Max@Math Revolution wrote:[GMAT math practice question]

$$If\ \ x>y>0,\ is\ y<2?$$

1) $$\frac{1}{x}=\frac{1}{2}$$
2) $$\frac{1}{x}+\frac{1}{y}=1$$

We are given that 0 < y < x and need to determine whether y < 2.

Statement One Alone:

1/x = 1/2

Simplifying the equation we have:

2 = x

Since y is less than x, we see that y must be less than 2. Statement one alone is sufficient to answer the question.

Statement Two Alone:

1/x + 1/y = 1

Multiplying the above equation by xy we have:

y + x = xy

y = xy - x

y = x(y - 1)

y/(y - 1) = x

Substituting into our given inequality of y < x we have:

y < y/(y-1) (see Note 1 below)

y^2 - y < y

y^2 < 2y (see Note 2 below)

y < 2

Statement two alone is sufficient to answer the question.

Notes:

1. Notice that when we multiply (y - 1) to both sides of y < y/(y - 1) to obtain y^2 - y < y, we don't need to worry about switching the inequality sign since y - 1 is always positive. Since y > 0, had y - 1 been negative (i.e., y - 1 < 0 or y < 1) , then 1/y would be greater than 1. That means 1/x must be negative in order for 1/x + 1/y = 1. However, since x is also positive, 1/x can't be negative. In other words, 1/y can't be negative.

2. Since y > 0, we can divide both sides of the inequality by y without switching the inequality sign.

Answer: D