Is the median of the 3 different integers equal to their average (arithmetic mean)?
1) The median of the 3 integers is 19
2) The range of the 3 integers is 19
can anybody explain this for me?
Is the median of the 3 different integers equal to their
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Sure, kyuhunl! (Nice problem, by the way.)kyuhunl wrote:Is the median of the 3 different integers equal to their average (arithmetic mean)?
1) The median of the 3 integers is 19
2) The range of the 3 integers is 19
can anybody explain this for me?
$$a < b < c\,\,{\rm{ints}}$$
$$b\,\,\mathop = \limits^? \,\,{{a + b + c} \over 3}$$
$$\left( 1 \right)\,\,b = 19\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {18,19,20} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {18,19,21} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,c - a = 19\,\,\,\, \Rightarrow \,\,\,\,\left( {a,b,c} \right) = \left( {a,b,a + 19} \right)$$
$$b\,\,\mathop = \limits^? \,\,{{2a + b + 19} \over 3}\,\,\,\, \Leftrightarrow \,\,\,\,2b\,\,\mathop = \limits^? \,\,2a + 19\,\,\,\, \Leftrightarrow \,\,\,\,b\,\,\mathop = \limits^? \,\,a + {{19} \over 2}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,$$
$$\left( * \right)\,\,\,a\,\,{\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\,a + {{19} \over 2}\,\, \ne {\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\,b \ne a + {{19} \over 2}\,\,\,\,\,\,\left( {b\,\,{\mathop{\rm int}} } \right)$$
The correct answer is therefore (B).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Let's take each statement one by one.kyuhunl wrote:Is the median of the 3 different integers equal to their average (arithmetic mean)?
1) The median of the 3 integers is 19
2) The range of the 3 integers is 19
can anybody explain this for me?
1) The median of the 3 integers is 19.
Say the three integers are: x, y and z such that x < y < z
Case 1: Say the integers are 1, 19, 20. Average ≠Median. The answer is No.
Case 2: Say the integers are 18, 19, 20. Average = Median. The answer is Yes.
No unique answer. Insufficient.
2) The range of the 3 integers is 19.
=> z = x + 19 and average = x + 19/2 = not an integer.
Since y = median is an integer (given) but the average is not an integer, the answer is No. Sufficient.
The correct answer: B
Hope this helps!
-Jay
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