Is z/(x+y) equal to z/x + z/y for nonzero x and y?

• A. True
• B. False
• C. True only if z=0
• D. True if x=y

Answer: (B)

The statement tries to treat division as if it distributes over addition, but division does not behave that way. In general, z/(x+y) is not equal to z/x + z/y.

To see why, suppose they were equal and z ≠ 0. Divide both sides by z to get 1/(x+y) = 1/x + 1/y = (x+y)/xy. Cross-multiplying gives xy = (x+y)^2, which simplifies to x^2 + xy + y^2 = 0. For real numbers with x and y not both zero, this is impossible, so no nonzero z can satisfy the equality for real x and y. Therefore the statement is false in general.

The only case that makes the two sides equal is when z = 0, in which case both sides are zero (assuming x+y ≠ 0 so the left side is defined). Since the question doesn’t fix z to be zero, the equality does not hold generally.