## Monday, May 14, 2012

### answer to last week's MGRE Math Beast Challenge

Correct! The answer is indeed 14.286%. Strangely enough, MGRE's initial reasoning is rather tortured. Instead of following their own advice about solving a problem quickly through the plug-and-chug method, they went for the brute-force approach and solved the problem with variables. It's only at the end that the gurus discuss plug-and-chug as an afterthought. Here's what MGRE wrote:
When calculating a percent change “from (original) to (new),” be careful to use the ratio (change/original), not (change/new) or (new/original).

Create some variables:
x = attendance in 2010
y = attendance in 2011
z = attendance in 2012

In 2012, attendance was greater than in 2011, and even greater than it had been in 2010. So, x < y < z.

The question asks for the percent change from 2010 to 2011, or [(y - x)/x]•100%. This can be rewritten as [(y/x)-1]•100%, so what we really need to find is y/x.

We are given two percent changes from one year to another, but watch out! The “from (original)” year is different for each percent given.

“In 2012, attendance at an annual sporting event was 5% greater than it was in 2011”:
Note that “5% greater than” a number means “105% of” that number.
z = 105% of y
z = 1.05y

“In 2012, attendance at an annual sporting event was ... 20% greater than it was in 2010”:
Note that “20% greater than” a number means “120% of” that number.
z = 120% of x
z = 1.2x

To find y/x, we’ll first set both z equivalents equal:
z = 1.05y = 1.2x
y/x = 1.20/1.05

The answer is [(y/x) - 1]•100% = (1.14285714 - 1)•100% = 14.285714%. Rounded to the nearest 0.001%, the final answer is 14.286%.

Alternatively, we could pick numbers. A smart number for z would be a multiple of 120 and 105 (reflecting 5% and 20% increases from an easy base of 100).

z = (105)(120)
y = (100)(120)
x = (105)(100)

The answer is [(y/x) - 1]•100% = {[(100•120)/(105•100)] - 1}•100% = 14.285714%. Rounded to the nearest 0.001%, the final answer is 14.286%
For what it's worth, my own method took me 45 seconds with an on-screen calculator.

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