Problem Solving #1

My favourit question from this weeks problem solving sheet was question #10.

The sum on nine consecutive numbers positive integers is 99. The largest of these integers is
(a)9 (b)11 (c)19 (d) 7 (e)15

When i first looked at this question i thought that the only way to solve it would be trial and error but that would take a long.
One way to solve it is with an arithmetic sequence. Which is what we have been learning for the past week.
the sequence would be

x+1, x+2, x+3, x+4.... x+9 =99

from that sequence you can get the following formula

9x+45=99
9x=54
x=6

you then use 'x' and plug it into the 9th term.

x+9
6+9 = 15

so the largest of the integers is 15.

What I especially liked about this question is that it has so many other way to solve it. I find it amazing because with these other ways you can also check you answers. I enjoy having to think hard to find these alternate answers. Some of these alternate ways are the following:

x-4, x-3, x-2, x-1, x, x+1, x+2, x+3, x+4

9x+99
x=11

and
x+4 =15
11+4 =15

and there you have your answer.

At first i was anoyed because i didnt know how to solve it and i was applying what i know to solve the question. I learned that you have look deeper and use what you have learned to solve questions like that. Thats what being good math is all about.