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

\(\frac{m+x}{n+x} > \frac{m}{n}\)

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

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

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

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"
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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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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.

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.

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!

Answer: C

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.

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?

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.

Note:
If \(\frac{a}{b} > \frac{c}{d}\) and all values are positive, then:
Rephrase 1: \(ad > bc\)
Rephrase 2: \(\frac{a}{c} > \frac{b}{d}\)

Alternate approach:

Statement 2: x > 0
Applying rephrase 2 to the question stem, we get:
\(\frac{m+x}{m} > \frac{n+x}{n}\)
\(1 + \frac{x}{m} > 1 > \frac{x}{n}\)
\(\frac{x}{m} > \frac{x}{n}\)
Applying rephrase 1, we get:
\(nx > mx\)
If we divide by positive value x, we arrive at the following rephrase of the question stem:
Is n > m ?
No way to determine whether n > m.
INSUFFICIENT.

Statement 1: m < n
Let m=1 and n=2.
Plugging these values into the question stem, we get:
\(\frac{1+x}{2+x} > \frac{1}{2}\) ?
If x=1, the answer is YES, since 2/3 > 1/2.
If x=-1, the answer is NO, since 0 < 1/2.
INSUFFICIENT.

Statements combined:
Rephrased question stem in Statement 2: Is n > m ?
Statement 1 indicates that the answer to this question is YES.
SUFFICIENT.

From the question data, we know that both m and n are positive; however, what the question does not tell us is which one of the two is bigger.

\(\frac{m }{ n}\) can be looked at as a ratio of two terms. A ratio of two terms increases / decreases depending on:

whether m is lesser than n or otherwise
whether a constant is being added or subtracted

If m<n, adding the same number to both terms of the ratio will increase its value and subtraction does the opposite.

If m>n, the contrary is true.

Therefore, we have to look for a relationship between m and n and also try to understand if we are adding or subtracting a constant

From statement I alone, m < n.
No information about the constant being added / subtracted.
Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, x > 0. This means a positive value is being added.
No information about the relationship between m and n.
Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:

From statement I, we understand that the ratio m/n is such that m<n.
From statement II, we know that x is a positive number.
Therefore, we are adding the same value to both the terms of a ratio m:n, where m<n. The resultant ratio will definitely be bigger,

Is \(\frac{(m+x)}{(n+x)}\) > \(\frac{m}{n}\)? YES.

The combination of statements is sufficient to answer the question. Answer option E can be eliminated.

The correct answer option is C.

Hope that helps!
Aravind B T

Hi Bunuel you shared a very simple way that I was not able to see. What is wrong with the following approach that I attempted:

rearranging the question:

If m > 0 and n > 0, is m+x/n+x>m/n? ==> bcoz we are adding same value to both num. and denom. I rephrased the question as is the new m/n > old m/n -- only possible when m/n is a proper fraction (as it tend to move towards 1 and increases in value). Therefore: is m<n (because its a proper fraction).?

Statement 1: gives the m<n - hence sufficient
statement 2: says nothing about m and n - hence insufficient.

Wondering where I'm going wrong.

Thanks

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