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Manager  Joined: 12 Dec 2012
Posts: 151
Location: Poland
Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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2
5 00:00

Difficulty:   45% (medium)

Question Stats: 63% (01:21) correct 37% (01:24) wrong based on 223 sessions

### HideShow timer Statistics Three sisters have an average (arithmetic mean) age of 25 years and a median age of 24 years. What is the minimum possible age, in years, of the oldest sister?

(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

f = age of the first sister, etc.
M = Median = 24
m = mean

$$\frac{f+s+t}{3}=25=m$$

$$M=s=24$$

$$f+s=51$$

My assumption:
The problem says about the oldest sister, so I assumed that:

f≤s<t

In case the younger sisters were twins. The book explains: The youngest sister must be less than or equal to 24 years old. Agreed.

However, the book also assumes that:

s≤t:

Of course, the oldest sister must be at least as old as the middle sister (...) Not agreed. (Further implications concerning M=24 that turn the ≤ into the < possibility are correct.)

My thinking is: If there is one sister that is the oldest, she must not be the same age as the younger sister(s). Hypothetically, she could be 9 months older and still born in the same year, but I think it is impossible from the medical point of view. Or the sisters could be a step sisters, etc. Nevertheless, this situation is purely hypothetical and - even if the point was to mislead the thinker - in my opinion this is a bit too much, because the very natural assumption for expression of an age is it be an integer.

Moreover, if there were a hypothetical sets:

A={24,25,25},
OR
B={25,25,25},

none of the elements in each set were the greatest (oldest), because "the greatest" is an expression inherently relative to some other objects.

Disputably, in the A'={24,25} we could point 25 as "the greatest", because it's the greatest within the set, even though inside the set it is only "the greater" element, because there is only one other element to be relatively smaller (24).

If you have a one brother, you never say "I am his oldest brother" (unless you're joking).
Instead, you will say: "I am his older brother.". Ain't right?

And if you have two siblings, you will say: "I am the oldest one (of us three).", won't you?

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Originally posted by HumptyDumpty on 14 Jan 2013, 06:16.
Last edited by stonecold on 15 Dec 2016, 15:35, edited 1 time in total.
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Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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This question is fairly simple. Note that in the question stem they do not say that each sister has a different age..meaning two could have the same age. If we have a median of 24 that means the middle sister is 24. If we have a mean of 25 that means that the total age of all 3 sisters is 75. If you take out the middle sisters age from the total you get 51...the youngest and oldest sisters must combine for 51 years.
The first answer choice that makes sense is 27 years for the oldest sister leaving the middle and youngest to be 24.
Manager  Joined: 12 Dec 2012
Posts: 151
Location: Poland
Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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Dear sambam,
Quote:
This post is about a doubtful assumption, perhaps an error in the book, which however doesn't influence the answer to the original question.

Solving the problem as such is not the purpose of my post.
I would be glad if you've read the whole story before answering .
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Joined: 12 Aug 2015
Posts: 2609
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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Nice Question.
Here is my Solution to this one ->

Let the Age of the three sister in ascending order be ->
S1
S2
S3

Mean = 25

$$Using Mean =Sum/#$$

Sum(3)=25*3=75

Hene S1+S2+S3=75

As #=3=Odd => Median = 2nd term = S2
S2=24

Now to minimise the largest term that is S3 we must maximise all other terms.
S1=S2=24

Hence 2*24+S3=75
S3=75-48=27

Hence D

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Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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The median is 24, which means that the age of the middle sister must be 24. So we have x, 24, y, where x is the age of the youngest sister and y is the age of the oldest sister. To find the minimum age of the oldest sister the other values should be maximized. We also know that the sum of all three ages is 75. Setting x as 24 (the highest value it can be if the median is 24) gives y as 27.

POE for A-C:

A) x must be 24 or lower (for 24 to be the mean). If y is 24 the sum of 75 cannot be met.
B) x must be 24 or lower (for 24 to be the mean). If y is 25 the sum of 75 cannot be met.
C) x must be 24 or lower (for 24 to be the mean). If y is 26 the sum of 75 cannot be met.
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Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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Posting official solution of this problem.
Attachments official_average.PNG [ 118.79 KiB | Viewed 1622 times ]

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Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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HumptyDumpty wrote:
Three sisters have an average (arithmetic mean) age of 25 years and a median age of 24 years. What is the minimum possible age, in years, of the oldest sister?

(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

a + b + c = 75

Given -

a + 24+ c = 75

Now, a + c = 51

Now comes the fun part, so check using options...

Here C is the oldest sister, and we need to minimize c by maximizing a..

(A) If c = 24 , a = 27 ( not Possible as a > c )
(B) If c = 25 , a = 26 ( not Possible as a > c )
(C) If c = 26 , a = 25 ( not Possible as a > b < c )

(D) If c = 27 , a = 24 ( possible as a ≤ b < c )
(E) If c = 28 , a = 23 ( Possible as but now, this is not the minimum value c can take )

Hence, answer must be (D) 27
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Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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HumptyDumpty wrote:
Three sisters have an average (arithmetic mean) age of 25 years and a median age of 24 years. What is the minimum possible age, in years, of the oldest sister?

(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

Since 3 sisters have an average age of 25 years, the sum of their ages is 75.

Since the median is 24, the two youngest sisters could both be 24 years old.

Thus, the minimum possible age of the oldest sister is 75 - (24 + 24) = 75 - 48 = 27.

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Re: Three sisters have an average (arithmetic mean) age of 25 ye  [#permalink]

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Top Contributor
HumptyDumpty wrote:
Three sisters have an average (arithmetic mean) age of 25 years and a median age of 24 years. What is the minimum possible age, in years, of the oldest sister?

(A) 24
(B) 25
(C) 26
(D) 27
(E) 28

If the average age is 25, we can write: (sum of all 3 ages)/3 = 25
This means, the sum of all 3 ages = 75

If the median is 24, then we can express the ages in ascending order as follows: __ , 24, __

Since the sum of the ages is 75, we can MINIMIZE the age of the oldest girl by MAXIMIZING the age of the youngest girl.
Well, the youngest girl cannot be older than the median age (24), but the "youngest" girl can also be 24 (so there are 2 youngest sisters)
We get: 24, 24, __

Finally, since the sum of all 3 ages = 75, the oldest girl must be 27 years old.

Cheers,
Brent

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_________________ Re: Three sisters have an average (arithmetic mean) age of 25 ye   [#permalink] 05 Feb 2019, 07:46
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