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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