Problem Solving

Please help! Thanks.

Quick Sell Outlet sold a total of 40 televisions, each of which was either a Model P TV or A Model Q TV. Each Model P sold for $p and each model Q sold for $q. The average selling price of the 40 televisions was $141. How many of the 40 televisions were Model P Televisions?

1 - Model P sold for $30 less than the Model Q Televisions
2 - Either p = 120 or q = 120.

Total revenue for all 40 televisions = 40*141 = 5640.
Since the average price = 141, one price must be LESS than 141, while the other price must be GREATER than 141.
(Unless p=q=141, which is highly unlikely.)

Statement 1: Model P sold for $30 less than Model Q.
Thus, p< 141, while q>141.
Check the ONLY case that also satisfies statement 2:

Case 1: p=120 and q=150.
To evaluate this case, use ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.

Step 1: Plot the 3 prices on a number line, with the prices for the two models on the ends and the average price in the middle.
P 120-------------141------------150 Q

Step 2: Calculate the distances between the prices.
P 120-----21------141-----9------150 Q

Step 3: Determine the ratio in the mixture.
The required ratio of Model P televisions to Model Q televisions is equal to the RECIPROCAL of the distances in red.
P:Q = 9:21 = 3:7 = 12:28.
Thus, if 12 Model P televisions are sold for $120 each, and 28 Model Q televisions are sold for $150 each -- for a total of 40 televisions -- the total revenue will be $5640:
(12*120) + (28*150) = 5640.

Case 2: Reverse the distances from Case 1 and plot the new prices for P and Q on the ends of the number line
P 132-----9------141-----21------162 Q

In this case, P:Q = 21:9 = 7:3 = 28:12.
Thus, if 28 Model P televisions are sold for $132 each, and 12 Model Q televisions are sold for $162 each -- for a total of 40 televisions, with a price difference of $30 between the 2 models -- the total revenue will still be $5640:
(28*132) + (12*162) = 5640.

Since both cases are possible, INSUFFICIENT.

Statement 2: Either p = 120 or q = 120.
Case 1 also satisfies statement 2.

Case 3: If p and q swap positions on the number line in Case 1 -- so that q=120 and p=150 -- the result will be that 12 Model Q televisions are sold for $120 each, while 28 Model P televisions are sold for $150 each.

Since both cases are possible, INSUFFICIENT.

Statements combined:
Only Case 1 satisfies both statements, implying that 12 Model P televisions are sold for $120 each.
SUFFICIENT.

The correct answer is C.

For two similar problems, check here:

https://www.beatthegmat.com/ratios-fract ... 15365.html