Quick Sell Outlet sold a total of 40 televisions, each of which was either a Model P television or a...

Understanding the Question

We need to find: How many of the 40 televisions were Model P televisions?

Given Information

- Total televisions sold: 40
- Each TV is either Model P (sold for \(\$\mathrm{p}\)) or Model Q (sold for \(\$\mathrm{q}\))
- Average selling price of all 40 TVs: \(\$141\)

What We Need to Determine

For sufficiency, we need enough information to find the exact number of Model P televisions sold. This is a value question - we need a specific number, not just yes/no.

Key Insight

This is a weighted average problem. The \(\$141\) average must lie between the two prices p and q. The key insight is that we can use "balance point" reasoning: if the average is closer to one price, more of that model was sold. Different price combinations could potentially give us different quantity mixes.

Analyzing Statement 1

Statement 1: The Model P televisions sold for $30 less than the Model Q televisions

This tells us that \(\mathrm{p} = \mathrm{q} - 30\), establishing a fixed \(\$30\) difference between the two prices.

Testing Different Scenarios

Let's test whether different price levels could give us different quantity mixes:

Scenario A: If \(\mathrm{q} = \$150\), then \(\mathrm{p} = \$120\)
- The average \(\$141\) is \(\$21\) above p and \(\$9\) below q
- Since \(\$141\) is much closer to \(\$150\) than to \(\$120\), more Model Q TVs must have been sold

Scenario B: If \(\mathrm{q} = \$180\), then \(\mathrm{p} = \$150\)
- The average \(\$141\) is now \(\$9\) below p and \(\$39\) below q
- Since \(\$141\) is closer to \(\$150\) than to \(\$180\), more Model P TVs must have been sold

Conclusion

Different price levels lead to different quantity mixes. Without knowing the actual prices, we cannot determine a unique answer for how many Model P televisions were sold.

Statement 1 is 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: Either p = 120 or q = 120

This gives us one concrete price, but we still don't know the other price.

Testing Different Scenarios

Case 1: If \(\mathrm{p} = \$120\) (Model P costs \(\$120\))
- We don't know q, but since the average is \(\$141\), q must be greater than \(\$141\)
- If \(\mathrm{q} = \$150\): The \(\$141\) average (closer to \(\$150\)) means more Model Q were sold
- If \(\mathrm{q} = \$200\): The \(\$141\) average (closer to \(\$120\)) means more Model P were sold
- Different values of q give different quantity mixes

Case 2: If \(\mathrm{q} = \$120\) (Model Q costs \(\$120\))
- This creates an interesting situation: the average is \(\$141\), which is above \(\mathrm{q} = \$120\)
- For the average to be higher than \(\$120\), p must be greater than \(\$141\)
- But wait - this means the cheaper Model Q costs \(\$120\) and the more expensive Model P costs over \(\$141\)
- Different values of p (say \(\$160\) or \(\$180\)) would still give different mixes

Conclusion

Knowing just one price doesn't uniquely determine the quantity mix.

Statement 2 is NOT sufficient.

This eliminates choice B.

Combining Statements

Let's see what happens when we use both statements together.

From Statement 1: \(\mathrm{p} = \mathrm{q} - 30\) (Model P costs \(\$30\) less)
From Statement 2: Either p = 120 or q = 120

Combined Analysis

Case 1: If \(\mathrm{p} = \$120\)
- Using Statement 1: \(\mathrm{q} = \mathrm{p} + 30 = \$120 + \$30 = \$150\)
- We now have concrete prices: \(\mathrm{p} = \$120\) and \(\mathrm{q} = \$150\)
- With these specific prices and the \(\$141\) average, we can determine the exact mix

Let's verify this makes sense:
- Average \(\$141\) is \(\$21\) above \(\$120\) and \(\$9\) below \(\$150\)
- Since \(\$141\) is closer to \(\$150\), more Model Q televisions were sold
- The exact calculation would show: 12 Model P and 28 Model Q televisions

[STOP - Sufficient!]

But let's check the other case to be thorough:

Case 2: If \(\mathrm{q} = \$120\)
- Using Statement 1: \(\mathrm{p} = \mathrm{q} - 30 = \$120 - \$30 = \$90\)
- This gives us prices: \(\mathrm{p} = \$90\) and \(\mathrm{q} = \$120\)
- But notice: Both prices (\(\$90\) and \(\$120\)) are BELOW the average (\(\$141\))
- This is mathematically impossible! A weighted average cannot exceed both component values

Why Together They Are Sufficient

Only Case 1 is valid, giving us unique prices of \(\mathrm{p} = \$120\) and \(\mathrm{q} = \$150\). With these specific prices and the known average of \(\$141\), we can determine exactly how many Model P televisions were sold.

The statements together are sufficient.

This eliminates choice E.

The Answer: C

Both statements together provide unique values for p and q, allowing us to determine the exact number of Model P televisions sold. Neither statement alone is sufficient.

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