GMAT Practice Question: The operation \otimes is defined for all nonzero numbers a and b by...
Question
The operation \otimes is defined for all nonzero numbers a and b by a \otimes b=\frac{a}{b}-\frac{b}{a}. If x and y are nonzero numbers, which of the following statements must be true? I. x \otimes x y=x(1 \otimes y) II. x \otimes y=-(y \otimes x) III. \frac{1}{x} \otimes \frac{1}{y}=y \otimes x
- I only
- II only
- III only
- I and II
- II and III
Topics: functions, expression formulas, must be true statements
Solution
Step 1
We recall the definition of the operation.
a \otimes b = \frac{a}{b} - \frac{b}{a}
Step 2
We evaluate x \otimes (xy) by substituting a = x and b = xy into the definition.
x \otimes (xy) = \frac{x}{xy} - \frac{xy}{x}
Step 3
We simplify the expression for x \otimes (xy).
x \otimes (xy) = \frac{\cancel{x}}{\cancel{x}y} - \frac{\cancel{x}y}{\cancel{x}} = \frac{1}{y} - y
Step 4
We evaluate 1 \otimes y by substituting a = 1 and b = y into the definition.
1 \otimes y = \frac{1}{y} - \frac{y}{1}
Step 5
We simplify the expression for 1 \otimes y.
1 \otimes y = \frac{1}{y} - y
Step 6
We multiply the simplified value by x.
x \times (1 \otimes y) = x \times (\frac{1}{y} - y) = \frac{x}{y} - x \times y
Step 7
We compare the results of x \otimes (xy) and x \times (1 \otimes y) to check Statement I.
x \otimes (xy) = \frac{1}{y} - y x \times (1 \otimes y) = \frac{x}{y} - x \times y
Step 8
We evaluate x \otimes y.
x \otimes y = \frac{x}{y} - \frac{y}{x}
Step 9
We evaluate y \otimes x.
y \otimes x = \frac{y}{x} - \frac{x}{y}
Step 10
We find the negative of y \otimes x and then simplify it.
-(y \otimes x) = -(\frac{y}{x} - \frac{x}{y}) = -\frac{y}{x} + \frac{x}{y} = \frac{x}{y} - \frac{y}{x}
Step 11
We observe that this matches x \otimes y, so Statement II is always true.
\frac{x}{y} - \frac{y}{x} = x \otimes y
Step 12
We evaluate \frac{1}{x} \otimes \frac{1}{y} by substituting a = \frac{1}{x} and b = \frac{1}{y} into the definition.
\frac{1}{x} \otimes \frac{1}{y} = \frac{\frac{1}{x}}{\frac{1}{y}} - \frac{\frac{1}{y}}{\frac{1}{x}} = \frac{y}{x} - \frac{x}{y}
Step 13
We compare this result with y \otimes x and see they are the same, so Statement III is always true.
\frac{y}{x} - \frac{x}{y} = y \otimes x
Answer
E
