A company makes bolts of only two sizes: small and large. Each small bolt that the company made last month...

Understanding the Question

We need to determine whether the company made more than 5,000 bolts last month.

Given Information

• Company makes only two types: small bolts and large bolts
• Small bolts: materials cost ≥ \(\mathrm{\$0.05}\), selling price = \(\mathrm{\$0.10}\)
• Large bolts: materials cost ≥ \(\mathrm{\$0.07}\), selling price = \(\mathrm{\$0.25}\)
• Let's call the number of small bolts "s" and large bolts "l"

What We Need to Determine

Is \(\mathrm{s + l > 5,000}\)?

Since this is a yes/no question, we need sufficient information to answer either "Yes, definitely more than 5,000" or "No, definitely not more than 5,000."

Key Strategic Insight

Notice that we're dealing with two opposing constraints:

• Material costs create a ceiling (you can't spend more than you have)
• Revenue requirements create a floor (you must sell enough to meet targets)

This tension between upper and lower limits will guide our entire analysis.

Analyzing Statement 1

Statement 1: The bolts used less than $500 worth of materials.

This creates a ceiling on production. Since each bolt uses materials, there's a maximum number we can produce with limited materials.

Testing the Boundaries

Let's check if we can produce MORE than 5,000 bolts:

• All small bolts strategy (uses least materials per bolt):
  • \(\mathrm{8,000 \text{ small bolts} × \$0.05 = \$400}\) in materials
  • \(\mathrm{\$400 < \$500}\) ✓ (within budget)
  • Total bolts: \(\mathrm{8,000 > 5,000}\)

Now let's check if we could produce FEWER than 5,000 bolts:

• Mixed production example:
  • 2,000 small bolts: \(\mathrm{2,000 × \$0.05 = \$100}\)
  • 1,000 large bolts: \(\mathrm{1,000 × \$0.07 = \$70}\)
  • Total materials: \(\mathrm{\$170 < \$500}\) ✓ (within budget)
  • Total bolts: \(\mathrm{3,000 < 5,000}\)

Conclusion for Statement 1

Since we found scenarios both above 5,000 (8,000 bolts) and below 5,000 (3,000 bolts), we cannot determine a definitive answer.

Statement 1 is NOT sufficient.

[STOP - Not Sufficient! Eliminate choices A and D]

Analyzing Statement 2

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

Statement 2: The sum of selling prices exceeded $1,300.

This creates a floor on production - we must have sold enough bolts to generate more than $1,300 in revenue.

Strategic Reasoning

To minimize the number of bolts while still hitting our revenue target, we should maximize the production of large bolts (they generate $0.25 each versus only $0.10 for small bolts).

Testing the extreme case - What if we made ONLY large bolts?

• Each large bolt sells for \(\mathrm{\$0.25}\)
• To exceed $1,300: We need more than \(\mathrm{\$1,300 ÷ \$0.25 = 5,200}\) large bolts

This is our key insight: Even in the most revenue-efficient scenario (100% large bolts), we need MORE than 5,200 bolts.

Why This Proves Sufficiency

Any other production mix would include small bolts, which generate less revenue per unit. Therefore:

• If we made some small bolts, we'd need even MORE total bolts to reach $1,300
• No matter what combination we choose, we must have made more than 5,000 bolts

Conclusion for Statement 2

The revenue requirement forces production above 5,000 bolts in ALL cases.

Statement 2 is sufficient.

[STOP - Sufficient! The answer is B]

The Answer: B

Statement 2 alone tells us the company definitely made more than 5,000 bolts, while Statement 1 leaves us uncertain.

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

Key Takeaway for Students

When facing constraints that create upper limits (like budget caps) versus lower limits (like minimum revenue), test the extreme cases first. This often reveals sufficiency without complex calculations.