Tuesday, June 5, 2012

this week's MGRE Math Beast Challenge

From here:

An item originally cost p dollars, where p > 0.

y – x > x – y

Quantity A
The price of the item if the original price were decreased by x%, increased by 35%, and then increased by y%

Quantity B
The price of the item if the original price were increased by x%, decreased by y%, and then increased by 35%

(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.

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



Kevin Kim said...

I'm pretty sure the answer to this is (D)-- cannot be determined. Here's why I think so.

First, let's figure out how to express Quantity A.

"the original price [p] decreased by x%..."

p - (x/100)p, or p - (xp/100)

"...increased by 35%..."

1.35(p - (xp/100))

"...and then increased by y%..."

((100 + y)/100)•[1.35(p - (xp/100))]

Yikes. But bear with me.

Next, let's try to express Quantity B.

"...original price [p] increased by x%..."

p + (xp/100)

"...decreased by y%..."

((100 - y)/100)•(p + (xp/100))

"...and then increased by 35%..."

1.35•[((100 - y)/100)•(p + (xp/100))]

Double yikes.

Let's set Quantities A and B opposite each other, once again using "Q" to mean "undefined relationship-- greater than, less than, equal to... we don't know."

A Q B, or, more exactly:

((100 + y)/100)•[1.35(p - (xp/100))]


1.35•[((100 - y)/100)•(p + (xp/100))]

We can divide by 1.35 on both sides, leaving us with this relationship:

((100 + y)/100)•(p - (xp/100))


((100p - yp)/100) + ((100xp - xpy)/10,000)

We can then multiply by 100 on both sides to eliminate the "100" in the denominator:

(100 + y)(p - (xp/100))


(100p - yp) + ((100xp + xpy)/100)

Using FOIL, Quantity A becomes:

100p - xp + yp - (xpy/100);

Quantity B, meanwhile, can be rewritten as:

100p - yp + xp + (xpy/100)

--so the relationship is now:

100p - xp + yp - (xpy/100)


100p - yp + xp + (xpy/100)

We can subtract 100p from both sides, then factor out p and eliminate it:

y - x - (xy/100) Q x - y + (xy/100)

At this point, or perhaps a step or two before, you may have noticed that Quantity B is the negative of Quantity A, and since we were given that

(y - x) > (x - y) [i.e., y > x],

we will be tempted to assume that, since Quantity B is always the negative of Quantity A, Quantity A is always greater. The problem is that, if Quantity A is negative, then B will be positive, making B greater than A.

To be sure, though, we can try plugging in some numbers for x and y. I'll use coordinate pairs to express these values, with "QuantA" and "QuantB" representing the results of plugging those values in.

Scenario 1: (0,1)
QuantA = +1
QuantB = -1
∴ A > B

Scenario 2: (-1,0)
QuantA = +1
QuantB = -1
∴ A > B

Scenario 3: (-2,-1)
QuantA = +0.98
QuantB = -0.98
∴ A > B

Scenario 4: (-100, 100)
QuantA = +300
QuantB = -300
∴ A > B

Scenario 5: (-200,-100)
QuantA = -100
QuantB = +100
∴ A < B UH-OH!!!!

Scenario 5 is the kicker. Since it's possible to find values that shift the balance such that B is greater than A, then all bets are off. I think the answer is (D).

If you beg to differ, and/or if you think you have a more economical way to solve this problem, I'm all ears.

Kevin Kim said...

Note, too, that I've taken the liberty of assuming that x and/or y could be negative. The problem offers no evidence to the contrary. Essentially, then, the problem boils down to this:

x Q -x

--and without knowing whether x is positive or negative, there's no way to define Q as > or < or =.