Let x, y, z be three different positive integers each less than 20.

How you choose these numbers depends on what you need.
Here, we need the value of the expression \(\frac{x − y}{−z}\) to be the smallest.

Another aspect of this questions is that the 3 numbers(x,y and z) need to be different and lesser than 20.
Of the available answer options, -18 is the smallest. But it is not possible to get -18.
The minimum value possible for the expression is \(\frac{2 − 19}{−1}\)(-17)

Hope that helps!

Think of \(\frac{x − y}{−z}\) as \(\frac{x − y}{(-1 * z)}\) = (-1) * \((\frac{x − y}{z})\)

In particular, focus on \((\frac{x − y}{z})\)

Why? Because one way to get the least possible negative value is to find the greatest possible positive value, and switch the sign ...

... which is exactly what -z will do.

In other words, the greatest positive value, once divided by -z, will turn into the most negative value, and hence the least possible value for the expression.

With a positive fraction, to get the greatest value, we need to minimize the denominator and maximize the numerator.

\(\frac{x − y}{z}\)

We have three different positive integers < 20.

To minimize denominator, let z = 1

To maximize the numerator, choose the greatest number possible (19) and subtract from it the smallest number possible (2, because we've already used 1 for the denominator).

Use original expression now, \(\frac{x − y}{-z}\), where x = 19, y = 2, and z = 1

\(\frac{19 - 2}{-1}\)

\(\frac{(17)}{(-1)}\) = -17

Answer B

Since z is positive, -z is negative. Thus, if we want the expression (x - y)/-z to be the smallest value possible, we want (x - y) to be positive. We see then that the entire expression will be negative. Furthermore, we want to make (x - y) the largest positive number possible and also make -z the largest negative number possible. Thus, we can let z = 1 so that -z = -1. Since x, y, and z are distinct, we can let x = 19 and y = 2, so that (x - z) = 17. Thus, the smallest possible value of (x - y)/-z = 17/-1 = -17.

Answer: B

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