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The arithmetic mean of a set of 12 numbers is 65, and the median......

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The arithmetic mean of a set of 12 numbers is 65, and the median......  [#permalink]

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New post Updated on: 29 Oct 2018, 22:39
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

56% (01:57) correct 44% (01:37) wrong based on 54 sessions

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The arithmetic mean of a set of 12 numbers is 65, and the median of the set is 40. What is the minimum possible value of the largest number of this set?

    A. 65
    B. 80
    C. 100
    D. 120
    E. 130

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Originally posted by EgmatQuantExpert on 25 Oct 2018, 00:26.
Last edited by EgmatQuantExpert on 29 Oct 2018, 22:39, edited 1 time in total.
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Re: The arithmetic mean of a set of 12 numbers is 65, and the median......  [#permalink]

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New post 25 Oct 2018, 02:38
1
I considered the set from number x1to x12

As the median of the set is 40 I have considered that all numbers from x1 to x7 are 40. Thus 40x7=280

On the other hand, the sum of all the set is 65*12=780

The subtract is
780-280=500

As we have 5 numbers between x8 to x12, I divided 500/5= 100

C. 100

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Re: The arithmetic mean of a set of 12 numbers is 65, and the median......  [#permalink]

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New post 29 Oct 2018, 00:45

Solution


Given:
    • The arithmetic mean of a set of 12 numbers = 65
    • Median of the set = 40

To find:
    • The minimum possible value of the largest number of the set

Approach and Working:
    • Given, the arithmetic mean of 12 numbers = 65
      o Implies, the sum of 12 numbers = 12 * 65 = 780
         \(t_1 + t_2 + … + t_6 + t_7 + …. + t_{11} + t_{12} = 780\)

    • We are also given that the median = 40,
      o That implies, \(t_6 + t_7 = 2 * 40\)

    • So, for the largest number to be minimum, all other numbers must take the maximum possible value
      o The maximum value of all the five numbers before the median term = the median itself (since they cannot be greater than the median value)
      o The maximum value of all the five numbers after the median term = the value of the largest number

    • Thus, 5 * 40 + 2 * 40 + 5 * x =780, where x is the largest number
      o Implies, \(x = \frac{500}{5} =100\)

    • Therefore, the minimum possible value of the largest number = 100

Hence, the correct answer is option C.

Answer: C

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Re: The arithmetic mean of a set of 12 numbers is 65, and the median...... &nbs [#permalink] 29 Oct 2018, 00:45
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