## Tuesday, January 3, 2012

### this week's MGRE Math Beast Challenge

My answer to last week's challenge, regarding the honeycombed hexagons, was indeed correct: Quantity A is greater. MGRE says:

Regular hexagons are very special—divide the hexagon with three diagonals (running through the center) and you will get six equilateral triangles. Why are the triangles equilateral? Since the sum of the angles in any polygon is (n – 2)(180), the sum for a hexagon is 720. Divide by 6 to get that each angle is 120. When you divide the hexagon into triangles, you split each 120 to make two 60 degree angles for each triangle. Any triangle that has two angles of 60 must have a third angle of 60 as well, since triangles always sum to 180.

Since the triangles are equilateral, we know that all of them have all sides equal to 2√3. For each triangle, we know that the height will always equal half the side times √3. (This is a good fact to simply memorize; however, had you not memorized this, you could divide each equilateral into two 30–60–90 triangles and use the side ratios of 1 : √3 : 2 for a 30–60–90 triangle.)

Therefore, the height of each triangle is simply √3 * √3 = 3.

Since the area of a triangle is [(1/2)bh], the area of each equilateral triangle is: (1/2)*(2√3)*3 = 3√3.

Since there are six such equilateral triangles in each hexagon, the area of each hexagon is 18.

Since there are six hexagons in the honeycomb, the total area of the figure is 108√3 = 187.061487....

This week, we've got another Data Interpretation challenge:

Go to it! My own answer will eventually appear in the comments.

_

#### 1 comment:

Kevin Kim said...

Here's what we know:

1. Pharm Div's Interest on Debt (IOD) is 4 times that of Chem Div's IOD.

2. In both cases, IOD is 2% of total budget.

THEREFORE, Pharm Div's total budget is 4 times that of Chem Div's.

Let's call Chem Div's total budget x, and Pharm Div's total budget 4x.

This means the raw amount of Pharm Div's payroll can be expressed as:

.26(4x), or 26% of 4x (the total raw amount of Pharm Div's budget)

Chem Div's payroll expenses can be expressed as:

.38x

The question is asking us what percent of Chem Div's payroll expenses are Pharm Div's, which means we need to set up the following ratio:

(.26(4x))/(.38x)

Obviously, the "x"es cancel, leaving us with

1.04/.38

which is

104/38

which is

52/19

which is approximately 2.74. Multiply that by 100 to get the percentage, and the answer would appear to be (E).

Without even doing any math, it's possible to know right away that Pharm's budget is much larger than Chem's, so whatever percentage we arrive at will be over 100, which means we can quickly eliminate choices (A) and (B).

Next week, we'll know the officially correct answer. In the meantime, don't fall into the trap of setting up your ratio incorrectly. If you reverse the fraction, such that your ratio is 19/52 (the reciprocal of 52/19), you'll end up with (A) as your answer. Such a mistake stems from a deadly misreading of the problem. Particularly when it comes to percentages, try to avoid such misreadings!