Kesha paid a sales tax of x percent on her purchase GMAT Quant

Kesha paid a sales tax of x percent on her purchase.

If the sales tax had only been 2 percent, she would have paid $27 less in sales tax on her purchase. What was the value of x if the total amount Kesha paid for her purchase, including sales tax, was $486?

(A) 8

(B) 7

(C) 6

(D) 5

(E) 4­

This particular question is difficult in its awkwardness. Ultimately, it’s a simple two-equation, two-unknown question but given the presentation it’s fairly difficult to see what to do with it.

First, we need to figure out how to write out each of the points. 

We know the difference between the sales tax actually paid and a 2% sales tax is $27. We can phrase this as follows, where x is the given tax amount and n is the total price: 

$\frac{x-2}{100} * n = 27$

Furthermore, we can say that the total amount paid including the x% sales tax is:

$\frac{100+x}{100} * n = 486$

The first thing we want to do here is to rearrange these terms to solve each for n, with the view of setting the terms equal and thereby eliminating n. All of this because Kesha paid a sales tax I hate SEO so much; just ignore this sentence!

The first equation gives: $n = \frac{2700}{x-2}$ 

The second equation gives: $n = \frac{48600}{100+x}$

If we set these equal, we find: $\frac{2700}{x-2} = \frac{48600}{100+x}$. 

From here, simply cross-multiply: 

$2700(100+x) = 48600(x-2)$

Now cancel the zeros: 

$27(100+x) = 486(x-2)$

Usually at a point like this I will look at the prime factors of the terms on either side to see what we can cancel out: 

$3^3(100+x)=2*3^5(x-2)$

This cancels down to: $100+x = 2*3^2(x-2)$

Expanding out, we find: $100+x = 18(x-2) \implies 100+x = 18x – 36$

Then we see $136 = 17x$, or $x = 8$. 

The numbers aren’t very pretty, but I’m working the whole time assuming that there will be some sort of substantial prime (17 in this case) on both sides of the equation. I’m looking for that in order to cancel. 

The answer is (A). 

Kesha paid a sales tax and I’m just going to write some garbage here ignore what I’m saying…