If Bob produces 36 or fewer items in a week, he is paid X dollars per item. If Bob produces...

Understanding the Question

What We Need to Determine

We need to find the exact number of items Bob produced last week. This is a value question - we need a single, specific number.

Given Information

• If Bob produces \(≤36\) items: He earns X dollars per item
• If Bob produces \(>36\) items: He earns X dollars for the first 36 items, then 1.5X dollars for each additional item
• There's a payment threshold at 36 items where overtime kicks in

Key Insight

The payment structure creates a "jump" at 36 items. This means the same total payment could potentially come from different production levels - either fewer items at a higher base rate, or more items at a lower base rate (with overtime compensation). This insight will guide our analysis throughout.

Analyzing Statement 1

Statement 1 tells us: Last week Bob was paid $480 total.

Testing Different Scenarios

Let's see if different base rates (X) could lead to different production quantities:

Scenario 1: What if Bob produced 32 items (below threshold)?

• If X = $15 per item: \(32 × \$15 = \$480\) ✓
• This works!

Scenario 2: What if Bob produced more than 36 items?

• If X = $10 per item, with overtime at $15:
• First 36 items: \(36 × \$10 = \$360\)
• Additional items needed: \((\$480 - \$360) ÷ \$15 = 8\) items
• Total items: \(36 + 8 = 44\) items ✓
• This also works!

Conclusion

Since we found two different production quantities (32 items and 44 items) that both result in $480 payment, we cannot determine a unique answer.

Statement 1 is NOT sufficient.

Analyzing Statement 2

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

Statement 2 tells us: This week Bob produced 2 more items than last week and was paid $510.

What This Reveals

We know:

• Last week: n items (unknown)
• This week: \(\mathrm{n} + 2\) items, paid $510
• The payment increased by some amount for 2 additional items

Testing Different Scenarios

The key insight is that we don't know whether Bob is working below or above the 36-item threshold:

Scenario 1: Both weeks below 36 items

• If n = 30 last week, then 32 items this week
• Each item pays the same rate X
• We'd need \(32\mathrm{X} = \$510\), so \(\mathrm{X} ≈ \$15.94\)
• This is possible!

Scenario 2: Both weeks above 36 items

• If n = 40 last week, then 42 items this week
• Both additional items earn overtime rate \(1.5\mathrm{X}\)
• Different values of n and X could satisfy this
• This is also possible!

Without knowing which scenario we're in or the value of X, we cannot determine n uniquely.

Statement 2 is NOT sufficient.

Combining Statements

Combined Information

From both statements together:

• Last week: n items, paid $480
• This week: \(\mathrm{n} + 2\) items, paid $510
• Payment increased by exactly $30 for 2 additional items

The Critical Insight

The $30 increase for 2 items averages to $15 per item. This is crucial - this $15/item could represent:

Case 1: Both weeks below threshold \((\mathrm{n} ≤ 34)\)

• Each item pays the base rate X
• If \(\mathrm{X} = \$15\): Last week 32 items ($480), this week 34 items ($510) ✓

Case 2: Both weeks above threshold \((\mathrm{n} > 36)\)

• Additional items pay overtime rate \(1.5\mathrm{X}\)
• If overtime rate = $15, then base rate \(\mathrm{X} = \$10\)
• With \(\mathrm{X} = \$10\): We can verify that \(\mathrm{n} = 44\) items also satisfies both conditions ✓

Why Multiple Solutions Exist

The payment structure allows the same payment increase ($15/item) to occur at different production levels - either as the base rate below the threshold or as the overtime rate above it. Since we've found two different values \((\mathrm{n} = 32\) and \(\mathrm{n} = 44)\) that satisfy both statements, the combined information still doesn't give us a unique answer.

The statements together are NOT sufficient.

The Answer: E

Even with both pieces of information, we cannot uniquely determine how many items Bob produced last week because the payment structure allows multiple valid scenarios.

Answer Choice E: "The statements together are not sufficient."