If a and b are Integers, and b > 0, Does (a - 1)/(b + 1) = a/b ? GMAT Data Sufficiency

Question: If a and b are integers, and b > 0, does (a - 1)/(b + 1) = a/b ?

(A) a = b − 4
(B) a = –b

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: B

Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

a and b are integers, and b > 0

Target question: Does (a - 1)/(b + 1) = a/b?
This is the process for rephrasing the target question.

Take the equation: (a - 1)/(b + 1) = a/b
Cross multiply to get: (b)(a - 1) = (a)(b + 1)
Expand both sides to get: ab - b = ab + a
Subtract ab from both sides to get: -b = a
Add b to both sides to get: 0 = a + b
Rephrased target question: Does a +b = 0?

Now let's consider 2 options that are provided:

Statement 1: a = b − 4
Let's TEST some values.
There are several values of a and b that satisfy statement 1. Here are two:

Case A: a = -2 and b = 2. In this case, a + b = (-2) + 2 = 0. So, the answer to the target question is YES, a+b = 0

Case B: a = -1 and b = 3. In this case, a + b = (-1) + 3 = 2. So, the answer to the target question is NO, a+b does NOT equal 0
Since we cannot answer the target question with certainty, statement 1 is Not Sufficient

Statement 2: a = –b
Add b to both sides to get: a + b = 0
The answer to the target question is YES, a+b = 0
Since we can answer the target question with certainty, statement 2 is Sufficient

Hence, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Approach Solution 2:
We are given that a and b are integers and b > 0.

It is asked to determine whether (a -1)/(b + 1) = a/b.

Since we also know b is greater than zero, we can multiply both sides by b, and we have:

ab - b = ab + a
-b = a

Therefore, if a = -b, then (a -1)/(b + 1) = a/b.

Statement One Alone:
a = b - 4

This does not mean a = -b. Statement one alone is not sufficient to answer the question.

Statement Two Alone:
a = -b

This is exactly what we have concluded in the stem analysis. Statement two alone is sufficient to answer the question.

Hence, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Approach Solution 3:
The problem statement informs that:
Given:

a and b are integers, and b > 0

Asked:

Is (a - 1)/(b + 1) = a/b?

Statement 1: a = b − 4
Substituting a for b−4, we get the question: is b − 5/b + 1 equal to b − 4/b?
If b = 1, then the answer to the question will be NO
If b=2, then the answer to the question will be YES
Hence statement 1 is Insufficient.

Statement 2: a = −b
Substituting a for −b, we get the question: – (b+1)/b+1 = −b/b.
Since b > 0, denominators can not be 0 and we can reduce both fractions to −1 = −1
Hence statement 2 is Sufficient

Therefore, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

“If a and b are integers, and b > 0, does (a - 1)/(b + 1) = a/b ?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Advanced Questions”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions include a problem statement followed by two factual statements. GMAT data sufficiency consists of 15 questions which are two-fifths of the total 31 GMAT quant questions.

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If a and b are Integers, and b > 0, Does (a - 1)/(b + 1) = a/b ? GMAT Data Sufficiency

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