If x and y are positive numbers, is (x + 1)/(y + 1) > x/y

If x and y are positive numbers, is \(\frac{x + 1}{y + 1} > \frac{x}{y}\)?

x & y are POSITIVE numbers

\(\frac{x + 1}{y + 1} > \frac{x}{y}\)

Lets simplify the equation, as we are given that x & y are positive integers we can multiply the R.H.S. with the L.H.S. as there will not be any impact on the inequality SIGN of the equation.

\((x + 1) * (y) > (y + 1) * (x)\)

\(xy + y > xy + x\)

Cancelling \(xy\) from both the sides.

\(y > x\)

So we want to answer, Is \(y > x\) ?

(1) \(x > 1\)

This does not tell us anything about y. Hence, Eq. (1) =====> NOT SUFFICIENT

(2) \(x < y\)

We can write it as y > x, this is what we have to prove. Hence, Eq. (2) =====> SUFFICIENT

Hence, Answer is B
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Target question: Is (x + 1)/(y + 1) > x/y ?

Given: x and y are positive numbers

This is a great candidate for rephrasing the target question.

We have the inequality: (x + 1)/(y + 1) > x/y
Since y is positive, we can multiply both sides of the inequality by y
Likewise, since y is positive, we know that y+1 is positive, which means we can multiply both sides of the inequality by (y+1)
When perform both of these multiplications we get: (x + 1)(y) > (x)(y + 1)
Expand to get: xy + y > xy + x
Subtract xy from both sides to get: y > x
So, we can now ask...
REPHRASED target question: Is y > x ?

Statement 1: x > 1
There's no information about y, so there's no way to determine whether or not y > x
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x < y
Perfect!!!
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

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x+1/y+1>x/y
cross multiply
xy +y > xy + x
subtract xy from both sides
y > x or x<y?

Ans B.

Sent from my SM-G935F using GMAT Club Forum mobile app

Might as well share that I picked C due to not Reading that x and y - positive. This would not allow crossmultiplication.

Posted from my mobile device

x and y are positive numbers
\(x,y>0 ;\\
(x+1)y>(y+1)x ;\\
y>x\)

(1) x > 1
X can be a proper fraction or improper fraction and y can be proper or improper fraction
NS
(2) x < y
Rephrased expression
Suff

Option B

We are given that x and y are positive numbers and need to determine whether (x + 1)/(y + 1) > x/y. Simplifying the question, we have:

Is y(x + 1) > x(y + 1)? (Note that we multiplied each side by y(y + 1), which is allowed because y > 0.)

Is yx + y > xy + x?

Is y > x?

Statement One Alone:

x > 1

Since we do not know anything about y, statement one alone is not sufficient to answer the question.

Statement Two Alone:

x < y

We see that statement two has answered the question; y is greater than x.

Answer: B
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(1) x <1 -- Insufficient
(2) x< y
let's understand:

Effects of addition
if the ratio is less than 1: After adding some positive number to the numerator and denominator, the resulting ration increases
eg 2/3
adding 10 to numerator and denominator
2+10 / 3+10 =12/13 > 2/3

if the ratio is greater than 1: After adding some positive number to the numerator and denominator, the resulting ration decreases eg. 7/5
7+20 /5 +20 = 27/25 <75

Hence B is correct

please correct me if I am wrong

Here, x and y are positive numbers (i.e., \(x>0\) and \(y>0\))
If this is the case then \(\frac{x}{y}\) must be positive. If \(\frac{x}{y}\) is positive, then \(\frac{x + 1}{y + 1}\) must be positive too..
So, the question is:
\(\frac{x + 1}{y + 1} > \frac{x}{y}\)?
I can answer this question if I know value of x and y (whatever the value it is!).

In statement 2:
Here is given that \(x < y\).
I don't care what about the specific value of x and y here. I know from this statement that y is greater than x. If the statement even says that x is greater than y, i still don't care because statement NEVER lies!. Without any calculation in statement 2, I can surely say that it gives a definite YES or a definite NO, but NOT the simultaneous YES and NO. So, sufficient.


In statement 1:
There is no value of y. So, insufficient.
Is my understanding wrong Bunuel, IanStewart?
Thanks__
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It might be a silly question, but before ruling Statement 1 out, can't we use the fee values of Y (as the question stem says Y is a positive number) ?
Use few positive values of Y and plug in the question to see if we get all Yes or all No?

cross multiply the given expression,
y(x+1)>x(y+1)
xy+y>xy+x
y>x ???

a. x>1. no info on Y. Not sufficient (Cancel AD)
b. y>x. Sufficient.
Answer B

Answer: Option B

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I get that we can simplify this to Y>X by multiplying.

But shouldn't the sign of inequality also be reversed according to the multiplication rule of inequality. Why it doesn't change ??

To what rule are you referring, SkDv12 ?

Posted from my mobile device

avigutman This is my mistake. As soon as I revised, I knew my error.

"Multiplying or dividing an inequality by a negative number reverses the order of the inequality". It is only if multiplication and division is done by a -ve number. At the time, I was remembering it as with any number.

Your mistake shows that you memorized the "rule" without exploring the reasoning behind it. That approach to studying will significantly limit your score potential, SkDv12. Attaching two images from my book.