Quick Sell Outlet sold a total of 40 televisions, each of which was either a Model P television or a Model Q television. Each Model P television sold for $p and each Model Q sold for $q . The average (arithmetic mean) selling price of the 40 televisions was $141 . How many of the 40 televisions were Model P televisions? the Model P televisions sold for $30 less than the Model Q televisions Either p = 120 or q = 120

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

GMAT Data Sufficiency : (DS) Questions

Source: Official Guide

Data Sufficiency

DS - Money

HARD

...

Notes

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Quick Sell Outlet sold a total of 40 televisions, each of which was either a Model P television or a Model Q television. Each Model P television sold for $\mathrm{\$p}$ and each Model Q sold for $\mathrm{\$q}$. The average (arithmetic mean) selling price of the 40 televisions was $\mathrm{\$141}$. How many of the 40 televisions were Model P televisions?

the Model P televisions sold for $\mathrm{\$30}$ less than the Model Q televisions
Either $\mathrm{p = 120}$ or $\mathrm{q = 120}$

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Solution

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."

Answer Choices Explained

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

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