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# Set R contains five numbers that have an average value of 55

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Re: Largest possible range in Set R [#permalink]

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27 May 2014, 01:16
russ9 wrote:
Bunuel wrote:
Orange08 wrote:
Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78
77 1/5
66 1/7
55 1/7
52

{$$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$, $$a_5$$}
As mean of 5 numbers is 55 then the sum of these numbers is $$5*55=275$$;
The median of the set is equal to the mean --> $$mean=median=a_3=55$$;
The largest number in the set is equal to 20 more than three times the smallest number --> $$a_5=3a_1+20$$.

So our set is {$$a_1$$, $$a_2$$, $$55$$, $$a_4$$, $$3a_1+20$$} and $$a_1+a_2+55+a_4+3a_1+20=275$$.

The range of a set is the difference between the largest and smallest elements of a set.

$$Range=a_5-a_1=3a_1+20-a_1=2a_1+20$$ --> so to maximize the range we should maximize the value of $$a_1$$ and to maximize $$a_1$$ we should minimize all other terms so $$a_2$$ and $$a_4$$.

Min possible value of $$a_2$$ is $$a_1$$ and min possible value of $$a_4$$ is $$median=a_3=55$$ --> set becomes: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

Hi Bunuel,

Since the statement says that the median = mean, aren't we supposed to assume that it's an evenly spaced set? If so, wouldn't a2 and a4 be different from a1 and a3?

For evenly spaced set mean = median, but the reverse is not necessarily true. Consider {1, 1, 2, 2, 4} --> mean = median = 2, but the set is not evenly spaced.
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Re: Largest possible range in Set R [#permalink]

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27 May 2014, 16:24
Bunuel wrote:
russ9 wrote:

Hi Bunuel,

Since the statement says that the median = mean, aren't we supposed to assume that it's an evenly spaced set? If so, wouldn't a2 and a4 be different from a1 and a3?

For evenly spaced set mean = median, but the reverse is not necessarily true. Consider {1, 1, 2, 2, 4} --> mean = median = 2, but the set is not evenly spaced.

thanks for clarifying.

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Re: Set R contains five numbers that have an average value of 55 [#permalink]

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28 May 2015, 08:24
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Re: Set R contains five numbers that have an average value of 55 [#permalink]

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29 May 2016, 05:04
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Re: Set R contains five numbers that have an average value of 55 [#permalink]

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14 Aug 2017, 21:24
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Re: Set R contains five numbers that have an average value of 55   [#permalink] 14 Aug 2017, 21:24

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# Set R contains five numbers that have an average value of 55

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