Bunuel wrote:
Let x, y, z be three different positive integers each less than 20. What is the smallest possible value of expression \(\frac{x − y}{−z}\) is
(A) –18
(B) –17
(C) –14
(D) –11
(E) –9
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
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