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

Given: sum of six numbers = 6*75

(1) None of the six numbers is less than 75 --> all 6 numbers must be 75, because if even one of them is more than 75 then the sum will be more than 6*75. Sufficient.

(2) None of the six numbers is greater than 75 --> the same here: all 6 numbers must be 75, because if even one of them is less than 75 then the sum will be less then than 6*75. Sufficient.

Answer: D.
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Video solution from Quant Reasoning:

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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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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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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.

So answer is D.

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

Hence, Answer is D
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We are given that the average of 6 numbers in a list is 75. Using this information we see that the sum of the 6 numbers is 6 x 75 = 450. We need to determine how many numbers in the list are equal to 75.

Statement One Alone:

None of the six numbers is less than 75.

Recall that the sum of the 6 numbers is 6 x 75 = 450.

Using the information in statement one we know that all the numbers in the list must be greater than or equal to 75.

However, if any of the numbers in our list are greater than 75, we would need another number to be less than 75 in order for the 6 numbers to sum to 450.

For example, let’s say we had the following 6 numbers:

75, 75, 75, 75, 75, 76

The sum of these numbers is 451, which is greater than the required sum of 450.

The only way to get our list to sum to 450 is if one of the 75s were reduced to 74. However, this is not possible, according to the information provided in statement one. Thus, all the numbers in the list must equal 75. Statement one is sufficient to answer the question.

Statement Two Alone:

None of the six numbers is greater than 75.

We can apply similar logic to the information provided in statement two as we did with the information in statement one.

Once again, we know that the sum of the 6 numbers is 6 x 75 = 450.

Using the information in statement two we know that all the numbers in the list must be less than or equal to 75.

Thus, if any of the numbers in our list are less than 75, we would need another number to be greater than 75 in order for the 6 numbers to sum to 450.

For instance, let’s say we had the following 6 numbers:

75, 75, 75, 75, 75, 74

The sum of these numbers is 449, which is less than the required sum of 450.

The only way to get our list to sum to 450 is if one of the 75s were increased to 76. However, this is not possible, according to the information provided in statement two. Thus, all the numbers in the list must equal 75. Statement two is sufficient to answer the question.

Answer: D
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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

Answer: D

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