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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