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# If the average of six numbers is 75, how many of the numbers are equal

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If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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18 May 2011, 08:17
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If the average of six numbers is 75, how many of the numbers are equal to 75

1) None of the six number is less than 75
2) None of the six numbers is greater than 75

OG12#DS136
Official explanation:
1) 5(75) + (75+x) = 6(75) + x, which is greater than (6)(75), contrary to the fact that sum is equal to 6*75.

2) 5(75) + (75-x) = 6(75) -x, which is less than 6*75, contrary to the fact that sum is equal to 6*75

I don't understand the Bold X... why its added in 1 and subtracted in 2 from the right side ?

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Re: If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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18 May 2011, 08:25
2
The easiest way to answer this is just by looking at statements, you dont need to actually solve the statement -

there are 6 numbers and average of them is 75. Meaning you will immediately think that some are lower than 75 and some are higher... to get the average of 75. 75 CANNOT be the 1st of 6 numbers OR last of 6 numbers.

but if you now see st-1 ==> None of 6 numbers is less than 75 meaning 75 is 1st in the list of 6 numbers... to get the average now ALL numbers have to be 75.

St - 2 ==> same explanation.

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Re: If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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18 May 2011, 09:06
1
shrive555 wrote:
If the average of six numbers is 75, how many of the numbers are equal to 75

1) None of the six number is less than 75
2) None of the six numbers is greater than 75

Official explanation:
1) 5(75) + (75+x) = 6(75) + x, which is greater than (6)(75), contrary to the fact that sum is equal to 6*75.

2) 5(75) + (75-x) = 6(75) -x, which is less than 6*75, contrary to the fact that sum is equal to 6*75

I don't understand the Bold X... why its added in 1 and subtracted in 2 from the right side ?

X is the quantity by which
1. exceeds 75, hence 75 + x
2. less than 75,hence 75-x.

Consider this example, 3 numbers, average = 3. How many numbers are equal to 3.
sum of 3 numbers = average * 3 = 3*3 = 9

1. none of the 3 numbers are lesser than 3.
means - numbers can be 3,3,3 or 4,3,4 in the second case the sum 11 > 9.
X as mentioned in the OG explanation is 4-3 = 1. (75+x here)

2. none of the 3 numbers are greater than 3.
means - numbers can be 3,3,3 or 2,3,2 in the second case the sum is 7 < 9.
X as mentioned in the OG explanation is 3-2 = 1. (75-x here).
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Re: If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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18 May 2011, 09:18
2
1) None of the six number is less than 75
1) 5(75) + (75+x) = 6(75) + x, which is greater than (6)(75), contrary to the fact that sum is equal to 6*75.
it is using indirect proof method.

See it this way, let 5 numbers be all 75 and 6th number be x.
75+75+75+75+75+x =375+x----(1)
and Sum of 6 numbers all 75 is 6*75=450-----(2)
Equating
375+x=450
x=75
Now suppose 5 numbers are all 76 and 6th number is x
5*76+x=380+x
Again equating
380+x=450
x=70 which contradicts statement 1 i.e None of the six number is less than 75

2) None of the six numbers is greater than 75
2) 5(75) + (75-x) = 6(75) -x, which is less than 6*75, contrary to the fact that sum is equal to 6*75

Again, let 5 numbers be all 75 and 6th number be x.
75+75+75+75+75+x =375+x----(1)
and Sum of 6 numbers all 75 is 6*75=450-----(2)
Equating
375+x=450
x=75
Now suppose 5 numbers are all 74 and 6th number is x
5*74+x=370+x
Again equating
370+x=450
x=80 which contradicts statement 2 i.e None of the six numbers is greater than 75

OA. D.
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Re: If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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19 May 2011, 21:26
Thanks All !! That really helped
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If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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28 Oct 2017, 10:36
1
Moderators, Bunuel , would you please move this question to DS?
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Posts: 51035
Re: If the average of six numbers is 75, how many of the numbers are equal  [#permalink]

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28 Oct 2017, 10:39
genxer123 wrote:
Moderators, Bunuel , would you please move this question to DS?

______________
Done. Thank you.
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Re: If the average of six numbers is 75, how many of the numbers are equal &nbs [#permalink] 28 Oct 2017, 10:39
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