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Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:

There are two paths to the solution: using algebra and picking numbers.

Algebra

Many algebraic manipulations are possible. However, we should be guided by the form of the answer choices, which contain only x‘s and y‘s. Thus, we should look for ways to express a in terms of only x and y (not b); likewise, we want b in the same terms, without a.

Again, we have ab = x and a/b = y.

If we solve the first equation for b and plug into the second equation, here is what we get:

b = x/a
Solve first eq. for b

a/(x/a) = y
Substitute x/a for b in second eq.

a²/x = y
Bring the a up to the numerator (dividing by x/a is the same as multiplying by a/x)

a² = xy
Multiply both sides by x

In fact, we could have gotten this result directly if we had multiplied the two original equations together:

ab = x
a/b = y       Multiply each side (canceling the b‘s)
a² = xy

Similarly, we can get b² = x/y either by substitution (solving one equation for a and plugging into the other equation) or by dividing the first equation by the second, canceling the a‘s:

ab = x
a/b = y       Divide each side
b² = x/y

(Note that ab divided by a/b is the same as ab times b/a. The a‘s cancel, leaving b².)

Now that we have a² = xy and b² = x/y, we can substitute into the expression we are looking for:

a² – b² = xy – x/y
Substitute the expressions above.

This does not match any answer choice, so we should manipulate further:

= x(y – 1/y)
Pull out the common factor x

= x(y²/y – 1/y)
Make a common denominator of y

= x(y² – 1)/y
Pull out the common divisor y (that is, y in the denominator)

= x(y – 1)(y + 1)/y
Factor the special product y² – 1 into (y – 1)(y + 1)

This expression matches the first answer choice.
The correct answer is A.

Plugging Numbers

Choose suitable numbers for a and b. Here, we should use small numbers and make a divisible by b. For instance, let a = 6 and b = 2.

Then, from ab = x, we have x = (6)(2) = 12.
Likewise, from a/b = y, we have y = 6/2 = 3.

Finally, we compute our target number:
a² – b² = 6² – 2² = 36 – 4 = 32.

Now, compute each answer choice and compare the result to the target of 32. Note that you should stop computing as soon as you see tha you are not going to get 32, but you should check all the answer choices.

(A) x(y – 1)(y + 1)/y = (12)(3 – 1)(3 + 1)/3
= (12)(2)(4)/3 = (4)(2)(4) = 32 CORRECT

(B) xy – y/x = (12)(3) – 3/12 = 36 – 1/4 = stop here (result is too big) INCORRECT

(C) x² – y² = 12² – 3² = 144 – 9 = stop here (result is too big) INCORRECT

(D) y(x² – 1/x²) = 3(12² – 1/12²) = 3(144 – 1/144) = stop here (result is too big) INCORRECT

(E) (y + x)(y – x)/x = (3 + 12)(3 – 12)/12 = stop here (result is negative)

Again, the correct answer is A.

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