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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. # A set has exactly five consecutive positive integers.  Question banks Downloads My Bookmarks Reviews Important topics
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A set has exactly five consecutive positive integers. [#permalink]
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Question Stats: 44% (01:29) correct 55% (01:25) wrong based on 88 sessions

A set has exactly five consecutive positive integers.

 Quantity A Quantity B The percentage decrease in the average of the numbers when one of the numbers is dropped from the set 20%

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Re: A set has exactly five consecutive positive integers. [#permalink]
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Carcass wrote:
A set has exactly five consecutive positive integers.

 Quantity A Quantity B The percentage decrease in theaverage of the numbers whenone of the numbers is droppedfrom the set 20%

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

Here

let's take the numbers from 1 to 5

So the average of the numbers = $$\frac{15}{5} =3$$

Now if we remove a number what will be the average? to get that we need to take the greatest number from the series

So in this case , 5 is the greatest

So average of 1 ,2 ,3 ,4 =$$\frac{10}{4}= 2.5$$

Therefore the percentage decrease = $$\frac{{3 - 2.5}}{3} * 100 = 16.66%$$, this the maximum value we can get if we remove from the set.

Hence QTY A < QTY B

*** it can be any 5 consecutive positive numbers the maximum value will be 16.66%****
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Re: A set has exactly five consecutive positive integers. [#permalink]
tricky wording. Only the decreases are issues .
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Re: A set has exactly five consecutive positive integers. [#permalink]
How do we know which number is to be excluded ? Ans can change depending on the number we exclude

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Re: A set has exactly five consecutive positive integers. [#permalink]
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Expert's post
Carcass wrote:

A set has exactly five consecutive positive integers.

 Quantity A Quantity B The percentage decrease in the average of the numbers when one of the numbers is dropped from the set 20%

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

Here is an algebraic solution that allows us to examine all possible cases.

Let x = the smallest integer in the set
So, x + 1 = the next consecutive integer
x + 2 = the next integer
x + 3 = the next integer
x + 4 = the greatest integer

Average $$= \frac{x+(x+1)+(x+2)+(x+3)+(x+4)}{5}=\frac{5x+10}{5}=x+2$$

Key concept: In order to get the greatest DECREASE in the average, we must remove the biggest number in the set.
So we'll remove (x+4) from the set
The new average $$= \frac{x+(x+1)+(x+2)+(x+3)}{4}=\frac{4x+6}{4}=x+1.5$$

Percent decrease = (100)(old - new)/old

We get: percent decrease $$= \frac{(100)[(x+2)-(x+1.5)]}{x+2}$$

$$= \frac{(100)[0.5]}{x+2}$$

$$= \frac{50}{x+2}$$

Notice that the percent increase depends on the value of x.
For example, if $$x=8$$, then the percent decrease $$= \frac{50}{8+2}= \frac{50}{10}=5%$$, which is LESS THAN 20%

Also notice that, in order to maximize the percent decrease, we must minimize the value of x.
Since we're told x is a positive integer, the smallest possible value of x is 1.
When $$x=1$$, then the percent decrease $$= \frac{50}{1+2}= \frac{50}{3}≈16.666...%$$
This means 16.666...% is the GREATEST possible value of Quantity A.

This means Quantity B will always be greater than Quantity A

Cheers,
Brent
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Re: A set has exactly five consecutive positive integers. [#permalink]
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Expert's post
Alpha14 wrote:
How do we know which number is to be excluded ? Ans can change depending on the number we exclude

@GreenlightTestPrep

Great question!
Notice that, if we remove the middle number, then the percent decrease in the average is zero.
So, in this case, Quantity B will be greater.
At this point our goal should be to MAXIMIZE the percent decrease. This is achieved by removing the biggest number in the set.
As I show in my post above, removing the biggest number in the set will always yield a percent decrease that is less than 20%.

Cheers,
Brent
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Re: A set has exactly five consecutive positive integers. [#permalink]
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Carcass wrote:

A set has exactly five consecutive positive integers.

 Quantity A Quantity B The percentage decrease in the average of the numbers when one of the numbers is dropped from the set 20%

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

Let the 5 numbers be: $$x, x+1, x+2, x+3, x+4$$

Since the numbers are in Arithmetic Progression (i.e. consecutive numbers have a constant gap), we have:
Mean = Median = 3rd term = $$x+2$$

(Note: You can always add up the terms and check the average)

One of the 5 numbers will be dropped

Maximum change in mean will occur if either of the 2 extreme terms, i.e. $$x$$ or $$x+4$$ is dropped. The average will decrease if $$x+4$$ is dropped (Note: if $$x$$ is dropped, the average would actually increase. Also, if the middle number, i.e. $$x+2$$ is dropped, there will be no change in the mean)

If $$x+4$$ is dropped: The 4 terms are: $$x, x+1, x+2, x+3$$

=> New Mean = Median = $$[(x+1)+(x+2)]/2 = x+1.5$$

=> Percent decrease in mean = $$[{(x+2)-(x+1.5)}/(x+2)] * 100$$ = $$[50/(x+2)]%$$

The above percent will be maximum if the value of $$x$$ is minimum, i.e. $$x=1$$

=> Maximum percent decrease = $$[50/(1+2)]% = 16.67%$$

Thus, Quantity B is greater than Quantity A

Note: Some important results that come up here:

In an Arithmetic Progression i.e. consecutive terms having a constant difference of $$d$$:

#1. Mean = Median
#2. The average remains unchanged if the middle term (or both middle terms) are removed
#3. The maximum change in mean occurs when one of the extreme terms is removed

#4. The maximum change in mean (when one of the extreme terms is removed) = $$d/2$$, where $$d$$ is the constant difference between consecutive terms
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_________ Re: A set has exactly five consecutive positive integers.   [#permalink] 28 Jan 2020, 09:55
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