math

What you are asking is equivalent to asking why (a + x)/(b + y) is always between x/y and a/b. First of all, this is not correct if y and b have opposite signs. For instance, if you let x = 2, y = -3, a = 1, and b = 2, then (a + x)/(b + y) = 3/-1 = -3, and -3 is not between x/y = -2/3 and a/b = 1/2.

If b and y have the same sign, then (a + x)/(b + y) is indeed between x/y and a/b. The term (a + x)/(b + y) is called the mediant of x/y and a/b. If a, b, x, and y are all positive, here's how you can prove that (a + x)/(b + y) is between x/y and a/b: suppose x/y < a/b. Since x, y, a, and b are positive, this is equivalent to:

⇒ bx < ay

Add xy to each side of this inequality:

⇒ bx + xy < ay + xy

⇒ x(b + y) < y(a + x)

⇒ x < y(a + x)/(b + y)

⇒ x/y < (a + x)/(b + y)

This is one side of the inequality we want. For the other side, add ab to each side of the inequality bx < ay:

⇒ bx + ab < ay + ab

⇒ b(x + a) < a(y + b)

⇒ x + a < a(y + b)/b

⇒ (x + a)/(y + b) < a/b

Combining the two inequalities, we obtain x/y < (a + x)/(b + y) < a/b.

If x/y > a/b, then we can obtain x/y > (a + x)/(b + y) > a/b through a similar reasoning (just change the direction of every inequality sign you see above).

For the case where a, b, x, and y are not necessarily positive, google "The Mediant Inequality".