Which of the following options must be truth?

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Thu Sep 07, 2017 3:00 pm

If m, p, s and v are positive, and m/p < s/v, which of the following must be between m/p and s/v.

I. (m+s)/(p+v).
II. (ms)/(pv).
III. s/v - m/p.

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

What properties can we use to find the right option quickly? What happen if the numbers are negative?

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Source: Beat The GMAT — Problem Solving | Thu Sep 07, 2017 3:00 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

Which of the following options must be truth?

by GMATGuruNY » Thu Sep 07, 2017 3:12 pm

If m, p, s and v are positive, and m/p < s/v, which of the following must be between m/p and s/v?

I. (m+s)/(p+v)
II. ms/pv
III. s/v - m/p

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

Case 1: m=1, p=2, s=1 and v=1
In this case, m/p = 1/2 and s/v = 1/1 = 1.
Eliminate any statement that does not yield a value between 1/2 and 1.

I: (m+s)/(p+v) = (1+1)/(2+1) = 2/3.
Since 2/3 is between 1/2 and 1, hold onto I.

II: ms/pv = (1*1)/(2*1) = 1/2.
Since 1/2 is NOT between 1/2 and 1, eliminate any answer choice that includes II.
Eliminate C and E.

III: s/v - m/p = 1/1 - 1/2 = 1/2.
Since 1/2 is NOT between 1/2 and 1, eliminate any remaining answer choice that includes III.
Eliminate D.

Test whether Statement I holds true when m/p and s/v are VERY CLOSE.
Case 2: m=9, p=10, s=1, and v=1.
In this case, m/p = 9/10 = 9/10 and s/v = 1/1 = 1.

I: (m+s)/(p+v) = (9+1)/(10+11) = 10/11.
Since 10/11 is between 9/10 and 1, statement I holds true.

Since statement 1 holds true even when the distance between m/p and s/v is extremely small, we should be satisfied:
Statement I must yield a value between m/p and s/v.

The correct answer is B.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

Which of the following options must be truth?

by GMATGuruNY » Thu Sep 07, 2017 3:33 pm

Vincen wrote:If m, p, s and v are positive, and m/p < s/v, which of the following must be between m/p and s/v.

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

Rule:
If a/b < c/d and all four values are positive, then a/c < b/d.

Below is an algebraic proof for Statement I, given that m/p < s/v and that all four values are positive.

Given condition:
m/p < s/v

Applying the rule in blue, we get:
m/s < p/v

Since Statement I refers to m+s and p+v, we want to add s to the value of m on the left side and v to the value of p on the right side.
To this end, add s/s to the left side and v/v to the right side, the equivalent of adding 1 to each side:
m/s + s/s < p/v + v/v

Put each side over a common denominator:
(m+s)/s < (p+v)/v

Applying the rule in blue, we get:
(m+s)/(p+v) < s/v.

Given condition, rephrased:
s/v > m/p

Applying the rule in blue, we get:
s/m > v/p

Since Statement I refers to m+s and p+v, we want to add m to the value of s on the left side and p to the value of v on the right side.
To this end, add m/m to the left side and p/p to the right side, the equivalent of adding 1 to each side:
s/m + m/m > v/p + p/p

Put each side over a common denominator:
(m+s)/m > (p+v)/p

Applying the rule in blue, we get:
(m+s)/(p+v) > m/p.

Thus:
m/p < (m+s)/(p+v) < s/v.

Last edited by GMATGuruNY on Fri Sep 08, 2017 5:07 am, edited 1 time in total.

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Thu Sep 07, 2017 9:34 pm

Vincen wrote:If m, p, s and v are positive, and m/p < s/v, which of the following must be between m/p and s/v.

I. (m+s)/(p+v).
II. (ms)/(pv).
III. s/v - m/p.

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

What properties can we use to find the right option quickly? What happen if the numbers are negative?

Given that m, p, s, and v are positive, and m/p < s/v.

We have to find out of the following must be between m/p and s/v.

Let's take each of them one by one.

Statement I: (m+s)/(p+v)

We know that m/p < s/v

Multiplying both the sides by p/s, we get m/s < p/v

Adding '1' to both the sides, we have

m/s + 1 < p/v + 1

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

By cross-mutiplying, we get (m + p)/(s + v) < s/v

We are not yet sure wether (m + p)/(s + v) > m/p.

Again, we know that m/p < s/v

Multiplying both the sides by p/s, we get m/s < p/v

Taking the reciprocal of above inequality

We have s/m > v/p

Adding '1' to both the sides, we have

s/m + 1 > v/p + 1

(s+m)/m > (v+p)/p

By cross-multiplying, we get (m+s)/(p+v) > m/p

Thus, m/p < (m+s)/(p+v) < s/v. Statement 1 is correct.

Statement II: (ms)/(pv)

We know that m/p < s/v

Multiplying above inequality by s/v, we get ms/pv < (s/v)^2

If s/v > 1, then (s/v)^2 > s/v, and then either ms/pv < s/v or ms/pv > s/v. This is a Could be true type of situation and not a Must be true type.

Statement III: s/v - m/p

We know that m/p < s/v

Transposing m/p, we have, 0 < s/v - m/p. The value of (s/v - m/p) may or may not be between m/p and s/v.

Say s/v = 3 and m/p = 2, then s/v - m/p = 1. The value of (s/v - m/p) does not lie between m/p and s/v.

The correct answer: B

Hope this helps!

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