scores: all 0 (lowest possible)
• 11th score: 80 (the median)
• Last 10 scores: all 80 (minimum to satisfy ≥ 80)
• Sum = \(10(0) + 80 + 10(80) = 0 + 80 + 800 = 880\)
• Average = \(880/21 \approx 41.9\)
• Is 41.9 > 75? NO

Scenario 2 - Maximum possible average:

• First 10 scores: all 80 (maximum while still ≤ 80)
• 11th score: 80 (the median)
• Last 10 scores: all 100 (maximum possible)
• Sum = \(10(80) + 80 + 10(100) = 800 + 80 + 1,000 = 1,880\)
• Average = \(1,880/21 \approx 89.5\)
• Is 89.5 > 75? YES

Since we get different answers (NO in scenario 1, YES in scenario 2), Statement 1 alone 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 minimum of the 21 test scores was 70.

This means all 21 scores are at least 70. Let's test the extremes:

Scenario 1 - Minimum possible average:

• All 21 scores = 70 (everyone scored the minimum)
• Sum = \(21(70) = 1,470\)
• Average = 70
• Is 70 > 75? NO

Scenario 2 - Maximum possible average:

• 1 score = 70 (just one student at minimum)
• 20 scores = 100 (everyone else scored perfectly)
• Sum = \(70 + 20(100) = 70 + 2,000 = 2,070\)
• Average = \(2,070/21 \approx 98.6\)
• Is 98.6 > 75? YES

Again, we get different answers (NO in scenario 1, YES in scenario 2), so Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Now we use BOTH statements together:

• The median score is 80 (Statement 1)
• The minimum score is 70 (Statement 2)

This means:

• All scores are ≥ 70 (no score can be below 70)
• The 11th score = 80 (the median)
• The first 10 scores are between 70 and 80 (inclusive)
• The last 10 scores are ≥ 80

Let's find the minimum possible sum with these constraints:

• First 10 scores: 70 each (minimum allowed)
• 11th score: 80 (fixed as the median)
• Last 10 scores: 80 each (minimum for scores ≥ 80)
• Minimum sum = \(10(70) + 80 + 10(80) = 700 + 80 + 800 = 1,580\)

The minimum average = \(1,580/21 \approx 75.24\)

Since even the minimum possible average (75.24) is greater than 75, the answer must always be YES.

[STOP - Sufficient!] Together, the statements are sufficient.

The Answer: C

Both statements together guarantee that the average exceeds 75, but neither statement alone is sufficient.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."