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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

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. Answer (C)
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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

1

This post received KUDOS

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.

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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