Given:
Quantity A: \( x^{- (c - d)}\)
Quantity B: \(y^{(d - c)}\)

Since \(-(c - d) = -1(c - d) = -c + d = d - c\), we can the rewrite Quantity A as follows:
Quantity A: \( x^{(d - c)}\)
Quantity B: \(y^{(d - c)}\)

Let's test some possible values of c, d, x and y

case i: Since \(x < y < 1\) and \(c > d\), one possible set of values is x = -1, y = 0.5, c = 2 and d = 1
We get:
Quantity A: \( x^{(d - c)}=(-1)^{(1 - 2)}=(-1)^{(-1)} = \frac{1}{(-1)^1} = \frac{1}{-1}=-1\)

Quantity B: \( y^{(d - c)}=(0.5)^{(1 - 2)}=(0.5)^{(-1)} = \frac{1}{(0.5)^1} = \frac{1}{0.5}=2\)
In this case, Quantity B is greater


case ii: Since \(x < y < 1\) and \(c > d\), one possible set of values is x = 0.25, y = 0.5, c = 2 and d = 1
We get:
Quantity A: \( x^{(d - c)}=(0.25)^{(1 - 2)}=(0.25)^{(-1)} = \frac{1}{(0.25)^1} = \frac{1}{0.25}=4\)

Quantity B: \( y^{(d - c)}=(0.5)^{(1 - 2)}=(0.5)^{(-1)} = \frac{1}{(0.5)^1} = \frac{1}{0.5}=2\)
In this case, Quantity A is greater

Answer: D

Cheers,
Brent



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