Problem Solving

e-GMAT

Set S contains consecutive natural numbers from 1 to 100, in increasing order. S_1 and S_2 denote the sum of first n even numbers and last n odd numbers of set S, respectively. What is the value of n, if S_1:S_2 = 16:85?

A. 5
B. 10
C. 15
D. 20
E. 25

OA C.

AAPL wrote:e-GMAT

Set S contains consecutive natural numbers from 1 to 100, in increasing order. S_1 and S_2 denote the sum of first n even numbers and last n odd numbers of set S, respectively. What is the value of n, if S_1:S_2 = 16:85?

A. 5
B. 10
C. 15
D. 20
E. 25

For any evenly spaced set:
sum = (count)(median)

Sum of the first n even numbers = n * median of the first n even numbers
Sum of the last n odd numbers = n * median of the last n odd numbers

Since the ratio of the two sums = 16/85, we get:
(n * median of the first n even numbers) / (n * median of the last n odd numbers) = 16/85
(median of the first n even numbers)/(median of the last n odd numbers) = 16/85

Implication:
The median of the first n even numbers = 16, with the result that the first n even numbers are as follows:
2, 4, 6, 8, 10, 12, 14, 16...
Since there are 7 numbers below the median of 16, there must be 7 numbers above the median of 16.
Thus, n = (7 numbers below the median) + (median of 16) + (7 numbers above the median) = 15 numbers.

The correct answer is C.

AAPL wrote:e-GMAT

Set S contains consecutive natural numbers from 1 to 100, in increasing order. S_1 and S_2 denote the sum of first n even numbers and last n odd numbers of set S, respectively. What is the value of n, if S_1:S_2 = 16:85?

A. 5
B. 10
C. 15
D. 20
E. 25

$${{{S_{\,{\rm{even}}}}} \over {{S_{\,{\rm{odd}}}}}} = {{16} \over {85}}\,\,\,\,\,;\,\,\,\,\,\,? = n$$
It´s easy to find a PATTERN!
$$n = 2\,\,\,\, \to \,\,\,{{2 + 4} \over {97 + 99}} = {{2 \cdot 3} \over {2 \cdot 98}} = {3 \over {98}}\,\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{numerator}} = n + 1 \hfill \cr
\,{\rm{denominator}} = 100 - n \hfill \cr} \right.$$
$$n = 3\,\,\,\, \to \,\,\,{{2 + 4 + 6} \over {95 + 97 + 99}} = {{3 \cdot 4} \over {3 \cdot 97}} = {4 \over {97}}\,\,\,\,\left\{ \matrix{
\,{\rm{numerator}} = n + 1 \hfill \cr
\,{\rm{denominator}} = 100 - n \hfill \cr} \right.$$
$$n = 4\,\,\,\, \to \,\,\,{{2 + 4 + 6 + 8} \over {93 + 95 + 97 + 99}} = {{4 \cdot 5} \over {4 \cdot 96}} = {5 \over {96}}\,\,\,\left\{ \matrix{
\,{\rm{numerator}} = n + 1 \hfill \cr
\,{\rm{denominator}} = 100 - n \hfill \cr} \right.$$
$$n = 5\,\,\,\, \to \,\,\,{{2 + 4 + 6 + 8 + 10} \over {91 + 93 + 95 + 97 + 99}} = {{5 \cdot 6} \over {5 \cdot 95}} = {6 \over {95}}\,\,\,\,\left\{ \matrix{
\,{\rm{numerator}} = n + 1 \hfill \cr
\,{\rm{denominator}} = 100 - n \hfill \cr} \right.$$
$${\rm{Hence}}\,\, \ldots \,\,\,\,\,\,\left\{ \matrix{
\,{\rm{numerator}} = n + 1 = 16 \hfill \cr
\,{\rm{denominator}} = 100 - n = 85 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,? = 15$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.