answer to last week's MGRE Math Beast Challenge

Well, nuts. The answer to last week's MGRE Math Beast Challenge isn't (D); it's (A). [Never pick (D), Kevin!] Here's MGRE's multi-pronged explanation:

This problem could be solved through logic, algebraically, or by plugging in numbers. For all three solutions, our first task is to simplify y – x > x – y. Notice that it has like terms that can be combined – it would be very bad to neglect to simplify this before plowing ahead with the problem!

y – x > x – y
y > 2x – y
2y > 2x
y > x

So, y is greater than x.

The logic solution is certainly the fastest. Since all of the percent changes in Quantity A and Quantity B are changes through multiplication, order doesn’t matter. Thus, the 35% increase on both sides can be ignored – it is the same on both sides, and the order in which this occurs doesn’t matter.

Additionally, the order in which the other changes occur doesn’t matter. Also, the price p is a positive number that is the same on both sides, so it can be ignored as well.

All that’s left is: Quantity A decreases a smaller percent and increases a larger percent. Quantity B increases a smaller percent and decreases a larger percent. Quantity A is definitely greater.

Or, algebraically:

Since y is greater than x, Quantity A is positive and Quantity B is negative.

Finally, plugging in numbers would also work. To make things easy, make p = 100, and make x and y easy percents, like 10 and 50, making sure y is greater than x.

Of course, we still had to simplify y – x > x – y in order to pick valid numbers, and this method is even faster if we realize we can ignore the 35% change on both sides.

See sample solution with p = 100, x = 10, and y = 50 below. (To decrease by 10%, multiply by 0.9. To increase by 50%, multiply by 1.5. To decrease by 50%, multiply by 0.5. To increase by 10%, multiply by 1.1).

Quantity A
100(0.9)(1.5) = 135

Quantity B
100(0.5)(1.1) = 55

Quantity A will be greater no matter what numbers you choose, provided that you make y > x.

The correct answer is A.

I was so close to the above conclusion, dammit. I had successfully deduced that Quantity B was the negative of Quantity A, but not that A was always positive and B was always negative. In my own explanation, I had even mentioned that it would be tempting to pick (A). I should have followed my instincts, I guess. But where did I go wrong in my math, such that (A) produced a negative result in my own calculations?


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rock and roll! got last week's MGRE problem correct!

I admit I felt shaky about my solution to last week's MGRE Math Beast Challenge problem, but I turned out to be correct. Here's MGRE's explanation:

Let’s put the words into equation form:

(Total Income – Exclusion)(Tax Rate) = Income Tax.

The question asks about total income, so we’ll solve the equation for Total Income:

Total Income – Exclusion = [Income Tax / Tax Rate]
Total Income = [Income Tax / Tax Rate] + Exclusion

To maximize total income, we’ll minimize Tax Rate (smaller denominator→larger value) and maximize the Exclusion:

Maximum Total Income = [$8700/0.15] + $9800 = $58,000 + $9,800 = $67,800.

To minimize total income, we’ll maximize Tax Rate (larger denominator→smaller value) and minimize the Exclusion:

Minimum Total Income = [$8700/0.35] + $5200 = $24,857.14 + $5,200 = $30,057.14.

The correct answers are C, D, E, and F.

The letters C, D, E, and F correspond to the values I had selected. MGRE arrived at the exact same range that I had arrived at, too: roughly $30,057 at the bottom end, and $67,800 at the top end.

Triomphe!


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