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The average (arithmetic mean) of three positive integers

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Bunuel
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The average (arithmetic mean) of three positive integers [#permalink] New post   29 Jun 2022, 01:30
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The average (arithmetic mean) of three positive integers \(a, b,\) and \(c\) is 2. Is the range of the three numbers greater than 2?

(1) \(a - b=1\)

(2) \(c\) is the largest of the three numbers.


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Re: The average (arithmetic mean) of three positive integers [#permalink] New post   29 Jun 2022, 02:14
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Pre-think
a + b + c = 6
And all three numbers are positive integers.

(1) Only
a = b +1
If b=1, then a=2, c=3
If b=2, then a=3, c=1
There are no other possible combinations.
In either case, range is 2.
We can answer "No" to the question "Is the range of the three numbers greater than 2".
Sufficient.

(2) Only
We already know the range can be 2.
If c=4, then a=b=1. The range is 3.
We cannot answer the question with certainty.
Insufficient.

The answer is thus (A).
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Re: The average (arithmetic mean) of three positive integers [#permalink] New post   29 Jun 2022, 08:18
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Bunuel wrote:
The average (arithmetic mean) of three positive integers \(a, b,\) and \(c\) is 2. Is the range of the three numbers greater than 2?

(1) \(a - b=1\)

(2) \(c\) is the largest of the three numbers.


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Average is 2 .. so sum must be 6 ..

a+b+c = 6

Constraint .. a,b, c are all positive integer


(1) \(a - b=1\)
a = b+1
the numbers are b+1, b, c
2b+c = 5
b, c can be (1,3) or (2, 1)
corresponding values for (a, b, c ) = (2, 1,3) or (3, 2, 1)

both have range 2. So we can definitely answer that range is not greater than 2.

Sufficient



(2) \(c\) is the largest of the three numbers.
if c is greatest, a and b minimum can be 1 and so c will be 4. range =3
if a = 1 and b = 2, c = 3.. range = 2
Insufficient.

Ans A
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Re: The average (arithmetic mean) of three positive integers [#permalink] New post   11 Oct 2022, 08:59
Bunuel wrote:
The average (arithmetic mean) of three positive integers \(a, b,\) and \(c\) is 2. Is the range of the three numbers greater than 2?

(1) \(a - b=1\)

(2) \(c\) is the largest of the three numbers.


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From the information we know that (a+b+c)/ 3 = 2 -------> a + b + c = 6

Statement 1:
a-b = 1
Case 1 ---------- a = 2, b = 1, c = 3 ------- range = 2
Case 2 ---------- a = 3, b = 2 and c = 1 ----- range = 2
There can be no other case hence the range is 2
SUFFICIENT

Statement 2:
C is the largest of 3 integers
Case 1: c = 3, b = 2 and a = 1 -------- range = 2
Case 2: c = 4, b = 1 and a = 1 ------- range = 3
We already heave a conflict hence this statement wont help
NOT SUFFICIENT

Answer A
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Re: The average (arithmetic mean) of three positive integers [#permalink]
11 Oct 2022, 08:59
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Bunuel
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