Is \(\frac{x+1}{y+1} > \frac{x}{y}\) GMAT Data Sufficiency

Question: Is\(\frac{x+1}{y+1}>\frac{x}{y}\)?

1. \(0 < x < y\)
2. \(xy > 0\)

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

Answer:
Approach Solution (1):
S1: 0 < x < y
This tells us that y is positive, which means y + 1 is also positive
This means we can safely take the inequality \(\frac{x+1}{y+1}>\frac{x}{y}\) and safely multiply both sides by y
When we do so, we get: \(y*\frac{x+1}{y+1}>x\)
We can also multiply both sides by y + 1 to get:\(y*(x+1)>x(y+1)\)
Expand to get: xy + y > xy + x
Subtract xy from both sides, we will get: y > x
Since S1 tells us that y > x, the answer to the target question is a definitive yes
Since we can answer the target question with certainty, statement 1 is sufficient
S2: xy > 0
Let’s test some values
There are several values of x and y that satisfy statement 2 (xy > 0).
Here are two:
Case a: x = 1 and y = 1
In this case,\(\frac{x+1}{y+1}=\frac{(1+1)}{(1+1)}=\frac{2}{2}=1, and \frac{x}{y}=\frac{1}{1}=1\)
So the answer to the target question is NO,\(\frac{x+1}{y+1}\)is not greater than x/y
Case b: x = -3 and y = -2
In this case,\(\frac{x+1}{y+1}=\frac{(-3+1)}{(-2+1)}=\frac{-2}{-1}=2, and \frac{x}{y}=\frac{-3}{-2}=\frac{3}{2}\)
So, the answer to the target question is yes,\(\frac{x+1}{y+1}\)Is greater than x/y?
Since we cannot answer the target question with certainty, S2 is not sufficient
Correct option: A

Approach Solution (2):
This can be re-written as:\(\frac{x+1}{y+1}-\frac{x}{y}>0\)
Simplifying this further, we get:\(\frac{y(x+1)-x(y-1)}{(y+1)(y)}>0\rightarrow\frac{xy+y-xy+x}{y(y+1)}>0\rightarrow\frac{x+y}{y(y+1)}>0\)
The rephrased question now reads:\(\frac{x+y}{y(y+1)}>0\)
(1) 0 < x < y
Since x and y both are positive, the expression will always be positive
Hence sufficient
(2) xy > 0
Here both x and y can be positive or negative. If x and y are negative, the expression is not positive. We don’t get a unique answer.
Hence insufficient
Correct option: A

Approach Solution (3):
(1) Multiplying both sides by y (y + 1) (note that both are positive), gives yx + y > yx + x and cancelling out yx gives y > x
This is exactly what we are told in our statement
Sufficient
(2) If y (y + 1) is positive we can repeat the same process as above to get y > x, but since all we know is that both x, y are positive and both are negative but do not know which is larger we cannot answer.
If y (y + 1) is negative we will get the expression y < x, but for the same reason as above, we still cannot answer.
Insufficient
Correct option: A

“Is \(\frac{x+1}{y+1}-\frac{x}{y}\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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