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x, y, and z are integers with 3 ≤ x < y < z ≤ 30 and y is a prime number. What is the value of x + y + z?

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x, y, and z are integers with 3 ≤ x < y < z ≤ 30 and y is a prime number. What is the value of x + y + z?

by Max@Math Revolution » Tue Jan 14, 2020 12:48 am
[GMAT math practice question]

x, y, and z are integers with 3 ≤ x < y < z ≤ 30 and y is a prime number. What is the value of x + y + z?

1) 1/x + 1/y = 1/2 + 1/z
2) 2xy = z
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Source: — Data Sufficiency |
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Re: x, y, and z are integers with 3 ≤ x < y < z ≤ 30 and y is a prime number. What is the value of x + y + z?

by Max@Math Revolution » Wed Jan 15, 2020 11:07 pm
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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Condition 2)
x = 3, y = 5, z = 2*3*5 = 30 are unique solutions as it is the only combination of numbers that works within the given conditions of 3 ≤ x < y < z ≤ 30 and y is a prime number. If x and y are larger numbers than z is greater than 30. We then have x + y + z = 3 + 5 + 30 = 38.
Since condition 2) yields a unique solution, it is sufficient.

Condition 1)
Since 3 ≤ x < y < z ≤ 30, we have 1/30 ≤ 1/z < 1/y < 1/x ≤ 1/3 when we take reciprocals.
Since we have 1/x + 1/y = 1/2 + 1/z, we have 1/2 < 1/x + 1/y < 1/x + 1/x = 2/x or 1/2 = 2/4 < 1/x.
Thus x < 4 and we have x = 3.
Since we have 1/2 = 1/3 + 1/y, we have 1/6 < 1/y or y < 6.
Since 3 < y < 6 and y is a prime number, we have y = 5.
1/z = 1/x + 1/y – 1/2 = 1/3 + 1/5 – 1/2 = 10/30 + 6/30 – 15/30 = 1/30 or z = 30.
Then, x + y + z = 3 + 5 + 30 = 38.

Since condition 1) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.
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