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# If m, p, s and v are positive, and m/p < s/v, which of the following

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Manager
Joined: 04 May 2014
Posts: 162
Location: India
WE: Sales (Mutual Funds and Brokerage)
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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01 Oct 2017, 05:19
Picking nos is the best approach.
m,p,s & v are +ve.
m/p<s/v.
Let m/p=1/2=0.5 ie m=1, p=2
and s/v=4/5=0.8 ie s=4 and v=5
0.5<0.8
Plugin the values in the answer choices.
I. (m+s)/(p+v)=(1+4)/(2+5)=5/7=0.71 this is between 0.5 and 0.8
II. ms/pv=4/10=0.4 not between 0.5 and 0.8
III. s/v−m/p=4/5-1/2=0.3 again not between 0.5 and 0.8.
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Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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08 Dec 2017, 11:28
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$
II. $$\frac{ms}{pv}$$
III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

Let’s analyze each statement using specific values for the variables.

We can let m = 2, s = 3, p = 4, and v = 5. Thus:

m/p = 2/4 = 0.5 and s/v = 3/5 = 0.6.

Notice that m/p = 0.5 is less than s/v = 0.6. Now let’s analyze each statement.

I. (m+s)/(p+v)

(2 + 3)/(4 + 5) = 5/9 = 0.555… is between m/p = 0.5 and s/v = 0.6.

II. (ms)/(pv)

(2 x 3)/(4 x 5) = 6/20 = 0.3 is NOT between 0.5 and 0.6.

III. s/v - m/p

3/5 - 2/4 = 0.6 - 0.5 = 0.1 is NOT between 0.5 and 0.6.

From the above, we see that only statement I is true. However, this was illustrated by using one set of numbers (m = 2, s = 3, p = 4, and v = 5). It’s possible that it could be false when we use another set of values for m, s, p, and m.

However, we can prove that (m+s)/(p+v) is between m/p and s/v; that is, we can prove that m/p < (m+s)/(p+v) < s/v regardless of the values we use for m, s, p, and m, as long as the values are positive.

Notice that m/p < (m+s)/(p+v) < s/v means m/p < (m+s)/(p+v) and (m+s)/(p+v) < s/v. Also, keep in mind that we are given that m/p < s/v, which is equivalent to mv < ps.

Let’s prove that m/p < (m+s)/(p+v):

m/p < (m+s)/(p+v) ?

m(p+v) < p(m + s) ?

mp + mv < mp + ps?

mv < ps ? (YES)

Since mv < ps is true, m/p < (m+s)/(p+v) is true. Finally, let’s prove that (m+s)/(p+v) < s/v:

(m+s)/(p+v) < s/v ?

v(m+s) < s(p+v)?

mv + sv < sp + sv ?

mv < ps ? (YES)

Again, since mv < ps is true, (m+s)/(p+v) < s/v is true. Thus we have shown that m/p < (m+s)/(p+v) < s/v is always true.

Answer: B
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Senior Manager
Joined: 29 Jun 2017
Posts: 318
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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22 Dec 2017, 09:29
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$

II. $$\frac{ms}{pv}$$

III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

THIS IS HARD
the point tested here is that we need to change from ratio from to factor form
m/p<s/v
multiple two sides with s.v
mv<sp

from this point you can solve the problem.
Intern
Joined: 24 Oct 2017
Posts: 11
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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07 Jan 2018, 06:02
One very easy way to make I correct is as below
Any number can be expressed in form of A/1 (lets say --3/2 it can be written as 1.5/1)
so basically if you see option 1 is asking us A/1 < (A+B)/(2) < B/1

The average is always in btw A & B

II & III can simply be rejected by taking values 1,2,3,4.
Intern
Joined: 18 Apr 2018
Posts: 2
If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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19 Apr 2018, 00:36
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$

II. $$\frac{ms}{pv}$$

III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

I. if $$\frac{m+s}{p+v}$$ is to be more than $$\frac{m}{p}$$,

==> $$\frac{m}{p}$$<$$\frac{m+s}{p+v}$$
==> pm+mv<pm+ps and since all numbers are positive,
==> $$\frac{m}{p}$$<$$\frac{s}{v}$$ which is true as per question stem.

if $$\frac{m+s}{p+v}$$ is to be less than $$\frac{s}{v}$$,
==> $$\frac{s}{v}$$>$$\frac{m+s}{p+v}$$
==> sp+sv>mv+sv
==> $$\frac{m}{p}$$<$$\frac{s}{v}$$ which is true as per question stem.

II. if $$\frac{ms}{pv}$$ is to be more than $$\frac{m}{p}$$,

==> $$\frac{s}{v}$$ must be greater than 1, which is not conclusive as per stem of question.

III. if $$\frac{s}{v} - \frac{m}{p}$$ is to be more than $$\frac{m}{p}$$,

==> $$\frac{s}{v} must be greater than 2 X [m]\frac{m}{p}$$, which is not conclusive as per stem of question.

So only (I) can be derived with 100% confidence.
Hence Answer must be B.
Intern
Joined: 22 Jan 2018
Posts: 22
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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11 May 2018, 12:36
Bunuel wrote:
siriusblack1106 wrote:
which of the following 'must be' between m/p and s/v?

I didn't understand the question. What does the question mean here by 'must be'?

"Must be" means for any (possible) values of m, p, s and v. So, $$\frac{m}{p}< (option) <\frac{s}{v}$$, must hold true fo any positive values of m, p, s and v.

Must or Could be True Questions to practice: http://gmatclub.com/forum/search.php?se ... tag_id=193

Hope it helps.

Do you have a more elegant solution to this problem Bunuel? Most of the answers here are either choosing numbers or doing complicated calculations that I would find difficult to try during the test.
Intern
Joined: 07 May 2015
Posts: 22
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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13 May 2018, 16:44
DarkBlizzard wrote:
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$
II. $$\frac{ms}{pv}$$
III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

This question is quite easy:
III. is clearly out, because this will be smaller than m/p.
II. is clearly out. Test this with m/p=0 or s/v=1
I. Struggled a minute here, but then I got this idea:

If we pick a number for m/p and a number for s/v: For example: m/p = 1/3 and s/v = 3/4

Now, I we try to get these numbers to the same denominator we get:
4/12 < 9/12
If these numbers are added ab (4+9)/(12+12), we always get a number that is higher than m/p but lower than s/v. This is how I solved this

This works for every number pair. We add the same denominator, but an enumerator which is bigger than the original one. Therefore this has to be bigger than m/p but smaller than s/v! :wink:

is III clearly out? 1 < 100, but 100-1 is in between the 2. am i misreading something?

I picked 1/8 < 2/3 to plug in, and III does work here. with 1/2 < 3/4 however it does not.
Senior Manager
Joined: 31 Jul 2017
Posts: 368
Location: Malaysia
WE: Consulting (Energy and Utilities)
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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14 May 2018, 05:57
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$

II. $$\frac{ms}{pv}$$

III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

As the question asks for Must, we can take test with only one set of value -

s = 5, m = 4, p,v = 1

Only Statement II Satisfies. Hence, Option B.
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Intern
Joined: 07 May 2015
Posts: 22
If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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14 May 2018, 07:16
rahul16singh28 wrote:
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$

II. $$\frac{ms}{pv}$$

III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

As the question asks for Must, we can take test with only one set of value -

s = 5, m = 4, p,v = 1

Only Statement II Satisfies. Hence, Option B.

Yeah I understand. I guess what I'm saying is that picking the right numbers is essential on this one, as if the wrong numbers are picked, III works too. Just realized that there isn't a I and III option though, so it were down to I and III one could pick new numbers
Manager
Joined: 29 Sep 2017
Posts: 77
Location: United States
Concentration: Strategy, Leadership
GPA: 3.3
WE: Consulting (Consulting)
Re: If m, p, s and v are positive, and m/p < s/v, which of the following [#permalink]

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15 May 2018, 12:17
bp2013 wrote:
rahul16singh28 wrote:
imhimanshu wrote:
If m, p, s and v are positive, and $$\frac{m}{p} <\frac{s}{v}$$, which of the following must be between $$\frac{m}{p}$$ and $$\frac{s}{v}$$

I. $$\frac{m+s}{p+v}$$

II. $$\frac{ms}{pv}$$

III. $$\frac{s}{v} - \frac{m}{p}$$

A. None
B. I only
C. II only
D. III only
E. I and II both

As the question asks for Must, we can take test with only one set of value -

s = 5, m = 4, p,v = 1

Only Statement II Satisfies. Hence, Option B.

Yeah I understand. I guess what I'm saying is that picking the right numbers is essential on this one, as if the wrong numbers are picked, III works too. Just realized that there isn't a I and III option though, so it were down to I and III one could pick new numbers

Which is why when picking numbers, it's always best to choose at least 2 set. I picked: 1, 2, 3, 9 for easy math and they made B true and everything else false. If you have time, best to verify with a second set of numbers or algebraically.
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Re: If m, p, s and v are positive, and m/p < s/v, which of the following   [#permalink] 15 May 2018, 12:17

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# If m, p, s and v are positive, and m/p < s/v, which of the following

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