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Bunuel
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 13 Apr 2021, 04:00
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B
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65% (hard)

Question Stats:

60% (02:08) correct 40% (02:29) wrong based on 235 sessions

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If the average (arithmetic mean) of 3, 8 and w is greater than or equal to w and smaller than or equal to 3w, how many integer values of w exist?

(A) Five
(B) Four
(C) Three
(D) Two
(E) One
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B
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Jitu20
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 13 Apr 2021, 04:11
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IMO B

Assume values, start with 4/5, then quickly we can realize that 6,7,8,9 are the only integers that satisfy the conditions.

I started with 1 :D
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 13 Apr 2021, 06:35
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Bunuel
If the average (arithmetic mean) of 3, 8 and w is greater than or equal to w and smaller than or equal to 3w, how many integer values of w exist?

(A) Five
(B) Four
(C) Three
(D) Two
(E) One
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Given: \(\frac{3 + 8 + w}{3 }>= w\)
11 + w >= 3w
11 >= 2w (Equation 1)

Also, given: \(\frac{3 + 8 + w}{3 }<= 3w\)
11 + w <= 9w
11 <= 8w (Equation 2)

Combining 1 & 2:
2w <= 11 <= 8w
This equation will be satisfied with w ranging from 2 to 5(inclusive) = 4
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 04 Mar 2023, 11:29
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With the same equation i.e 11>= 2w, I have the values 1 to 5 that satisfy it, then why B and not C?
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 22 Jul 2024, 06:16
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w<=(11+w)/3<=3w

3w<=11+w<=9w

at w=1, w=0,w=-1 doesn't satisfy
at w=2 -->6<=13<=18 (satisfies)
similarly satisfies for w=3,4,5
Total satisfied cases four, hence option (B) - FOUR
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ShenotShih1017
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 25 Jul 2024, 20:27
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The average of the 3 numbers is going to be:

(3+8+W)/3 >\= W

Multiply both sides by 3:

(3+8+W) >/= 3W

Simplify:

5.5 >/= W

Same with second inequality:

(3+8+W)/3 </= 3W

11 </= 8W

11/8 </= 8W

11/8 = 1.375

Count the integers between 1.375 and 5.5: 2, 3, 4, 5.

B is the answer.

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Natashaairan
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If the average (arithmetic mean) of 3, 8 and w is greater than or equa 27 Jul 2024, 02:19
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Solutions are 2, 3, 4,5
So, four ans B

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