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

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

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Updated on: 03 Apr 2019, 21:35
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66% (01:43) correct 34% (01:39) wrong based on 350 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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14 Apr 2011, 14:31
7
1
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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10 May 2010, 14:38
3
4
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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27 Jul 2011, 10:00
6
$$\frac{m+x}{n+x} > \frac{m}{n}$$

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

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

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

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

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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.

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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06 Mar 2012, 07:30
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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06 Mar 2012, 08:08
2
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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06 Mar 2012, 11:17
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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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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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
1
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

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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
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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25 Mar 2019, 23:44
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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