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

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Question Stats: 60% (01:56) correct 40% (02:09) wrong based on 225 sessions

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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

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Math Expert V
Joined: 02 Sep 2009
Posts: 55228
The average height of the four Torres towers is 800 feet. If the four  [#permalink]

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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.

Answer: C.
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Math Expert V
Joined: 02 Sep 2009
Posts: 55228
Re: The average height of the four Torres towers is 800 feet. If the four  [#permalink]

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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.

Answer: C.

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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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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

My answer is A, Please clarify if i am wrong
Math Expert V
Joined: 02 Sep 2009
Posts: 55228
Re: The average height of the four Torres towers is 800 feet. If the four  [#permalink]

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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

My answer is A, Please clarify if i am wrong

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

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Here is the answer of the question
Attachments IMAG0316.jpg [ 931.67 KiB | Viewed 1906 times ]

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

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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.

Answer: C.

Bunuel,

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

Thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 55228
Re: The average height of the four Torres towers is 800 feet. If the four  [#permalink]

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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.

Answer: C.

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 V
Joined: 02 Sep 2009
Posts: 55228
Re: The average height of the four Torres towers is 800 feet. If the four  [#permalink]

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1
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.

Answer: C.

Bunuel,

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

Thanks

For more on this topic:

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

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