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What is the ratio of the area of A to the area of B in the figure?

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What is the ratio of the area of A to the area of B in the figure?

by Max@Math Revolution » Tue Jun 09, 2020 4:13 am

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[GMAT math practice question]

What is the ratio of the area of A to the area of B in the figure?
6.9DS.png
1) The biggest triangle consists of 6 different isosceles right triangles.
2) PQ = 4
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Source: — Data Sufficiency |
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Re: What is the ratio of the area of A to the area of B in the figure?

by Max@Math Revolution » Thu Jun 11, 2020 12:30 am

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6.9DS(A).png
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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.

Let’s look at the condition 1). It tells us that the ratio of the areas of triangles A and B is 3:8, as shown below:

Assume PQ = x.
Then PQ = PR = x and SR = SP = x/√2.
We have TS = TP = SP/√2 = x/2.
US = UT = ST/√2 = x/2√2.
Then UR = US + SP = x/2√2 + x/√2 = 3x/2√2.
The area of triangle A is (1/2)(3x/2√2)^2 = 3x2/16.
The area of triangle B is (1/2)x^2 = x^2/2.
Thus, the ratio of the areas of A to B is 3x^2/16 to x^2/2 or 3:8.

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 2)
We don’t assume anything about PR, RU, and so on.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition 1) ALONE is sufficient

Therefore, A is the answer.
Answer: A
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