The mean of (54,820)^2 and (54,822)^2 = : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 25 Feb 2017, 05:03

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# The mean of (54,820)^2 and (54,822)^2 =

Author Message
TAGS:

### Hide Tags

Director
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 536
Location: United Kingdom
GMAT 1: 730 Q49 V45
GPA: 2.9
WE: Information Technology (Consulting)
Followers: 75

Kudos [?]: 3070 [1] , given: 217

The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

17 Nov 2011, 15:45
1
KUDOS
20
This post was
BOOKMARKED
00:00

Difficulty:

45% (medium)

Question Stats:

63% (02:19) correct 37% (01:15) wrong based on 438 sessions

### HideShow timer Statistics

The mean of (54,820)^2 and (54,822)^2 =

(A) (54,821)^2
(B) (54,821.5)^2
(C) (54,820.5)^2
(D) (54,821)^2 + 1
(E) (54,821)^2 – 1
[Reveal] Spoiler: OA

_________________

Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

Manager
Joined: 31 May 2011
Posts: 88
Location: India
GMAT Date: 12-07-2011
GPA: 3.22
WE: Information Technology (Computer Software)
Followers: 1

Kudos [?]: 51 [3] , given: 4

Re: Mean of 2 numbers [#permalink]

### Show Tags

17 Nov 2011, 20:53
3
KUDOS
mean of (54,820)^2 and (54,822)^2

=> [(54,820)^2 +(54,822)^2]/2 (using a^2 + b^2 = (a-b)^2 + 2ab)
=> [(54,82-54,820)^2 + 2*54,820*54,822]/2
=> [2^2 + 2*54,820*54,822]/2
=> [2 + 54,820*54,822]
=> [2 + (54,821-1)*(54,821+1)] (using (a+b)(a-b) = a^2-b^2)
=> [2 + 54,821^2 - 1]
=> 54,821^2 +1

and D
Intern
Joined: 24 Jan 2011
Posts: 6
Followers: 0

Kudos [?]: 2 [1] , given: 1

Re: Mean of 2 numbers [#permalink]

### Show Tags

10 Sep 2012, 22:53
1
KUDOS
(54820)^2 + (54822)^2 = (54821-1)^2 + (54821+1)^2
= 2 (54821)^2 + 2

So mean = (54821)^2 + 1
(D)

-----------------------
Thanks
Prashant
Math Expert
Joined: 02 Sep 2009
Posts: 37113
Followers: 7254

Kudos [?]: 96583 [6] , given: 10770

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

11 Sep 2012, 01:02
6
KUDOS
Expert's post
10
This post was
BOOKMARKED
enigma123 wrote:
The mean of (54,820)^2 and (54,822)^2 =

(A) (54,821)^2
(B) (54,821.5)^2
(C) (54,820.5)^2
(D) (54,821)^2 + 1
(E) (54,821)^2 – 1

APPROACH #1:

Test some small numbers: $$\frac{2^2+4^2}{2}=10=3^2+1$$ or: $$\frac{4^2+6^2}{2}=26=5^2+1$$.

APPROACH #2:

Say $$54,821=x$$, then $$\frac{54,820^2+54,822^2}{2}=\frac{(x-1)^2+(x+1)^2}{2}=x^2+1=54,821^2+1$$.

APPROACH #3:

The units digit of $$54,820^2+54,822^2$$ is $$0+2=4$$. Now, since $$54,820^2+54,822^2$$ must be a multiple of 4, then $$\frac{54,820^2+54,822^2}{2}$$ must have the units digit of 2. Only answer choice D fits.

_________________
Senior Manager
Joined: 13 Aug 2012
Posts: 464
Concentration: Marketing, Finance
GMAT 1: Q V0
GPA: 3.23
Followers: 26

Kudos [?]: 443 [2] , given: 11

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

10 Dec 2012, 02:04
2
KUDOS
Shrink the monster technique.
The two numbers are two consecutive even numbers raised to 2. Shrink the monster and think of baby numbers such as 0 and 2.

$$mean=\frac{{0^2 + 2^2}}{2}=2$$

Substitute you baby numbers to the answer choices.

(A) $${0+1}^{2}=1$$ Eliminate!
(B) $${0+1.5}^{2}=2.25$$ Eliminate!
(C) $${0+0.5}^{2}=0.25$$ Eliminate!
(D) $${0+1}^{2}+1=2$$ Bingo!
(E) $${0+1}^{2}-1=0$$ Eliminate!

_________________

Impossible is nothing to God.

Intern
Joined: 24 Apr 2012
Posts: 48
Followers: 0

Kudos [?]: 22 [2] , given: 1

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

10 Dec 2012, 03:42
2
KUDOS
Ans: Replace 54820 by x and 54822 by (x+2) and then solve by algebra. Mean of x^2 and (x+2)^2 is x^2+2x+2 which can be written as x^2+2x+1+1. So the answer is (D)
_________________

www.mnemoniceducation.com

Math Expert
Joined: 02 Sep 2009
Posts: 37113
Followers: 7254

Kudos [?]: 96583 [0], given: 10770

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

03 Jul 2013, 00:23
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Algebra: algebra-101576.html

DS Algebra Questions to practice: search.php?search_id=tag&tag_id=29
PS Algebra Questions to practice: search.php?search_id=tag&tag_id=50

Special algebra set: new-algebra-set-149349.html

_________________
Current Student
Joined: 24 Nov 2012
Posts: 177
Concentration: Sustainability, Entrepreneurship
GMAT 1: 770 Q50 V44
WE: Business Development (Internet and New Media)
Followers: 44

Kudos [?]: 286 [0], given: 73

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

04 Jul 2013, 02:58
Let 54820 = x

Equations reduces to

{x^2 + (x+2)^2}/2 = (x^2 + x^2 + 4x + 4)/2 = x^2+2x+2 = x^2 + 2x + 1 + 1 = (X+1)^2 + 1 = 54821^2 + 1
_________________

You've been walking the ocean's edge, holding up your robes to keep them dry. You must dive naked under, and deeper under, a thousand times deeper! - Rumi

http://www.manhattangmat.com/blog/index.php/author/cbermanmanhattanprep-com/ - This is worth its weight in gold

Economist GMAT Test - 730, Q50, V41 Aug 9th, 2013
Manhattan GMAT Test - 670, Q45, V36 Aug 11th, 2013
Manhattan GMAT Test - 680, Q47, V36 Aug 17th, 2013
GmatPrep CAT 1 - 770, Q50, V44 Aug 24th, 2013
Manhattan GMAT Test - 690, Q45, V39 Aug 30th, 2013
Manhattan GMAT Test - 710, Q48, V39 Sep 13th, 2013
GmatPrep CAT 2 - 740, Q49, V41 Oct 6th, 2013

GMAT - 770, Q50, V44, Oct 7th, 2013
My Debrief - http://gmatclub.com/forum/from-the-ashes-thou-shall-rise-770-q-50-v-44-awa-5-ir-162299.html#p1284542

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13960
Followers: 591

Kudos [?]: 167 [0], given: 0

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

24 Jul 2014, 06:56
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Manager
Joined: 27 May 2014
Posts: 85
Followers: 0

Kudos [?]: 19 [0], given: 21

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

24 Jul 2014, 19:13
Bunuel how do you know the expression is a multiple of 4? Wouldn't I need the tens and unit digit? Not just units?
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1858
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Followers: 49

Kudos [?]: 1989 [1] , given: 193

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

24 Jul 2014, 19:40
1
KUDOS
$$\frac{54820^2 + 54822^2}{2}$$

$$= \frac{54820^2 + (54820 + 2)^2}{2}$$

$$= \frac{54820^2 + 54820^2 + 2 * 54820 * 2 + 2 * 2}{2}$$

$$= 54820^2 + 2 * 54820 + 1 + 1$$

$$= (54820 + 1)^2 + 1$$

$$= 54821^2 + 1$$

_________________

Kindly press "+1 Kudos" to appreciate

Math Expert
Joined: 02 Sep 2009
Posts: 37113
Followers: 7254

Kudos [?]: 96583 [0], given: 10770

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

24 Jul 2014, 23:30
bankerboy30 wrote:
Bunuel how do you know the expression is a multiple of 4? Wouldn't I need the tens and unit digit? Not just units?

54,820^2 + 54,822^2 = even^2 + even^2 = multiple of 4 + multiple of 4 = multiple of 4.

Hope it's clear.
_________________
Intern
Joined: 27 Mar 2013
Posts: 4
Followers: 0

Kudos [?]: 6 [0], given: 209

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

26 Jul 2014, 11:42
enigma123 wrote:
The mean of (54,820)^2 and (54,822)^2 =

(A) (54,821)^2
(B) (54,821.5)^2
(C) (54,820.5)^2
(D) (54,821)^2 + 1
(E) (54,821)^2 – 1

So the question is avg of (x-1)^2 and (x+1)^2. Solve it and you get x^2 + 1. x here is 54,821 so the answer is option D.
Intern
Joined: 09 Jun 2014
Posts: 5
Followers: 0

Kudos [?]: 0 [0], given: 38

The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

29 Sep 2014, 21:33
Let a = 54820
Then, we need to find
$$\frac{a^2 + (a + 2)^2}{2}$$ => $$\frac{a^2 + a^2 + 4a + 4}{2}$$

$$\frac{2a^2+ 4a + 4}{2}$$ => $$\frac{2(a^2 + 2a + 2)}{2}$$

$$a^2 + 2a + 1 + 1$$ => $$(a + 1)^2 + 1$$

Substitute for a

$$(54820 + 1)^2 + 1$$ => $$(54821)^2 + 1$$

Intern
Joined: 09 Jun 2014
Posts: 5
Followers: 0

Kudos [?]: 0 [0], given: 38

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

29 Sep 2014, 21:52
enigma123 wrote:
The mean of (54,820)^2 and (54,822)^2 =

(A) (54,821)^2
(B) (54,821.5)^2
(C) (54,820.5)^2
(D) (54,821)^2 + 1
(E) (54,821)^2 – 1

Let a = 54820
Then, we need to find
$$\frac{a^2 + (a + 2)^2}{2}$$ => $$\frac{a^2 + a^2 + 4a + 4}{2}$$

$$\frac{2a^2+ 4a + 4}{2}$$ => $$\frac{2(a^2 + 2a + 2)}{2}$$

$$a^2 + 2a + 1 + 1$$ => $$(a + 1)^2 + 1$$

Substitute for a

$$(54820 + 1)^2 + 1$$ => $$(54821)^2 + 1$$

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13960
Followers: 591

Kudos [?]: 167 [0], given: 0

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

02 Oct 2015, 09:14
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Intern
Joined: 05 Jun 2015
Posts: 26
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
Followers: 0

Kudos [?]: 6 [0], given: 444

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

14 Mar 2016, 01:04
Bunuel wrote:

APPROACH #3:

The units digit of $$54,820^2+54,822^2$$ is $$0+2=4$$. Now, since $$54,820^2+54,822^2$$ must be a multiple of 4, then $$\frac{54,820^2+54,822^2}{2}$$ must have the units digit of 2. Only answer choice D fits.

Hi,

Concerning last digits, I only know that we can add or multiply last digits of various numbers to determine the last digits of their sum or product. Can somebody tell me the rule for division of last digits (as presented by Bunuel)?

Another question: when and only when a multiple of 4 is divided by 2, the last digit of the resulting quotient will always be 2? Is there any other rules to memorize?

Thank you very much!
Verbal Forum Moderator
Joined: 02 Aug 2009
Posts: 4367
Followers: 357

Kudos [?]: 3686 [1] , given: 106

The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

14 Mar 2016, 01:14
1
KUDOS
Expert's post
truongynhi wrote:
Bunuel wrote:

APPROACH #3:

The units digit of $$54,820^2+54,822^2$$ is $$0+2=4$$. Now, since $$54,820^2+54,822^2$$ must be a multiple of 4, then $$\frac{54,820^2+54,822^2}{2}$$ must have the units digit of 2. Only answer choice D fits.

Hi truongynhi

Concerning last digits, I only know that we can add or multiply last digits of various numbers to determine the last digits of their sum or product. Can somebody tell me the rule for division of last digits (as presented by Bunuel)?

Quote:
Another question: when and only when a multiple of 4 is divided by 2, the last digit of the resulting quotient will always be 2? Is there any other rules to memorize?

Thank you very much!

Hi,

Quote:
Another question: when and only when a multiple of 4 is divided by 2, the last digit of the resulting quotient will always be 2? Is there any other rules to memorize?

This is not CORRECT..
In the given context, last digit 4 and multiple of 4 are interlinked, so just stating one that multiple of 4 when divided by 2 gives 2 will be wrong

reason-

-
8 is a multiple of 4 and will give last digit as 4 and NOT 2..

what is meant here is
we have seen that the last digit of SUM is 4 and this SUM is multiple of 4..
A number Multiple of 4, with last digit 4 is 04,24, 44,64,84 as last two digits, so when div by 2, in each case last digit is 2..
the number cannot be ending in 14,34 etc as Property of 4 is last two digits should be div by 4 if the entire number is to be div by 4..

_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 1926
Followers: 59

Kudos [?]: 426 [0], given: 477

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

10 Dec 2016, 03:03
Here is my solution to this one -->

Let 54822=t
Hence sum of the two numbers becomes => (t-2)^2+t^2 = 2t^2+4-4t
Hence mean => t^2+2-2t => t^2-2t+1+1 => (t-1)^2+1 = 54821^2 +1

Hence D

Great Official Question.

_________________

Give me a hell yeah ...!!!!!

Manager
Joined: 05 Dec 2016
Posts: 60
Followers: 0

Kudos [?]: 1 [0], given: 5

Re: The mean of (54,820)^2 and (54,822)^2 = [#permalink]

### Show Tags

20 Dec 2016, 05:54
One more cyclicity question.
Fisrt, we should find out the unit digit of the mean of 54820^2 and 54822^2.
To figure it out we need find out last digits of both numbers. Last number of 54820^2 will be 0, last number of 54822^2 will be 4.
Mean will be (0+4)/2 = 2, thus unit digit of the mean number will be 2.
Analyzing all options, it is clear that only one option gives us this value, Option D 54821^2+1
Re: The mean of (54,820)^2 and (54,822)^2 =   [#permalink] 20 Dec 2016, 05:54

Go to page    1   2    Next  [ 21 posts ]

Similar topics Replies Last post
Similar
Topics:
10 If the mean of set S does not exceed mean of any subset of 11 15 Jan 2012, 07:28
1 PS : Mean & SD 2 17 Aug 2010, 11:35
14 If the mean of set S does not exceed mean of any subset of 44 01 May 2010, 23:12
Mean and median 4 29 Dec 2009, 04:51
29 The mean of (54,820)^2 and (54,822)^2 = 13 14 Jun 2008, 21:27
Display posts from previous: Sort by