Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000
III. At least one of the homes was sold for less than $130,000.

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

This question is from the GMAT prep tests 1 and 2 so if you haven't done those yet skip this one for now. You're given average and median and with those two pieces of information need to make some inferences. Know your basic GMAT content. There shouldn't be any confusion about fundamental statistics terms, average, median, mean, and standard deviation. For standard deviation you don't need to know the formula or how to calculate it but should have a strong understanding of what it measures.

OK, back to the question. What does it mean that the median sale prices was $130,000? Well, median is a spatial measurement. It's a location: the middle of the set. For an odd set it's just the middle number (in this case the 8th). For an even set it is the average of the two middle numbers. OK, so if you have a median of $130,000 in a 15 number set then it must be that 7 numbers are equal to or below $130,000 and 7 numbers are equal to or above above $130,000. With that information alone none of the statements MUST be true. They all COULD be true.

I. At least one of the homes was sold for more than $165,000. Sure. All of the houses above the median could have been sold for $1.2 million. Or not. They could all have been sold for $140,000. There's nothing limiting this.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000 Same idea as statement one. Easy to have home in that range or not.
III. At least one of the homes was sold for less than $130,000. All of the homes could have been sold at the median or you could have 7 houses below and 7 above. There a variety of ways to make this true or not true.

The average sale price being $150,000 also doesn't allow us to commit to any of the statements. They still all COULD be true.

I. At least one of the homes was sold for more than $165,000. With the mean at $150,000 we know that the balance of things ends up at $150,000 but how we get there is completely unknown. You could have 14 houses sell for $1 and the 15 house sell for way above $150,000 so the average balances at $150,000. You could have all 15 houses sell for $150,000. Again, there many ways to do this.

II. At least one of the homes was sold for more than $130,0000 and less than $150,000 Same idea as statement one. Easy to have home in that range or not.
III. At least one of the homes was sold for less than $130,000. All of the homes could have been sold at the average ($150,000) or you could have a home sold for less than $130,000 with at least one of the other houses above the average to balance out to the average.

 

Putting the mean and median together it becomes clear that at least some of houses need to be above $130,000 in order to create a mean of $150,000. That puts a shadow over statement III. It doesn't seem that we any limitations on the downside. Our main concern is having enough $'s above the median to balance things out to $150,000. Statement II has the same issue. There's nothing to limit the sales prices that are above the mean to the range of $130,000 and $150,000. With that we can pick A.

But how do we prove that at least one of the homes was sold for more than $165,000? Let's say we maximize the value of the houses to the left of the median. To do that we'd make them all equal the median at $130,000. If you have 7 houses at $130,000 what do the 7 houses above the median have to be to balance to a mean of $150,000? There's an equal number of houses (7 and 7) so just think of it as one house at $130,000. What does the other house have to be: $170,000. Add those and divide by two and you get $150,000. But, you might say, what about the median? We didn't count that. No, we didn't. But will the median at $130,000 make it so the $170,000 is higher or lower? We're adding more weight to the $130,000 side so the $170,000 would need to be increased to compensate. At $170,000 we're already clearing the $165,000 line from statement I so adding the median just pushes us further in that direction. Again, A MUST be true.

GMAT Question of the Day - Problem Solving - Average/Word Problem

For the past 300 days machine X has produced an average of 78 units of part K. Today machine X produced Z units of part K bringing machine X's total production of part K to 23,468 units. What is the value of Z?

A. 34

B. 36

C. 68

D. 72

E. 74

Answer Show

GMAT Question of the Day Solution

GMAT weighted average questions are very similar to GMAT Rate/Work questions. For both question types you can use a T to organize the information. The T will help you make inferences. It can also help you to define the question. In a classic GMAT weighted average question you will have two scenarios and a total. You can organize those scenarios into three T's. Once you've done that this question of the day breaks down to simple arithmetic. Here's an Official GMAT question on which we also use the T method to organize the information: A boat traveled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then traveled the same distance downstream at an average speed of (v+3) miles per hour

GMAT Question of the Day Problem Solving Average Word Problem Solution

More GMAT Average Rate Practice Questions:

Distance, Work and Rate - Average Rate 1

Distance, Work and Rate - Average Rate 2

For more GMAT distance, work, and rate practice visit GMAT Question of the Day.

GMAT Question of the Day - Problem Solving - Word Problem/Ratio/Algebra

Alice and Carl each completed a portion of a certain job. Alice completed 20% of the job while Carl completed the rest. If they were each paid the same amount, what proportion of her payment must Alice give to Carl so that they each are paid the amount of the total that is proportionate with the amount of work that they each performed?

A. 1/2

B. 2/3

C. 3/5

D. 3/4

E. 5/8

Answer Show

GMAT Question of the Day Solution

The GMAT Quant section has a whole bunch of word problems so it's important to feel comfortable with them. These questions tend to have a tough image but are often simple in concept (this question of the day has this feel). One thing that seems to help GMAT tutoring students: Taking time to carefully read the question. You do not need to solve the question as you read. You don't even have to take notes. The important thing is to process the information. To understand what the question is asking you to do. This is the time to make a broad plan for solving the question.

In this case Alice will need to give Carl part of her payment so that the pay per work is equal. Sounds like we can set up an equation with Carl on one side and Alice on the other. Pick a variable for the original payment (let's call it x) and pick a variable for the amount that Alice needs to lose from her side and that Carl will gain (let's call this y). Solve for y in terms of x and that will tell you the proportion of her payment that Alice needed to give to Carl.

You might be thinking: how do I deal with the different proportions of the job done by Alice and Carl? Well - look at it this way. We're dealing with 20% and 80% (.2 and .8). So the ratio is 1:4. Carl did 4 units of work to Alice's 1 unit of work. So Carl's money divided by his effort of 4 units must be equal to Alice's money divided by her effort of one unit.

GMAT Question of the Day Word Problem Algebra Solution Diagram

GMAT Question of the Day - Data Sufficiency - Ratio/Weighted Average

The ratio of the number of students in the math department, history department, and science department is 3 to 7 to 10 respectively. Is the average height of students from all of the departments less than 60 inches?

(1) The sum of the heights of all the students is less than 100 feet.

(2) This math department has 7 fewer students than the science department.

Answer Show

GMAT Question of the Day Solution

This GMAT question of the day presents another threshold question. In this case we want to know whether we are above or below 60 inches. From the given information we know that the minimum number of students is 20 (3 + 7 + 10).

Statement (1) If we divide max height (100 feet or 1,200 inches) by min students (20) we get 60. That's right at the threshold. Because we have an inequality we must be below this threshold. Sufficient.

Statement (2) From this we can infer the number of students but have no information on their heights. Insufficient.

GMAT Question of the Day - Data Sufficiency - Weighted Average/Ratios

Each Doctor at a certain medical practice is either a specialist or a generalist. What is the ratio of generalists to specialists?

1) The average (arithmetic mean) number of procedures performed per year by specialists is 50 less than the average (arithmetic mean) number of procedures performed per year by generalists.

2) The average (arithmetic mean) number of procedures performed by generalists is 40 more than the average (arithmetic mean) number of procedures performed by all of the doctors at the practice.

Answer Show

GMAT Question of the Day Solution:

The question is asking about ratios/proportions so think about picking numbers. It should be clear that each piece of information on it's own is insufficient. There are too many moving parts. Try a few different sets of numbers to prove this to yourself. Putting things together the results are locked into a ratio of 1:4. Again, pick numbers for this. You know that there must be more specialists than generalists so assume that there's only one generalist and that there are x specialists. Also from statement 1 you know that the average for generalists is 50 more than the average for specialists. So you can pick numbers to represent this. I chose 150 for the generalists and 100 for the specialists. Set up the equation and solve for x.

GMAT Question of the Day Data Sufficiency Average/Ration Solution

 

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