The question stem specifies that the median of all houses sold was $450,000. The median of a set of numbers is the value that appears in the middle, when the numbers are arranged in increasing order. The question asks whether the average price of the houses sold was greater than $400,000. To calculate this average, you'd need to know the sum of the price of all of the houses sold divided by the number of houses sold.
For the median figure given, it's possible that an odd number of houses were sold; if so, then one of those houses sold for exactly $450,000. For example, if three houses were sold, then the middle of the three sold for exactly $450,000. However, it's also possible that an even number of houses were sold. When there are an even number of values in the set, the median is equal to the average of the two numbers in the middle. So, for example, four houses could have been sold, in which case all you'd know is that the two middle houses averaged to a price of $450,000...but the actual prices for those two houses could be almost anything.
(1) INSUFFICIENT: Knowing that one house sold for $800,000 isn't enough information to find the average.
Case 1: House 1 sold for $100,000 and House 2 sold for $800,000. The median of these two is $450,000 and the average is also $450,000. In this case, the answer to the question is Yes, the average was greater than $400,000.
Case 2: House 1 and House 2 each sold for $1, House 3 and House 4 each sold for $450,000, and House 5 sold for $800,000. The median is $450,000. The average is approximately $1,700,000 / 5, which equals something between $300,000 and $400,000. In this case, the answer to the question is No, the average was not greater than $400,000.
Because there are Yes and No answers, this statement is not sufficient to answer the question.
(2) INSUFFICIENT: If three houses were sold, then the middle house cost exactly $450,000.
Case 1: House 1 sold for $1, House 2 sold for $450,000, and House 3 also sold for $450,000. In this case, the average is less than $400,000, and the answer to the question is No.
Case 2: House 1 sold for $1, House 2 sold for $450,000, and House 3 sold for $5 million. In this case, the average is greater than $400,000, and the answer to the question is Yes.
Because there are Yes and No answers, this statement is not sufficient to answer the question.
(1) AND (2) SUFFICIENT: Exactly three houses sold and the most expensive one sold for $800,000.
Case 1: House 1 sold for $1, House 2 sold for $450,000, and House 3 sold for $800,000. In this case, the average is about $1,250,000 / 3, which equals a little more than $400,000. In this case, the answer to the question is Yes.
Case 2: The only value that you can change is House 1's price and the only thing you can do is to increase it. Since the average was already greater than $400,000 when House 1 cost just $1, the average must be greater than $400,000 no matter how much House 1 cost.
The correct answer is (C): The two statements together are sufficient to answer the question, but neither statement is sufficient by itself.
