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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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Joined: 12 Sep 2015
Posts: 3853
Re: If m, p, s and v are positive, and m/p < s/v, which of the following  [#permalink]

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31 Jan 2019, 16:41
Top Contributor
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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

The key word here is MUST.
So, if we can show that a certain statement is NOT TRUE we can eliminate some answer choices.

GIVEN: m/p < s/v

So, let's see what happens when m = 1, p = 3, s = 1 and v = 2
We get the inequality: 1/3 < 1/2, which works.

Now test the 3 statements:

I) (m+s)/(p+v) = (1+1)/(3+2) = 2/5
Since it IS the case that 1/3 < 2/5 < 1/2, statement I COULD be true

II) ms/pv = (1)(1)/(3)(2) = 1/6
Here, 1/6 < 1/3 < 1/2
In other words, 1/6 is NOT between 1/3 and 1/6
So, statement II need not be true.
ELIMINATE II

III) s/v - m/p = 1/2 - 1/3 = 1/6
Here, 1/6 < 1/3 < 1/2
In other words, 1/6 is NOT between 1/3 and 1/6
So, statement III need not be true.
ELIMINATE III

At this point, only answer choices A and B remain.
We COULD try testing more values in the hopes that statement I may not be true.
Or we can try to convince ourselves that statement I IS true.
Let's try the latter.

I) (m+s)/(p+v)
Is it the case that (m+s)/(p+v) is BETWEEN m/p and s/v?
Let's first see whether (m+s)/(p+v) < s/v
Since v is POSITIVE, we can safely multiply both sides by v to get: v(m+s)/(p+v) < s
Since (p+v) is POSITIVE, we can safely multiply both sides by (p+v) to get: v(m+s) < s(p+v)
Expand to get: vm + sv < sp + sv
Subtract sv to get: vm < sp
Divide both sides by p to get: vm/p < s
Divide both sides by v to get: m/p < s/v
So, our (m+s)/(p+v) < s/v turns into the inequality m/p < s/v, which is GIVEN information.
Since we know that the inequality m/p < s/v is true, it must also be the case that the inequality (m+s)/(p+v) < s/v is also true

Using the same technique to show that m/p < (m+s)/(p+v) [I'll leave it to you to do that ]

Since we can show that m/p < (m+s)/(p+v) < s/v, we can conclude that statement I is true.

Cheers,
Brent
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Joined: 17 Jul 2016
Posts: 55
If m, p, s and v are positive, and m/p < s/v, which of the following  [#permalink]

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03 Mar 2019, 13:48
Let's analyze statement 1.

Let me first ask you this. What would happen IF we had the fraction ([m+am]/[p+ap])?
We could factor 1+a and get m/p.

Now let ma=s.
Now let's compare the fraction ma/pa to s/v. The numerators are equal. V has to be less than pa. If two positive fractions have the same numerator the fraction
with the lesser denominator is greater. So pa>v.
Now let's compare ([m+am]/[p+ap]) to ([m+s]/[p+v]). ([m+s]/[p+v]) must be greater since the numerators are equal but p+ap is greater than p+v.

We can use very similar reasoning to show that ([m+s]/[p+v]) is less than s/v. So it must lie between m/p and s/v.

Statement 2 can easily be shown to be false using the numbers .9 and .1

Statement 3 can be shown to be false using the numbers .9 and .8
Director
Joined: 29 Jun 2017
Posts: 776
Re: If m, p, s and v are positive, and m/p < s/v, which of the following  [#permalink]

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13 Apr 2019, 20:32
all the number are positive, we multiple all the inequalities to find the answer
this can be done is 1 minutes.
Re: If m, p, s and v are positive, and m/p < s/v, which of the following   [#permalink] 13 Apr 2019, 20:32

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