GMAT Tip: Replacement Scheme

Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:

Problem

x is replaced by 1 – x everywhere in the expression 1/x – 1/(1 – x), with x ≠ 0 and x ≠ 1. If the result is then multiplied by x² – x, the outcome equals

(A) x + 1

(B) x – 1

(D) 2x – 1

(E) 1 – 2x

Solution

We can attack this problem by doing Direct Algebra. First, carry out the replacement. That is, literally replace every x in the expression with 1 – x, putting parentheses around the 1 – x in order to preserve proper order of operations:

Original: 1/x – 1/(1 – x)

Replacement:

1/(1 – x) – 1/(1 – (1 – x))

Now simplify the second denominator: (1 – (1 – x)) = (1 – 1 + x) = x

So the replacement expression becomes this:

1/(1 – x) – 1/x

This should make sense. If we replace x by 1 – x, then it turns out that we are also replacing 1 – x by x (since 1 – (1 – x) = x). Thus, the denominators of the original expression are simply swapped.

Now we can either combine these fractions first (by finding a common denominator) or go ahead & multiply by x² – x, as we are instructed to. Let’s take the latter approach.

[1/(1 – x) – 1/x] (x² – x)

Instead of FOILing this product right away, we should factor the expression x² – x first. If we do so, we will be able to cancel denominators quickly.

x² – x factors into (x – 1)x. We can now rewrite the product:

[1/(1 – x) – 1/x] (x – 1)x

= (x – 1)x/(1 – x) – (x – 1)x/x

The second term, (x – 1)x/x, becomes just x – 1 after we cancel the x’s.

Since (x – 1) = –(1 – x), we can rewrite the first term as –(1 – x)x/(1 – x) and then cancel the (1 – x)’s, leaving –x.

So, the final result is

–x – (x – 1) = –x – x + 1 = 1 – 2x

This is the answer.

Separately, since this is a Variables In Choices problem, we could instead pick a number and calculate a target. Since 0 and 1 are disallowed, let’s pick x = 2. We are told that x should be replaced by 1 – x, so we calculate 1 – x = –1 and put in –1 wherever x is in the original expression.

1/x – 1/(1 – x) = 1/(–1) – 1/(1 – (–1))

= –1 – ½

= –3/2

Now multiply this number by x² – x = 2² – 2 = 2. We get –3 as our target number.

Finally, we plug x = 2 into the answer choices and look for –3:

(A) x + 1 = 2 + 1 = 3

(B) x – 1 = 2 – 1 = 1

(D) 2x – 1 = 2(2) – 1 = 3

(E) 1 – 2x = 1 – 2(2) = –3

We can eliminate choices A, B, and D, but to choose between C and E, we would need to pick another number. For instance, if we pick x = 3, we get a target of –5. Only E fits this target.

The correct answer is (E).

For more information on ManhattanGMAT, download Clear Admit’s independent guide to the leading test preparation companies here. This FREE guide includes coupons for discounts on test prep services at ten different firms!

Read the full article: GMAT Tip: Replacement Scheme

SPONSORS

POPULAR ARTICLES

Related Articles