If x and y are positive integers, is x/y < (x+5)/(y+5)?

Concept: When the same positive number is added to the numerator as well as denominator of a positive fraction, the fraction tends toward 1. If the original fraction is smaller than 1, it increases in value. If the original fraction is greater than 1, it decreases in value. Try a few values say 2/3. Add 1 to both to get 3/4 which is closer to 1 than 2/3. Another value say 5/4. Add 1 to both to get 6/5 which is less than 5/4.

We want to know that when we add 5 to both x and y of the fraction x/y, do we get a smaller fraction or a larger fraction. This depends on whether x/y is less than 1 or greater than 1.

(1) y = 5
Doesn't tell us whether x/y is greater than 1 or less than 1.

(2) x > y
This tells us that x/y > 1. So when we add 5 to both numerator and denominator, the fraction will decrease in value. This gives us that x/y > x+5/y+5.
Sufficient alone.

Answer (B)

Target question: Is x/y < (x+5)/(y+5)?

This is a great candidate for REPHRASING the target question.
Aside: We have a free video with tips on rephrasing the target question (below)

If y is a positive integer, then we can be certain that y and (y+5) are both POSITIVE. This allows us to safely multiply both sides of our inequality by y and (y+5).

Take x/y < (x+5)/(y+5) and multiply both sides by y to get: Is x < (y)(x+5)/(y+5)?
Then take x < (y)(x+5)/(y+5) and multiply both sides by (y+5) to get: Is x(y+5) < (y)(x+5)?
Expand to get: Is xy + 5x < xy + 5y?
Subtract xy from both sides to get: Is 5x < 5y?
Divide both sides by 5 to get: Is x < y?

We now have a very simple, REPHRASED target question.....
REPHRASED target question: Is x < y?

Now onto the statements!

Statement 1: y = 5
Since we have no information about x, there's no way to tell whether or not x < y.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x > y
Perfect!
If x > y then we can answer our REPHRASED target question with certainty: NO, x is definitely not less than y.
So, statement 2 is SUFFICIENT

Answer = B

RELATED VIDEO

Simplify (we can cross multiply, since x and y are positive)

xy+5x<xy+5y
5x<5Y
So basically the question asks if x<y

st1. nothing about x-> insf.
st2. clear answer-> suff.

Answer B

MAGOOSH OFFICIAL SOLUTION:

We are adding the same number, 5, to both the numerator and the denominator, so the value of x/y will move closer to 1. All we need to determine is whether x/y is greater than 1 or less than 1.

Statement #1: y = 5. Here, we have a definite value for y, but zero information about x. If y = 5, some fractions (1/5) can be less than one, while others (7/5) will be greater than one. Either is possible. Since both are possible, we can’t give a definitive answer to the prompt. This statement, alone, by itself, is insufficient.

Statement #2: x>y. Dividing both sides of this inequality by y, we get (x/y) > 1. This means x/y must be a fraction greater than 1, which means the resultant fraction (x + 5)/(y + 5) must be closer to one, which means the resultant fraction must be smaller. Therefore, we can definitively say: the answer to the prompt question is, “No.” Because we can give a definite answer to the prompt, we have sufficient information. This statement, alone, by itself, is sufficient.
Statement #1 is insufficient and Statement #2 is sufficient.

Answer = B.
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according to the solution
x>y
lets take x=2 y=1
2<7/6
its not valid
Can someone give the solution?

Hi ShravyaAlladi
See here the question is asking us whether x/y<x+5/y+5
There are two ways to solve this
First one => Using the concept of Fractions and decimals => Adding the same thing number to the numerator and the denominator just bring the number closer to one. Hence the question is really asking us if x/y is a proper fraction or an improper one.
Here if x>y => x/y will be an improper fraction
Hence x+5/y+5 will be less than the x/y so the answer is always a NO.
Second using the concept of inequality we can cross multiply to change the sides and the stem of the question reduces to is x/y<1

AS per your query you are absolutely right
The reason B is the answer is that no matter what be the values of x and y ; if x>y => x/y>1
so if x=2 y=1 => the answer is a NO
The answer to the question is always a NO

Refer to this to learn this concept => https://magoosh.com/gmat/2012/gmat-short ... nominator/
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Hello!

I understand this concept, however, when I first went through this question, I answered D.

The reason is that for (1), I substituted y=5 into the equation, which lead me to x<5. And if x<5 then this statement is sufficient as that would make x<y.

Could someone please explain the flaw in my reasoning?

Thank you!

If x and y are positive integers, is x/y < (x+5)/(y+5)?


x/y < (x+5)/(y+5)

0 < (x+5)/(y+5) - x/y

0 < y(x+5) - [x(y+5)]/ y(y+5)....looks ugly but simplifying we get

0 < (5y-5x) / y(y+5)

0 < 5(y-x) / y(y+5)

As y & x are positive integers, the denominator is always positive.

teh question repharsed to be

is y-x> 0?? or is y>x

(1) y = 5

No info about x. We do not know sign of (y-x)

Insufficient

(2) x > y

Directly we can say: y-x<0 means it is ALWAYS negative. Answer is ALWAYS No

Sufficient

Answer: B

\(\frac{x}{y}<\frac{x+5}{y+5}\)
=> \(x<y?\)

1. y=5, no information about x. -> INSUFFICIENT.
2. x>y -> SUFFICIENT.

B.
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1) Insufficient since we need the value of both x and y to determine x/y. So we can know if x/y is less than 1 or more than one.
2) sufficient
Answer: B

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