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?
(1) The Model P televisions sold for $30 less than the Model Q televisions.
(2) Either p=120 or q=120.
Correct answer: C.
I was able to solve this problem but it took me a considerable amount of time and certainly not under 2 minutes. I am interested to see how the experts here will solve this problem in the shortest amount of time possible.
Thank you.
Data Sufficiency Problem
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Since the average price = 141, one price must be LESS than 141, while the other price must be GREATER than 141.TariqOmar wrote: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?
(1) The Model P televisions sold for $30 less than the Model Q televisions.
(2) Either p=120 or q=120.
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.
In this case, P=12 and Q=28, for a total of 40 televisions.
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
Here, P:Q = 21:9 = 7:3 = 28:12.
Thus, P=28 and Q=12, for a total of 40 televisions.
Since P can be different values, 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 alligation will yield the following:
Q : P = 28:12.
Thus, Q=28 and P=12, for a total of 40 televisions.
Since P can be different values, INSUFFICIENT.
Statements combined:
Only Case 1 satisfies both statements.
In Case 1, P=12.
SUFFICIENT.
The correct answer is C.
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Solution:TariqOmar wrote: ↑Wed May 09, 2018 3:26 amQuick 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?
(1) The Model P televisions sold for $30 less than the Model Q televisions.
(2) Either p=120 or q=120.
Correct answer: C.
Statement One Alone:
the Model P televisions sold for $30 less than the Model Q televisions
Suppose 20 Model P televisions were sold for $126 each and suppose 20 Model Q televisions were sold for $156 each. Then, the average selling price of 40 televisions is (20*126 + 20*156)/40 = (2,520 + 3,120)/40 = 5,640/40 = $141. In this scenario, 20 of the 40 televisions were Model P.
Suppose, on the other hand, that 28 Model P televisions were sold for $132 and 12 Model Q televisions were sold for $162. Then, the average selling price of the 40 televisions is (28*132 + 12*162)/40 = (3,696 + 1,944)/40 = 5,640/40 = $141. In this scenario, 28 of the 40 televisions were Model P.
Statement one alone is not sufficient. Eliminate answer choices A and D.
Statement Two Alone:
Either p=120 or q=120
Suppose 10 model P televisions were sold for $120 and 30 model Q televisions were sold for $148. Then, the average selling price of a television was (10*120 + 30*148)/40 = (1,200 + 4,440)/40 = 5,640/40 = $141. In this scenario, 10 model P televisions were sold.
Suppose, on the other hand, that 30 model P televisions were sold for $120 and 10 model Q televisions were sold for $204. Then, the average selling price of a television was (30*120 + 10*204)/40 = (3,600 + 2,040)/40 = 5,640/40 = $141. In this scenario, 30 model P televisions were sold.
Statement two alone is not sufficient. Eliminate answer choice B.
Statements One and Two Together:
If Model Q television sells for $120, then a Model P television must sell for 120 - 30 = $90 since statement one tells us that a Model P television sells for $30 less than a Model Q television. Then, the average selling price of a television will be between 90 and 120, which is inconsistent with the given fact that the average selling price of a television is $141. Thus, it must be true that a Model P television sells for $120 and a Model Q television sells for 120 + 30 = $150.
If we let n be the number of Model P televisions sold, then the number of Model Q televisions sold must be 40 - n. We can create the following equation:
(120*n + 150*(40 - n))/40 = 141
120n + 6,000 - 150n = 141*40
-30n = 5,640 - 6,000
-30n = -360
n = 12
So, the number of Model P televisions sold was 12. Statements one and two together are sufficient.
Answer: C
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