Last year, in a certain housing development, the average (arithmetic m

The total price of all 20 house was 160,000 x 20 = $3,200,000. We want to determine whether there are more than 9 houses with prices less than the average price of $160,000.

Statement One Alone:

Last year the greatest price of one of the 20 houses was $219,000.

That means the total price of the remaining 19 houses is $3,200,000 - 219,000 = $2,981,000. It’s possible that more than 9 houses were priced less than $160,000. For example, say 14 houses were priced $150,000 each for a total of $2,100,000. Then the last 5 houses could have been priced (2,981,000 - 2,100,000)/5 = 881,000/5 = $176,200 each. However, it’s also possible that no more than 9 houses were priced less than $160,000. For example, say 9 houses were priced $150,000 each for a total of $1,350,000. Then the last 10 houses could have been priced (2,981,000 - 1,350,000)/10 = 1,631,000/10 = $163,100 each.

Statement one alone is not sufficient.

Statement Two Alone:

Last year the median of the prices of the 20 houses was $150,000.

That means the average price of the 10th and 11th houses (assuming the prices of the houses are listed in ascending order) was $150,000. It is possible that the prices of these two houses were $150,000. If that is the case, then there will be at least 11 houses with prices less than the average price of $160,000. It is also possible that the price of the 10th house is less than $150,000 and the price of the 11th house is more than $150,000. If that is the case, we still have at least 10 houses with prices less than the average price of $160,000. Therefore, in either case, we have more than 9 houses with prices less than the average price.

Statement two alone is sufficient.

Answer: B
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Questions dealing with average, mean, and std can often be solved with logic, without explicit calculations.
We'll look for such an answer, a Logical approach.

(1) Knowing only one of the 20 values isn't enough, we can, for example, use one house to balance out the $219,000 and divide the rest however we like. They could all be exactly $160,00 (answering the question with NO) or could have more than 9 be a bit less than $160,000 and the rest a bit more than $160,000) (giving YES)
Insufficient.

(2) If the median was $150,000 then either 10 or 11 houses cost less than or equal to $150,000 (depending on if the two middle values were equal or not) meaning the answer to our question must be YES - at least 9 houses had less than $160,000.

(B) is our answer.
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Solution

Given:

• We are given that the average price of 20 houses is $160,000.

To find:

• We need to find if the price of 9 of the 20 houses is less than the average price i.e $160,000 or not.

Approach and Working:

Statement 1: Last year the greatest price of one of the 20 houses was $219,000.
This does not give us any information about the pricing of remaining 19 houses. Thus, we cannot say whether 9 of the 20 houses are priced less than $160,000 or not.

Statement 2: Last year the median of the prices of the 20 houses was $150,000.
In a set of 20 numbers, the median is the 10th and 11th term or median = 10th term + 11th term/2

Thus, we can get median= $150,000 in two cases:

• When the values of both, 10th and 11th term is $150,000.
• Or, when 10th term is as less from $150,000 as 11th term is greater than $150,000.

However, in both the cases at least 10 terms will be less than $150,000.

Hence, statement 2 is sufficient to get the answer.

Answer: B
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DavidTutorexamPAL

Is it at least 9 or 10?
It seems that it should be at least 10 houses had less than $160,000.
Am I missing anything?
Thanks__
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Hello Experts,
EMPOWERgmatRichC, VeritasKarishma, IanStewart, Bunuel, chetan2u, ArvindCrackVerbal, GMATGuruNY, AaronPond, GMATinsight, ccooley, RonPurewal
The wording of the statement 1 does not make sense to me. Could you help me, please?
Which one from the following could be better version?

a) Last year the greatest price of one of the 20 houses was $219,000
VS
b) Last year the greatest price of one of the 20 houses was $219,000

Doesn't version a) mean ''there are more than 1 houses which carry highest prices''? How can more than one house be highest at a time?
Thanks__
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Consider five houses with the following prices:
{100, 150, 200, 200, 200}
Of the three prices in the set above -- 100, 150, and 200 -- 200 is the greatest.
Three of the five houses have the greatest price (200).
Thus, it is possible for more than one house to have the greatest price.
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I agree that the wording of Statement 1 is not ideal, since it seems to be talking about just one house in the group, and what the greatest price of that one house was, as if the house was sold several times at different prices and $219,000 was the highest price at which it sold. It would be better if it read "Last year the greatest price of any of the 20 houses was $219,000". But it's still clear what Statement 1 means.

I don't agree with the post above that suggests Statement 1 implies that exactly one house sold for $219,000. It simply conveys that no house sold for more than $219,000.
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Hi Asad

The language of the statement as mentioned in Question is apt.

The question clearly intends to tell you that one of the 20 houses which is marked at highest price was 219000

I think it's more of a SC doubt than Quant Doubt. just kidding.

But yeah, the language as mentioned is perfect.
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IanStewart
So, do you want to mean that ''in these 20, the two prices (19th house and 20th house prices) could be $219,000 and $219,000, respectively''?
Thanks Ian...
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Hello Asad,

Statement I is probably intended to be confusing. We are all used to the idea of the superlative being ONE value and that’s probably what this statement uses in a way to make the statement sound confusing.

My 2 cents on this - imagine that there are 20 students in a class and you say the “the greatest height of one of the students is 160cm”. This means that there is a possibility that there are more than one students who are 160cm tall. The GREATEST height is 160 cm, so there cannot be a number greater than 160 that represents the height; there can be more than one student who has this number as his/her height.

On the other hand, if you say “The tallest of the 20 students is 170 cm tall”, this means that there is ONE unique student who is 170 cm tall.

So, it really depends on what the adjective “greatest” modifies, in such questions. As GMATinsight has mentioned, there’s a small element of SC involved here that can lead to better comprehension of the statement.
Hope that helps!

Yeah, not a fan of the wording.

(1) Last year the greatest price of one of the 20 houses was $219,000

We are talking about one of the houses. Looks like the price keeps fluctuating. So the greatest price last year that it reached was $219,000.
Doesn't look like we are comparing prices of different houses.

To do that, it has to be "any of the houses".

Taking the whole question into consideration, I understand that they are talking about one price point in the year for each house. So perhaps what they mean to say is that there was a house with price = 219k and no other house had a price more than this. In any case, it is obvious that stmnt 1 is not sufficient.
Thankfully, stmnt 2 is sufficient alone.
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Hi

Could anyone please advise of an efficient way of testing statement 1? I found that it took me awfully long to come up with the bookend scenarios: 1 house priced less than average and 19 houses priced less than average. EMPOWERgmatRichC could you please advise?

Thanks

Hi silverprince,

We're told that last year, in a certain housing development, the AVERAGE (arithmetic mean) price of 20 new houses was $160,000. We're asked if MORE than 9 of the 20 houses have prices that were LESS than the average price last year. This question can be solved with a bit of Arithmetic and TESTing VALUES (although you don't have to do much math to get to the solution).

To start, since the average price of the 20 houses was $160,000, we know that the SUM of those 20 prices is (20)($160,000) = $3,200,000. We're ultimately looking to see how we can distribute that total sum to see if more than 9.... OR exactly 9 or less than 9 of the houses are below the $160,000 threshold. I want to reiterate that you DO NOT need to be doing lots of math - as long as you understand how the 20 values "relate" to the $160,000 average.

(1) Last year the greatest price of one of the 20 houses was $219,000.

With Fact 1, we know that the HIGHEST PRICE was $59,000 more than the average... but what about the other 19 prices? To bring the average down to $160,000.....

-We could 'distribute' that $59,000 overage equally to the remaining 19 houses and make ALL 19 below average in price... and the answer to the question is YES.
-We could 'distribute' that $59,000 overage equally to 8 of the 19 houses and make those 8 below average in price...and keep the other 11 AT the exact average price... and the answer to the question is NO.
Fact 1 is INSUFFICIENT

(2) Last year the median of the prices of the 20 houses was $150,000.

With Fact 2, we're dealing with the MEDIAN of the group. If we put the prices of the 20 houses in order from least to greatest, the median would be the AVERAGE of the two "middle prices" (re: the 10th and 11th prices). Since the median price is BELOW the AVERAGE price, we know that 1 OR both of the 10th and 11th prices are BELOW the average.

For example,
-IF the two prices were $150,000 and $150,000, then they are BOTH below the average
-IF the two prices were $130,000 and $170,000, then ONE price is below the average and one is above the average
-IF the two prices were $140,000 and $160,000, then ONE price is below the average and one IS the average

In ALL THREE situations, since we're dealing with the MEDIAN, we know that there are 9 other prices that are equal to (or below) the lower price of that pair (remember, we had to put the numbers in order from least to greatest before we got the median). Thus, we KNOW that those 9 prices are below the average AND we have 1 or 2 additional prices that are below the average (depending on the prices that you chose for the two 'median' houses). Thus, there will ALWAYS be either 10 or 11 houses that are below the average... so the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Thanks EMPOWERgmatRichC this is very helpful and shows that actually we don't actually need to calculate the possible prices for the bookend scenarios in statement 1, we just need to understand the logic of how the prices of the houses relate to the average.

For anyone looking at this thread, I would highly recommend searching for replies / posts by [b]EMPOWERgmatRichC[/b]. In my experience, they have consistently provided the best approach to solving problems and have taught me a completely new way of thinking about the GMAT.

For those who, like me, are super confused with Statement A, here is an approach I used that finally satisfied me:

Keeping the question in mind:
"Did more than 9 of the 20 houses have prices that were less than the average price last year"

Given the highest value that can be taken is $219k, let's take two cases:
1. Finding the maximum price of the first 8 (a number less than 9) houses (arranged in increasing order of price), and
2. Finding the maximum price of the first 10 (a number greater than 9) houses
If both these values are lower than $160k, we can be sure that Statement 1 can't get a definite answer.

Case 1:
Price of 1st 8 = x each
Price of next 12 = 219 each (a value certainly more than the average $160)
8*x+219*12=20*160
=> x = 400-219*12/8 ~400-220*12/8 ~400-330 ~70
=> x= 70 (approx)

Case 2:
Price of 1st 10 = x each
Price of next 8 = 219 each
10*x+219*8=20*160
=> x =320-219*8/10 ~320-220*8/10 ~320-176 ~154
=> x = 154 (approx)

Case 1 and 2 shows that values lower than the average ($160) can occur both less than and more than 9 times.
Hence, reject A.

I struggled hard with this 600 level question that 76% folks did correctly. Phew!

P.S.: I tried using the logic used in solution of this Official question: https://gmatclub.com/forum/last-month-1 ... 28477.html

(1) The highest value of the 20 houses is 219,000. From this information, it is tough to know other distributions of numbers. Insufficient.

(2) The median is 150,000 so surely the first 9 houses have prices less than or equal to 150,000 prices and sum of the 10th and 11th prices are 300,000. As to get median numbers are arranged in ascending order so the 10th number (price) will also be 150,000.

Option 2 is sufficient.

Ans. B
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So for statement 2, the conclusion is YES, more than 9 houses had a price less than the median?

Since the first 9 are definitely below.

10th + 11th = 300K which means at minimum an additional house costs below 160k and possibly even both of them (in the event that they are both 150K)
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Answer: Option B

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