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

[Max@Math Revolution]

[GMAT math practice question]

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

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

[GMATGuruNY]

[GMAT math practice question]

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

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

Statement 1:
1/x = 1/2
x=2.
Substituting x=2 into x>y>0, we get:
2>y>0.
Thus, y<2, so the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
Test whether it's possible that y≥2.
Case 1: y=2
Substituting y=2 into 1/x + 1/y = 1, we get:
1/x + 1/2 = 1
1/x = 1/2
x=2.
In this case, x=y.
Not viable, since the prompt requires that x>y.

Case 2: y=3
Substituting y=3 into 1/x + 1/y = 1, we get:
1/x + 1/3 = 1
1/x = 2/3
x=3/2.
In this case, x < y.
Not viable, since the prompt requires that x>y.

The cases above illustrate that it is not possible that y≥2.
Thus, y<2, so the answer to the question stem is YES.
SUFFICIENT.

The correct answer is D.

An algebraic approach to Statement 2:
1/x + 1/y = 1
1/x = 1 - (1/y)
1/x = (y-1)/y
x = y/(y-1).

Substituting x = y/(y-1) into x>y, we get:
y/(y-1) > y
1/(y-1) > 1
1 > y-1
2 > y.
Thus, the answer to the question stem is YES.

[Max@Math Revolution]

=>
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 1 equation ( x > y ), D is most likely to be the answer.

Condition 1)
1/x = 1/2 implies that x = 2. Since x > y, we must have y < 2.
Condition 1) is sufficient.

Condition 2)
The original condition x > y > 0 implies that 1/x < 1/y.
Using 1/x + 1/y = 1 and 1/x < 1/y together, we can see that 1/y > 1/2.
Thus, 0 < y < 2.
Condition 2) is sufficient.

Therefore, the answer is D.

Note: The VA approach tells us that the answer is most likely to be D, since this is a CMT(Common Mistake Type) 4B question.

Condition 1) is easy to check, but condition 2) is more difficult to work with. If you can't figure out condition 2), you should choose D as the answer.

Answer: D

[Scott@TargetTestPrep]

[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