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If m/n < 0 and n - m < 0, which of the following must be true?

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If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   Updated on: 10 May 2020, 10:49
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GMATBusters’ Quant Quiz Question -7

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If m/n < 0 and n - m < 0, which of the following must be true?

i) m > 0
ii) n > 0
iii) 1/m > 1/n

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii
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Official Answer
C

Originally posted by GMATBusters on 09 May 2020, 18:57.
Last edited by GMATBusters on 10 May 2020, 10:49, edited 3 times in total.
Renamed the topic and edited the tags.
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 18:58
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Official Solution:



If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

m/n<0 means m & n are of opposite sign.
one is positive other is negative
nd n-m < 0 means m>n
So, m must be +ve so,
statement 1 must be true.

n<0
Statement 2 is wrong

if m>n
  • we can take reciprocal both sides without flipping the inequality sign when both sides are of opposite sign.
  • we can take reciprocal both sides with flipping the inequality sign when both sides are of the same sign.
  • if we don't know about the signs, we cannot take reciprocal.


Since, both are of opposite sign, 1/m > 1/n

iii) is true.
Hence, i & iii only are true.

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii

Answer C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 19:19
If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

m/n<0 means either one is negative and n-m < 0 means m>n
So, m must be +ve so, statement 1 must be true.
Statement 2 is wrong as it must be -ve.
1/m > 1/n
if m = 2 and n = -2
1/2 > -1/2, yes
if m = 1/5 and n =-2
5>-1/2, yes

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii

Ans. C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 19:19
If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

Now
1) m/n<0 means m and n are of opposite signs.
2) n-m < 0 means m > n

Case 1 – m >0 , n < 0
Condition 2) is always fulfilled. -> 3

Case 2 – m < 0, n > 0
Condition 2) is not satisfied.

So, the only acceptable solutions is Case 1 where m > 0 and n < 0.
i) m > 0 – Yes
ii) n > 0 – No
iii) 1/m > 1/n – Yes
So, Ans - C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 19:28
m/n=+/-<0 as given m>n ie m+ve ,n-ve
hence m>0,correct
and 1/m>1/n correct
Answer C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 20:04
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Kudos
given
m/n<0 and n-m<0
possible when
n>m and m is +ve and n is -ve
out of given options 1 & 3 are correct

check with m=1 and n = -2 and m = 1/2 and n = -3/2

OPTION C

If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 20:27
1. n-m<0 => n<m
2. m/n <0 => one of m or n has to be -ve => from 1. n is -ve and m is +ve

Statements:
i) m > 0 --> true
ii) n > 0 --> false
iii) 1/m > 1/n --> taking reciprocals-- -ve number will still be smaller.
------
ex-
-4 < 2==> -1/4 < 1/2
-------
so 3rd is true. as we already established m is +ve and n is -ve.

Answer: C) i & iii only
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   09 May 2020, 20:50
m/n<0 means that m and n have opposite signs.
n-m<0 means n<m.
Thus n should be negative and m should be positive.

Thus i) m>0 and iii) 1/m> 1/n must be true.

Answer: C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   Updated on: 10 May 2020, 08:55
Asked: If m/n<0 and n-m < 0, which of the following must be true?

m/n<0; m & n have opposite signs; (m>0 & n<0) or (m<0 & n>0)
n-m<0; n<m; m>0 & n<0

i) m > 0
MUST BE TRUE
ii) n > 0
MUST BE FALSE
iii) 1/m > 1/n
Since m>0;1/m>0 & n<0;1/n<0
1/m>0>1/n
MUST BE TRUE

IMO C


Originally posted by Kinshook on 09 May 2020, 21:51.
Last edited by Kinshook on 10 May 2020, 08:55, edited 1 time in total.
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 01:17
Quote:
If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii


Plug in and check

\(m = 1\) & \(n = -2\)

\(\frac{m}{n}<0 = -\frac{1}{2} < 0\) & \(n-m < 0 = -3 < 0\)


Now, Check the options -

i) \(m > 0\) ( True)

\(1 > 0\)

ii) \(n > 0\) (False)

\(-2 < 0\)

iii) \(\frac{1}{m} > \frac{1}{n}\) ( True)

\(1 > -\frac{1}{2}\)

Hence, Answer must be (C)
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 01:21
If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii

Solution:

Given:
m/n<0 and n-m < 0
From m/n< 0, either m or n is negative.
From n – m < 0,
n < m

From above both statements, it can be said that m > 0.
Answer: A
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 03:24
If m/n<0 and n-m < 0, which of the following must be true?
i) m > 0
ii) n > 0
iii) 1/m > 1/n

A) i only
B) ii only
C) i & iii only
D) ii & iii only
E) i, ii, iii

1st Statement for \(\frac{m}{n}<0\), either m<0 and n>0 or m>0 and n<0
2nd Statement for n-m < 0, there are 3 scenarios:
a) n+ and m+ with m>n
b) n- and m+ for all m,n values
c) n- and m- for n<m (e.g. n=-3 and m=-2)

n and m must meet both criteria. Hence, only scenario b (n <0 and m>0) that suffice the criteria
1/n <0 and 1/m >0. the answer is C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 03:44
01) m/n < 0
This means that m and n must have different sign, ie. m is +ve and n is -ve, and vice versa. Neither m nor n is zero.

02) n - m < 0 --> n < m
Because m and n must have different signs, it is logical that m is +ve and n is - ve.

Let's review one by one

i) m > 0
MUST BE TRUE. This is aligned with the explanation in 02)

ii) n > 0
FALSE. This contradicts with the explanation in 02)

iii) 1/m > 1/n
Because m is +ve and n is -ve, this MUST BE TRUE.

FINAL ANSWER IS (C) i & iii only

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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 03:57
Given that:
m / n < 0 ---(1)
n - m < 0 --- (2)

From (2):
n < m ---(3)

From (1) and (3):
m must be positive (i.e. m > 0) >> (i) is True
n must be negative (i.e. n < 0) >> (ii) is False

Since:
n < m --- (3) and
m is Positive
n is Negative

Then:
(divide by mn on both sides, flip sign once)
1/m > 1/n >> (iii) is True

Alternatively note that:
m is Positive >> 1/m is Positive
n is Negative >> 1/n is Negative
Therefore: 1/m > 1/n >> (iii) is True

Answer: (i) and (iii) must be true, C
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 04:14
IMO C

1) m and n has opposite signs for sure .
2) m is greater than n
(from Q.S.)

therefore
m has to be + ve
n has to be - ve

1 stmt is correct
m is greater than n
2 stmt is incorrect
3 stmt is correct since even the reciprocal of -ve is less than the reciprocal of positive

Posted from my mobile device
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink] New post   10 May 2020, 07:04
\(m/n < 0\) =>
i)m=+ve and n=-ve
or
ii) m=-ve and n=+ve

Since \(n<m\) => Case ii) is invalid.

Hence m > 0 and n < 0

i) m > 0 must always be true
ii) n > 0 will never be true
iii) Since n<m => 1/n > 1/m - Not true

Only i) must be true

Answer - A
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Re: If m/n < 0 and n - m < 0, which of the following must be true? [#permalink]
10 May 2020, 07:04
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