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# The arithmetic mean of a collection of 5 positive integers, not necess

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The arithmetic mean of a collection of 5 positive integers, not necess  [#permalink]

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26 Jul 2017, 11:52
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The arithmetic mean of a collection of 5 positive integers, not necessarily distinct, is 9. One additional positive integer is included in the collection and the arithmetic mean of the 6 integers is computed. Is the arithmetic mean of the 6 integers at least 10 ?

1. The additional integer is at least 14.

2. The additional integer is a multiple of 5.

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Re: The arithmetic mean of a collection of 5 positive integers, not necess  [#permalink]

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03 Sep 2018, 05:21
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carcass wrote:
The arithmetic mean of a collection of 5 positive integers, not necessarily distinct, is 9. One additional positive integer is included in the collection and the arithmetic mean of the 6 integers is computed. Is the arithmetic mean of the 6 integers at least 10 ?

1. The additional integer is at least 14.

2. The additional integer is a multiple of 5.

The solution below explores the homogeneity nature of the average.

$$\left( * \right)\,\,\,5\,\,{\text{ints}} \geqslant 1$$

$$\sum\nolimits_{\,5} {\, = } \,\,5 \cdot 9 = 45$$

$$\sum\nolimits_{\,5} {\, + \,\,x\,\,\left( {6{\text{th}}} \right)\,\,\,\,\mathop \geqslant \limits^? } \,\,\,\,6 \cdot 10\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\,\,\,\mathop \geqslant \limits^? \,\,\,15$$

$$\left( 1 \right)\,\,x \geqslant 14\,\,\,\left\{ \begin{gathered} \,x = 14\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \,x = 15\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\, + \,\left( * \right):\,\,\,\,\,x = 5,10,15, \ldots \,\,\,\,\,\,\left\{ \begin{gathered} \,x = 5\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \,x = 15\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\ \end{gathered} \right.$$

$$\left( {1 + 2} \right) + \,\left( * \right):\,\,\,\,x = 15,20,25, \ldots \,\,\,\,\, \Rightarrow \,\,x \geqslant 15\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,$$

The above follows the notations and rationale taught in the GMATH method.
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Re: The arithmetic mean of a collection of 5 positive integers, not necess  [#permalink]

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26 Jul 2017, 12:25
Answer should be "C" as with (1) and (2) the smallest of added no is 15 which makes the arith mean 10 (atleast).
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Re: The arithmetic mean of a collection of 5 positive integers, not necess  [#permalink]

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26 Jul 2017, 12:32
Ans C:
From Question Stem, we can determine the required range of value of 6th Integer as >=15, using Formulae for Arithmetic Mean.
To conclude this, we need to use both Statements 1 and 2.
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Re: The arithmetic mean of a collection of 5 positive integers, not necess  [#permalink]

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05 Sep 2019, 16:24
1
carcass wrote:
The arithmetic mean of a collection of 5 positive integers, not necessarily distinct, is 9. One additional positive integer is included in the collection and the arithmetic mean of the 6 integers is computed. Is the arithmetic mean of the 6 integers at least 10 ?

1. The additional integer is at least 14.

2. The additional integer is a multiple of 5.

Sum of original 5 numbers = 45
One number is added so now N=6. The question is asking if the sum of 6 numbers is >=60.
That means the number has to be >=15.

Statement 1: Not Sufficient
Statement 2: Number can be 5,10,15...

Both Combined: 15 is first such possible number. So Sufficient.

Hence C
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Re: The arithmetic mean of a collection of 5 positive integers, not necess   [#permalink] 05 Sep 2019, 16:24
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