If m, p, s and v are positive,

Please correct the typos in the problem so that it reads as follows:

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 an EXTREME case.
Case 2: m=10, p=1, s=100, and t=4.
In this case, m/p = 10/1 = 10 and s/t = 100/4 = 25.

I: (m+s)/(p+v) = (10+100)/(1+4) = 22.
Since 22 is between 10 and 25, statement I holds true.

Since statement 1 holds true when the distance between the values is small (m/p = 1/2 and s/v = 1) and when the distance between them is great (m/p = 10 and s/v = 25), we should be satisfied:
Statement I must yield a value between m/p and s/v.

The correct answer is B.

GMATGuruNY wrote:Please correct the typos in the problem so that it reads as follows:

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 an EXTREME case.
Case 2: m=10, p=1, s=100, and t=4.
In this case, m/p = 10/1 = 10 and s/t = 100/4 = 25.

I: (m+s)/(p+v) = (10+100)/(1+4) = 22.
Since 22 is between 10 and 25, statement I holds true.

Since statement 1 holds true when the distance between the values is small (m/p = 1/2 and s/v = 1) and when the distance between them is great (m/p = 10 and s/v = 25), we should be satisfied:
Statement I must yield a value between m/p and s/v.

The correct answer is B.

Corrected

But if you take m=1, p=2, s=3 and v=4 then,I agree still the ans is B.

Will it be ok, if we use plugging in must be true question.

Any alternate method.

For all positive numbers a, b, c, and d:
If a/b < c/d, then a/c < b/d.
If a/b > c/d, then a/c > b/d.

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)

Algebraic proof for Statement 1:

Given info:
m/p < s/v

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

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

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

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

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

Since m/p < (m+s)/(p+v) and (m+s)/(p+v) < s/v, (m+s)/(p+v) is between m/p and s/v.

NandishSS 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

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