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# If the average (arithmetic mean) of six numbers is 75, how

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Re: Average of 6 numbers [#permalink]
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GMATD11 wrote:
104) If the average of six numbers is 75, how many of the numbers are equal to 75?

a) None of the six numbers is less than 75
b) None of the six numbers is greater than 75

a)none is less than 75 so all numbers are 75 as averagege is 75
b) Noe is greater than 75 so all numbers are 75 as average is 75. If any number become less than 75 then average will not 75 it will less than 75.
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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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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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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 (arithmetic mean) of six numbers is 75, how [#permalink]
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If the average (arithmetic mean) of six numbers is 75, how many of the numbers are equal to 75?

$$Sum = 75 * 6$$

(1) None of the six numbers is less than 75.

If none of the six numbers is less than 75, then we can have only one possibility that all the 6 numbers are 75

Hence, (1) ===== is SUFFICIENT

(2) None of the six numbers is greater than 75.

If none of the six numbers is greater than 75, then we can have only one possibility that all the 6 numbers are 75

Hence, (2) ===== is SUFFICIENT

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If the average (arithmetic mean) of six numbers is 75, how [#permalink]
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GMATD11 wrote:
If the average (arithmetic mean) of six numbers is 75, how many of the numbers are equal to 75 ?

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

Target question: How many of the numbers are equal to 75 ?

Given: The average (arithmetic mean) of six numbers is 75

Statement 1: None of the six numbers is less than 75
One possible scenario is that the 6 number are {75, 75, 75, 75, 75, 75}. This satisfies the conditions that none of the six numbers is less than 75, AND the average is 75
Notice that if we increase ANY of the 6 values, the resulting average will be greater than 75
So, it must be the case that the 6 number are {75, 75, 75, 75, 75, 75}.
The answer to the target question is all 6 numbers are equal to 75
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: None of the six numbers is greater than 75.
One possible scenario is that the 6 number are {75, 75, 75, 75, 75, 75}. This satisfies the conditions that none of the six numbers is greater than 75, AND the average is 75
Notice that if we decrease ANY of the 6 values, the resulting average will be less than 75
So, it must be the case that the 6 number are {75, 75, 75, 75, 75, 75}.
The answer to the target question is all 6 numbers are equal to 75
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

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Re: If the average (arithmetic mean) of six numbers is 75, how [#permalink]
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Re: If the average (arithmetic mean) of six numbers is 75, how [#permalink]
GMATD11

If the average (arithmetic mean) of six numbers is 75, how many of the numbers are equal to 75 ?

(1) None of the six numbers is less than 75.
All numbers are 75.
SUFFICIENT

(2) None of the six numbers is greater than 75.
All numbers are 75.
SUFFICIENT

IMO D
Re: If the average (arithmetic mean) of six numbers is 75, how [#permalink]
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