In general, the greater the numerator, the greater the fraction, whereas the greater the denominator, the smaller the fraction, assuming all the numbers involved are positive.

As such, between the four combinations a/p, a/r, c/p and c/r, we can see that a/r will be the smallest value while c/p will get us the largest value (since a > c and r > p).

So, b/q can lie between a/r and c/p, and by looking at the inequality signs,

a/r < b/q ≤ c/p

Allowing a or p to be smaller than 0 means that the numbers will become negative.

Suppose we divide two expressions a/b so that a is negative. The more negative a gets, the more negative a/b gets.

Suppose we divide two expressions a/b so that b is negative. The less negative b gets, the more negative a/b gets (think why).

So, if a < 0 < b ≤ c and 0 < p ≤ q < r, the number b and q still remain positive but the lower limit will change.

Because a will be in the numerator, the more negative a gets, the more negative the result will be. For the denominator, however, the smaller it gets, the more negative the resulting fraction will be.

So, the lower limit must be a/p (but not including itself).

So, we have a/p < b/q ≤ c/p. The variable r will not be used here.

Instead, if 0 < a < b ≤ c and p < 0 < q ≤ r, then we must also be careful.

The closer the lower limit is to zero from the negative side, the more negative the fraction will get.

Think of 5 divided by -0.000001.

So here, the lower limit is infinity because the number can get really negative.

The upper limit remains c/p.

So, our new result is b/q ≤ c/p (without any lower limit for b/q).