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

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If m>0 and n>o, is (m+x)/(n+x) greater than m/n ? [#permalink]  14 Apr 2011, 02:14
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If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?

(1) m < n
(2) x > 0
[Reveal] Spoiler: OA
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Re: Another DS [#permalink]  14 Apr 2011, 02:50
whichscore wrote:
If m>0 and n>0, is (m+x)/(n+x) greater than m/n ?
1. m < n
2. x > 0

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

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

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

A: Does $$\frac{x(n-m)}{n(n+x)} > 0$$?

1. $$m<n$$

$$m-n<0$$
$$n-m>0$$

But, what if $$x<0 \hspace{2} & \hspace{2} |x|<n$$
Then the answer to the question would be No.

If $$x>0$$
Then the answer to the question would be Yes.
Not Sufficient.

2. $$x>0$$

Denominator is +ve.
Numerator must be +ve to make the expression true.

$$n-m>0$$
$$n>m$$ should be true to make the expression true. But, it is not given which of the two is greater.

If n>m, the answer would be Yes.
If n<m, the answer would be No.

Combining both statements:
Denominator becomes +ve.
Numerator also becomes +ve as m<n
Answer to the question is Yes.
Sufficient.

Ans: "C"
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Re: Another DS [#permalink]  14 Apr 2011, 03:26
1) Insufficient.
Lets consider m = 1 n = 2
If x = 0 The answer is NO
If x = 1 The answer is YES

2) Insufficient
consider m = n = 1 x = 1 The answer is NO
Consider m = 1 n = 2 x = 1 The answer is YES

1) + 2) Consider m = 1 n = 2 x = 1 The answer is YES. Sufficient

whichscore wrote:
If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?
1. m < n
2. x > 0
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Re: Another DS [#permalink]  14 Apr 2011, 04:24
As per (1)

m = 2, n = 3

m/n = 2/3

if x = -2, then 0/1 < 2/3

if x = 0 then (m+0)/(n +0) = m/n

if x = 2 then (2 + 2)/(3+2) = 4/5 > 2/3

(1) is insufficient

As per (2)

x > 0, but it's possible that m = n , and hence (m+x)/(n+x) = m/n

(2) is insufficient

(1) and (2) together

if x = 1, m = 2, n = 3

(2 + 1)/(3 + 1) = 3/4 hence m/n < (m+x)/(n+x)

if x = 2, m = 2, n = 3 then (2 + 2)/(3+2) = 4/5 > 2/3 hence m/n < (m+x)/(n+x)

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Re: Another DS [#permalink]  14 Apr 2011, 05:12
Rephrasing given expression we have ((n-m)x)/(n(n+x)) >0?

1. Not sufficient as we don't know anything about x.

2. Not sufficient as we don't know anything about m,n

Together we have n>m & x>0 . sufficient to answer.

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Re: Another DS [#permalink]  14 Apr 2011, 13:31
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Expert's post
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: Another DS [#permalink]  15 Apr 2011, 11:20
great explanation. thanks karishma.
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Re: If m>0 and n>o, is (m+x)/(n+x) greater than m/n ? 1. m [#permalink]  06 Mar 2012, 06:30
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 ? 1. m [#permalink]  06 Mar 2012, 07:08
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Expert's post
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 ? 1. m [#permalink]  06 Mar 2012, 10:17
Expert's post
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>o, is (m+x)/(n+x) greater than m/n ? [#permalink]  07 Mar 2012, 09:53
@flute, thanks for the solution.
Re: If m>0 and n>o, is (m+x)/(n+x) greater than m/n ?   [#permalink] 07 Mar 2012, 09:53
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