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# The average height of the four Torres towers is 800 feet. If the four

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The average height of the four Torres towers is 800 feet. If the four [#permalink]

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14 Jun 2015, 04:31
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The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet
[Reveal] Spoiler: OA

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The average height of the four Torres towers is 800 feet. If the four [#permalink]

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14 Jun 2015, 12:05
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reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

$$a_1\leq{a_2}\leq{a_3}\leq{a_4}$$;

The average height of the four Torres towers is 800 feet: $$a_1+a_2+a_3+a_4=4*800$$;

The median height of 900 feet: $$\frac{a_2+a_3}{2}=900$$ --> $$a_2+a_3=1800$$;

Substitute $$a_2+a_3=1800$$ into $$a_1+a_2+a_3+a_4=4*80$$:
$$a_1+1800+a_4=4*800$$;
$$a_1+a_4=1400$$.

To maximize $$a_1$$ we should minimize $$a_4$$. Minimum value of $$a_4$$ is 900 (the median value), thus the maximum value of $$a_1$$ is 1400 - 900 = 500.

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Posts: 44644
Re: The average height of the four Torres towers is 800 feet. If the four [#permalink]

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14 Jun 2015, 12:16
Bunuel wrote:
reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

$$a_1\leq{a_2}\leq{a_3}\leq{a_4}$$;

The average height of the four Torres towers is 800 feet: $$a_1+a_2+a_3+a_4=4*800$$;

The median height of 900 feet: $$\frac{a_2+a_3}{2}=900$$ --> $$a_2+a_3=1800$$;

Substitute $$a_2+a_3=1800$$ into $$a_1+a_2+a_3+a_4=4*80$$: $$a_1+1800+a_4=4*800$$ --> $$a_1+a_4=1400$$.

To maximize $$a_1$$ we should minimize $$a_4$$. Minimum value of $$a_4$$ is 900 (the median value), thus the maximum value of $$a_1$$ is 1400 - 900 = 500.

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Other Min/Max questions are HERE (DS) and HERE (PS).
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Re: The average height of the four Torres towers is 800 feet. If the four [#permalink]

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31 Oct 2016, 05:05
reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

Math Expert
Joined: 02 Sep 2009
Posts: 44644
Re: The average height of the four Torres towers is 800 feet. If the four [#permalink]

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31 Oct 2016, 05:14
sukeshap wrote:
reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

The OA is given in the original post and it's C, not A.

In the second post you can find a solution.

In the third post you can check similar questions to practice.
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Status: Quant Expert Q51
Joined: 02 Aug 2014
Posts: 65
Re: The average height of the four Torres towers is 800 feet. If the four [#permalink]

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31 Oct 2016, 06:06
Here is the answer of the question
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Joined: 13 Aug 2016
Posts: 2
The average height of the four Torres towers is 800 feet. If the four [#permalink]

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11 Dec 2017, 06:12
Bunuel wrote:
reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

$$a_1\leq{a_2}\leq{a_3}\leq{a_4}$$;

The average height of the four Torres towers is 800 feet: $$a_1+a_2+a_3+a_4=4*800$$;

The median height of 900 feet: $$\frac{a_2+a_3}{2}=900$$ --> $$a_2+a_3=1800$$;

Substitute $$a_2+a_3=1800$$ into $$a_1+a_2+a_3+a_4=4*80$$:
$$a_1+1800+a_4=4*800$$;
$$a_1+a_4=1400$$.

To maximize $$a_1$$ we should minimize $$a_4$$. Minimum value of $$a_4$$ is 900 (the median value), thus the maximum value of $$a_1$$ is 1400 - 900 = 500.

Bunuel,

Could you please explain why the median value must be the min value of a4?

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 44644
Re: The average height of the four Torres towers is 800 feet. If the four [#permalink]

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11 Dec 2017, 06:30
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Expert's post
Ashgmat96 wrote:
Bunuel wrote:
reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

$$a_1\leq{a_2}\leq{a_3}\leq{a_4}$$;

The average height of the four Torres towers is 800 feet: $$a_1+a_2+a_3+a_4=4*800$$;

The median height of 900 feet: $$\frac{a_2+a_3}{2}=900$$ --> $$a_2+a_3=1800$$;

Substitute $$a_2+a_3=1800$$ into $$a_1+a_2+a_3+a_4=4*80$$:
$$a_1+1800+a_4=4*800$$;
$$a_1+a_4=1400$$.

To maximize $$a_1$$ we should minimize $$a_4$$. Minimum value of $$a_4$$ is 900 (the median value), thus the maximum value of $$a_1$$ is 1400 - 900 = 500.

Bunuel,

Could you please explain why the median value must be the min value of a4?

Thanks

$$a_4$$ cannot be less than the median, which is the average of $$a_2$$ and $$a_3$$. So, basically we have the following list:
$$\{a_1=500; \ a_2=900; \ a_3=900; \ a_4=900\}$$.
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Math Expert
Joined: 02 Sep 2009
Posts: 44644
Re: The average height of the four Torres towers is 800 feet. If the four [#permalink]

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11 Dec 2017, 06:33
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Ashgmat96 wrote:
Bunuel wrote:
reto wrote:
The average height of the four Torres towers is 800 feet. If the four Torres towers have a median height of 900 feet, what is the greatest possible height of the shortest building of the four?

A. 100 feet
B. 300 feet
C. 500 feet
D. 600 feet
E. 800 feet

$$a_1\leq{a_2}\leq{a_3}\leq{a_4}$$;

The average height of the four Torres towers is 800 feet: $$a_1+a_2+a_3+a_4=4*800$$;

The median height of 900 feet: $$\frac{a_2+a_3}{2}=900$$ --> $$a_2+a_3=1800$$;

Substitute $$a_2+a_3=1800$$ into $$a_1+a_2+a_3+a_4=4*80$$:
$$a_1+1800+a_4=4*800$$;
$$a_1+a_4=1400$$.

To maximize $$a_1$$ we should minimize $$a_4$$. Minimum value of $$a_4$$ is 900 (the median value), thus the maximum value of $$a_1$$ is 1400 - 900 = 500.

Bunuel,

Could you please explain why the median value must be the min value of a4?

Thanks

For more on this topic:

14. Min/Max Problems

For other topics check:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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Re: The average height of the four Torres towers is 800 feet. If the four   [#permalink] 11 Dec 2017, 06:33
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