Five homes were recently sold in a certain town. If the average

Question

Five homes were recently sold in a certain town. If the average (arithmetic mean) sale price was $150,000 and the median sale price was $135,000, what is the least possible sale price for the most expensive home?

(A) $135,000

(B) $150,000

(C) $165,000

(D) $172,500

(E) $210,000

(A) $135,000

(B) $150,000

(C) $165,000

(D) $172,500

(E) $210,000

Bunuel · Accepted Answer

GMATNinja Video Solution:

https://www.youtube.com/watch?v=zp4axuvNEcU

Daphnee · Answer

This is my solution:
Let the sale of the 5 homes be: x1 ≤ x2 ≤ x3 ≤ x4 ≤ x5
We have: x1 + x2 + x3 + x4 + x5 = $150,000 x 5 = $750,000 (1)
and x3 = $135,000

For x5 to be min -> x1, x2, x4 need to be max
Since x1 ≤ x2 ≤ x3, the maximum possible value for both x1 and x2 is equal to the median, x3 = $135,000
and since x4 ≤ x5, the maximum possible value for x4 is equal to x5

So equation (1) become: $135,000*3 + 2*x5 = $750,000 -> x5 = $172,500 (D)
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adyanihusain · Answer

Just a question: If both x4 and x5 have the same value of $172,500, won't x5 no longer be considered "the most expensive home" since x4 has the same value?

egmat · Answer

Hi adyanihusain,

Good eye on that step in Daphnee's solution, where x4 and x5 both land at  $172,500 . Let me clear up the worry.

"The most expensive home" just means the highest sale price in the set - the maximum value.  It does not require that one home stand alone above all the others. A tie at the top is completely allowed.

Notice the way the prices were set up: x1 ≤ x2 ≤ x3 ≤ x4 ≤ x5. Those are  ≤  signs, not strict  <  signs, so two homes are free to share the same price. When x4 = x5 =  $172,500 , that  $172,500  is still the largest price anyone paid - so it  is  the price of the most expensive home. There just happen to be two homes at that top price.

Think of it this way: if you list any group of numbers and ask "what's the biggest one?", the answer is the maximum value, whether or not something else ties it. The question asks for the smallest that maximum can be, and forcing x4 up to meet x5 is exactly what pushes x5 as low as it can go.

Quick parallel to lock it in:  Three homes sell for  $100k ,  $200k ,  $200k . What's the price of the most expensive home? It's  $200k  - the tie doesn't disqualify it; the top price is still the top price.

So the logic holds, and  D ($172,500)  stands.

Answer: D

adyanihusain · Answer

Ah yes I see that logic now, thank you very much on clearing that up

Five homes were recently sold in a certain town. If the average

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Five homes were recently sold in a certain town. If the average

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