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

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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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Updated on: 30 Jul 2012, 05:17
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If m > 0 and n > 0, is (m+x)/(n+x) > m/n?

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

Originally posted by LM on 10 May 2010, 09:32.
Last edited by Bunuel on 30 Jul 2012, 05:17, edited 1 time in total.
Edited the question and added the OA.
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10 May 2010, 14:38
3
3
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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Director
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10 May 2010, 21:01
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.

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

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17 Jan 2013, 03:57
1
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
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Re: If m > 0 and n > 0, is (m+x)/(n+x) > m/n?  [#permalink]

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26 Feb 2013, 04:02
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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26 Feb 2013, 04:07
1
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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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02 May 2014, 19:24
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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03 May 2014, 00:17
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

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

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03 May 2014, 04:50
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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03 May 2014, 06:55
Yes, I realize this now, and that with (1), we know that the answer to this question statement is Yes.

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

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26 Jan 2017, 20:12
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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