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If Bob produces \(36\) or fewer items in a week, he is paid \(\mathrm{X}\) dollars per item. If Bob produces more than \(36\) items in a week, he is paid \(\mathrm{X}\) dollars per item for the first \(36\) items and \(\frac{3}{2}\) times that amount for each additional item. How many items did Bob produce last week?
We need to find the exact number of items Bob produced last week. This is a value question - we need a single, specific number.
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.
Statement 1 tells us: Last week Bob was paid $480 total.
Let's see if different base rates (X) could lead to different production quantities:
Scenario 1: What if Bob produced 32 items (below threshold)?
Scenario 2: What if Bob produced more than 36 items?
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.
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.
We know:
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
Scenario 2: Both weeks above 36 items
Without knowing which scenario we're in or the value of X, we cannot determine n uniquely.
Statement 2 is NOT sufficient.
From both statements together:
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 \((\mathrm{n} ≤ 34)\)
Case 2: Both weeks above threshold \((\mathrm{n} > 36)\)
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.
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."