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Director  Joined: 03 Sep 2006
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If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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Question Stats: 67% (01:40) correct 33% (01:39) wrong based on 521 sessions

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If m > 0 and n > 0, is $$\frac{m + x}{n + x} > \frac{m}{n}$$?

(1) m < n

(2) x > 0

Originally posted by LM on 10 May 2010, 09:32.
Last edited by Bunuel on 03 Apr 2019, 21:35, edited 2 times in total.
Edited the question and added the OA.
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Re: If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?  [#permalink]

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3
whichscore wrote:
If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?
1. m < n
2. x > 0

It is useful to understand that when you add the same positive number to the numerator and denominator of a positive fraction, it moves towards 1.

Say 1/2 = 0.5
I add 1 to both numerator and denominator, it becomes 2/3 = 0.67 i.e. closer to 1 than 1/2
Say 3/2 = 1.5
I add 1 to both numerator and denominator, it becomes 4/3 = 1.33 i.e. closer to 1

When you add the same negative number (or in other words, subtract the same positive number) to the numerator and denominator of a positive fraction, it moves away from 1.

Say 1/2 = 0.5
I add -1 to both numerator and denominator, it becomes 0 i.e. farther from 1 than 1/2
Say 3/2 = 1.5
I add -1 to both numerator and denominator, it becomes 2/1 i.e. farther from 1

Once you understand this, the question takes a few seconds.

1. m < n
The fraction m/n must be less than 1. So if x is positive, the fraction will increase and move towards 1. If x is negative, the fraction will decrease to move away from 1. Not sufficient.

2. x > 0
x is positive but we do not know whether the fraction is greater than 1 or less than 1. If m/n is greater than 1, it will decrease and move towards 1. If m/n is less than 1, it will increase and move towards 1.

Using both together, m/n is less than 1 and x is positive so we are adding the same positive number to both numerator and denominator. Hence (m+x)/(n+x) will be greater than m/n to move towards 1.
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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If m>0 and n>0, is (m+x)/(n+x) > m/n?

(1) m < n. No info about x. Not sufficient.
(2) x >0. No info about m and n. Not sufficient.

(1)+(2) As from the above two statements nominators and denominator of both fractions are positive, we can crossmultiply --> is $$\frac{m+x}{n+x}>\frac{m}{n}$$ --> is $$(m+x)n>(n+x)m$$ --> is $$mn+xn>mn+xm$$ --> is $$x(n-m)>0$$ --> as $$x>0$$ and $$n>m$$, then $$x(n-m)>0$$ is true. Sufficient.

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$$\frac{m+x}{n+x} > \frac{m}{n}$$

$$\frac{m+x}{n+x}-\frac{m}{n} > 0$$

$$\frac{mn+nx-mn-mx}{n+x}> 0$$

$$\frac{x(n-m)}{n+x}> 0$$

1. $$m<n$$

Thus, n-m>0

Now, if x>0; the fraction will be greater than 0.
If x<0 but |x|<n; the fraction will be less than 0.

2. $$x>0$$

If x>0;
m<n; the fraction will be greater than 0
m>n; the fraction will be less than 0

Together;
The fraction is greater than 0.

Ans: "C"
********************************************************************

Since, we don't know the sign of x; we can't cross multiply as x can be negative and |x|>n
***********************************************************************
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##### General Discussion
Director  Joined: 03 Sep 2006
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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Bunuel wrote:
If m>0 and n>0, is (m+x)/(n+x) > m/n?

(1) m < n. No info about x. Not sufficient.
(2) x >0. No info about m and n. Not sufficient.

(1)+(2) As from the above two statements nominators and denominator of both fractions are positive, we can crossmultiply --> is $$\frac{m+x}{n+x}>\frac{m}{n}$$ --> is $$(m+x)n>(n+x)m$$ --> is $$mn+xn>mn+xm$$ --> is $$x(n-m)>0$$ --> as $$x>0$$ and $$n>m$$, then $$x(n-m)>0$$ is true. Sufficient.

Did you score 60 in the Quant or are you working with the GMAC!!! Awesome dexterity in giving the solutions.
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Re: If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?  [#permalink]

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Why cant we cross multiply to get; mn + nx > mn + mx, simplify, resulting is n>m??
is this totally wrong?
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Re: If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?  [#permalink]

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dchow23 wrote:
Why cant we cross multiply to get; mn + nx > mn + mx, simplify, resulting is n>m??
is this totally wrong?

Cross-multiplying (m+x)/(n+x)>m/n would be wrong, since we don't know whether n+x is positive or negative: if n+x>0 then we would have as you've written mn+nx>mn+mx BUT if n+x<0 then when multiplying by negative number we should flip the sign of the inequity and write mn+nx<mn+mx.

General rule: never multiply or divide inequality by a variable (or by an expression with variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality.

Hope it's clear.
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Re: If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?  [#permalink]

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dchow23 wrote:
Why cant we cross multiply to get; mn + nx > mn + mx, simplify, resulting is n>m??
is this totally wrong?

Yes, cross multiplying is incorrect here. We do not know whether (n+x) is positive or negative. We know that m and n are positive but we know nothing about x. When you multiply an inequality by a positive number, the inequality sign stays the same but when you multiply an inequality by a negative number, the inequality sign flips. So before you multiply/divide an inequality by a variable, you need to know whether the variable is positive or negative.
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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LM wrote:
If m > 0 and n > 0, is (m+x)/(n+x) > m/n?

(1) m < n.
(2) x > 0.

I used plug in...

1.
let m=3 and n=4 and x = 1
$$\frac{m}{n} = \frac{3}{4}$$ while $$\frac{m+x}{n+x}= \frac{4}{5}$$
$$\frac{3}{4} < \frac{4}{5}$$ YES!

let m=3 and n=4 and x=-1
$$\frac{m}{n} = \frac{3}{4}$$ while $$\frac{m+x}{n+x}= \frac{2}{3}$$
$$\frac{3}{4} > \frac{2}{3}$$ NO!

thus, INSUFFICIENT!

2. x > 0
From statement 1 we tested m=3 and n=4 and x=1 (see that x>0 here) and we got YES!

let m=4 and n=3
$$\frac{m}{n} = \frac{4}{3}$$ while $$\frac{m+x}{n+x}= \frac{5}{4}$$
$$\frac{4}{3} > \frac{5}{4}$$ NO!

thus, INSUFFICIENT!

Together, we combine and using statement 1 where when x>0 we get YES!

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Director  Joined: 29 Nov 2012
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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Why can't we cross multiply in the original statement?
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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fozzzy wrote:
Why can't we cross multiply in the original statement?

Never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know its sign.

We don't know whether n+x is positive or negative, thus don't know whether we should flip the sign or not.
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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Hi Bunuel,

Why isn't the answer B given that in the below steps, (2) gives us the same information as in (1)?

(2)
Because we know that both m and n are positive and that x is positive, we can safely cross-multiply.
(m+x)*n > (n+x)*m
mn + xn > mn + xm
xn > xm
n > m
Because we now know that n > m, we can use the same steps that you used for C to answer the question and only (2) will be sufficient to answer the problem.
Please tell me where I am going wrong here.
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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LM wrote:
If m > 0 and n > 0, is (m+x)/(n+x) > m/n?

(1) m < n.
(2) x > 0.

Statement I is insufficient:

Let us say m = 4 and n = 5

Is (4+x)/(5+x) > 4/5?
(Take a hint from the second statement - Apply the negation test)

(4 - 4)/(5-4) is not greater than 4/5
(4 + 5)/(5+5) is greater than 4/5

Statement II is not sufficient:
(4 + 5)/(5+5) is greater than 4/5
(5 + 4)/(4 + 4) is not greater than 5/4

Combining is sufficient:
m > n and x is positive
Cross multiplying the inequality:
(mn + nx) > mn + mx
n > m which is true in statement I

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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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TooLong150 wrote:
Hi Bunuel,

Why isn't the answer B given that in the below steps, (2) gives us the same information as in (1)?

(2)
Because we know that both m and n are positive and that x is positive, we can safely cross-multiply.
(m+x)*n > (n+x)*m
mn + xn > mn + xm
xn > xm
n > m
Because we now know that n > m, we can use the same steps that you used for C to answer the question and only (2) will be sufficient to answer the problem.
Please tell me where I am going wrong here.

For (2) we don't know whether n>m.

The question asks whether (m+x)/(n+x) > m/n. For (2) when you simplify the question becomes is n>m? This is not given, that;s exactly what we need to find out.

Does this make sense?
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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Yes, I realize this now, and that with (1), we know that the answer to this question statement is Yes.

Posted from my mobile device
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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1
Since we know m and n both are +ve, so we can cross multiply m and n in the question.
So,
the question becomes,
Is (m+x)/(n+x)> m/n?
Is n(m+x)>m(n+x) ?
Is nm + nx > mn + mx ?
cancel out mn from both sides, gives us

Is nx > mx ? or Is x(n-m) > 0 ?

Now St 1 only:
1. m < n We don't know anything abt x to answer our new re-phrased question. Insufficient.

St 2 only:
2. X> 0 relation between m and n not known. So Insufficient.

Now combined,
We know x > 0 i.e +ve and m < n so nx > mx answer is yes.

We can test values here too now to confirm,
x = 1, n = 3, m= 2, so nx > mx is 1.3 > 2.1 ie. 3>2 so yes.

So if x was -ve . i.e x< 0 then the inequality would have been revered. So both the stmts combined are sufficient.
Hence C.
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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A quick hint for such questions :

when X>0, y>0

case 1.

x>y i.e x/y>1
and a>0

then, x+a/y+a < x/y

the reason for such a result is since x>y the percentage change (%) in y will be greater than x, hence it will result in a fraction less than x/y

case 2: x<y i.e x/y <1

a>0

now, (x+a)/(y+a) > x/y

same reason as above, here percentage change is greater in the numerator.

Now from statement 1 we are sure that m<n but this result is still not sufficient because the equality can change depending on the value of (x in the question stem)

i.e if x>y , x/y>1

but a < 0

then, (x+a)/(y+a) > x/y

thus, value statement 1 and 2 together are sufficient

here is a link which can be really helpful for such questions
https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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Adding a positive constant to both numerator and denominator of a proper fraction increases the fractions value as it moves closer to 1

However, adding a positive constant to both numerator/denominator of an improper fraction decreases the value.
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Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?  [#permalink]

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As m>0 and n>0 , I can say that m/n is definitely greater than 0.
which makes it m+x/n+x >0 ; hence the value now solely depends on the value of x ;
if X is positive ; the m+x/n+x >0 if x is negative its value depends on m,n etc..
Now from (2) I know x>0 which should help me arrive at conclusion l hence I feel the answer is B.
Can someone please correct me if I'm wrong ?

LM wrote:
If m > 0 and n > 0, is $$\frac{m + x}{n + x} > \frac{m}{n}$$?

(1) m < n

(2) x > 0 Re: If m > 0 and n > 0, is (m + x)/(n + x) > m/n?   [#permalink] 04 Oct 2019, 06:59
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