carcass wrote:
The arithmetic mean of a collection of 5 positive integers, not necessarily distinct, is 9. One additional positive integer is included in the collection and the arithmetic mean of the 6 integers is computed. Is the arithmetic mean of the 6 integers at least 10 ?
1. The additional integer is at least 14.
2. The additional integer is a multiple of 5.
The solution below explores the
homogeneity nature of the average.
\(\left( * \right)\,\,\,5\,\,{\text{ints}} \geqslant 1\)
\(\sum\nolimits_{\,5} {\, = } \,\,5 \cdot 9 = 45\)
\(\sum\nolimits_{\,5} {\, + \,\,x\,\,\left( {6{\text{th}}} \right)\,\,\,\,\mathop \geqslant \limits^? } \,\,\,\,6 \cdot 10\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\,\,\,\mathop \geqslant \limits^? \,\,\,15\)
\(\left( 1 \right)\,\,x \geqslant 14\,\,\,\left\{ \begin{gathered}
\,x = 14\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\,x = 15\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\)
\(\left( 2 \right)\, + \,\left( * \right):\,\,\,\,\,x = 5,10,15, \ldots \,\,\,\,\,\,\left\{ \begin{gathered}
\,x = 5\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\,x = 15\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\)
\(\left( {1 + 2} \right) + \,\left( * \right):\,\,\,\,x = 15,20,25, \ldots \,\,\,\,\, \Rightarrow \,\,x \geqslant 15\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\)
The above follows the notations and rationale taught in the GMATH method.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
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