## Tuesday, November 15, 2011

### this week's MGRE Math Beast Challenge problem

My answer will appear in the comments, but right now I have to make some cole slaw. Respect the cabbage. _

Kevin Kim said...

You know what? This may take a while. The problem seems straightforward if you plug in numbers, but since we don't know the starting conditions and we don't know the value of x right off the bat, that's a pain in the ass.

Let's pretend that the initial population is 100, and the initial x% change is 10%.

2007: 100
2008: 110 (i.e., 100*1.1)
2009: 132 (i.e., 110*1.2)
2010: 100 (given)

This means there was approximately a 24.24% decrease in the population between 2009 to 2010, reverting us to 2007 levels.

I've tried working out the algebra for the MGRE problem, but have gotten tangled. This is probably more a question of sloppiness than of lack of ability. The problem doesn't seem that tricky to set up.

Correct me if I'm wrong, but the solution seems to follow this basic path:

If A and B are two numbers in a chronological sequence such that the B result comes after the A result, and if B < A, then to find the percentage decrease, you subtract B from A, divide that total by A, and multiply by 100 so you can drop decimals and use the percent sign. Example:

A = 8
B = 6

A - B = 2

2/8 = .25

.25*100 = 25, i.e., 25% decrease. Try this again with easier-to-calculate numbers:

A = 100
B = 75

A - B = 25

25/100 = .25

.25*100 = 25, i.e., 25% decrease

For the MGRE problem, then, we can start by setting things up this way:

Let A represent the population level in 2007.

2007: A

2008: A + [(x/100)(A)]

Let B equal the 2008 figure.

2009: B + [(2x/100)(B)]

2010: A

So I guess what we need to do is:

(( - )/)*100

I'll be back with what I discover, but the above looks like an unholy mess to me.

Elisson said...

I just dusted off my old algebra and came up with (E).

Here's my work:

Let P = 2007 population.

[(100+X)/100]P = 2008 population where X is the percentage increase.

[100+2X)/100][(100+X)/100]P = 2009 population

[100+2X)/100][(100+X)/100] = 2010 population = 100/(100-Y) where Y is the percentage of decrease.

100-[1000000/(100+2X)(100+X)]= Y