1986 AJHSME Problems/Problem 25: Difference between revisions
5849206328x (talk | contribs) New page: ==Problem== Which of the following sets of whole numbers has the largest average? <math>\text{(A)}\ \text{multiples of 2 between 1 and 101} \qquad \text{(B)}\ \text{multiples of 3 betwee... |
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==Solution== | ==Solution== | ||
{{ | There seems to be no better way to solve this other than just find each of those, so that's what we do. | ||
From <math>1</math> to <math>101</math> there are <math>\left\lfloor \frac{101}{2} \right\rfloor = 50</math> (see [[Floor function]]) multiples of <math>2</math>, and their average is | |||
<cmath>\begin{align*} | |||
\frac{2\cdot 1+2\cdot 2+2\cdot 3+\cdots + 2\cdot 50}{50} &= \frac{2(1+2+3+\cdots +50)}{50} \\ | |||
&= \frac{2\cdot \frac{50\cdot 51}{2}}{50} \\ | |||
&= \frac{2\cdot 51}{2} \\ | |||
&= 51 \\ | |||
\end{align*}</cmath> | |||
Similarly, we can find that the average of the multiples of <math>3</math> between <math>1</math> and <math>101</math> is <math>51</math>, the average of the multiples of <math>4</math> is <math>52</math>, the average of the multiples of <math>5</math> is <math>52.5</math>, and the average of the multiples of <math>6</math> is <math>51</math>, so the one with the largest average is <math>\boxed{\text{D}}</math> | |||
==See Also== | ==See Also== | ||
[[1986 AJHSME Problems]] | [[1986 AJHSME Problems]] | ||
Revision as of 10:30, 24 January 2009
Problem
Which of the following sets of whole numbers has the largest average?
Solution
There seems to be no better way to solve this other than just find each of those, so that's what we do.
From 
to 
there are 
(see Floor function) multiples of 
, and their average is
Similarly, we can find that the average of the multiples of 
between and is 
, the average of the multiples of 
is 
, the average of the multiples of 
is 
, and the average of the multiples of 
is , so the one with the largest average is 
