What is the Value of w in Terms of x and y GMAT Problem Solving

Question:

What is the value of w in terms of x and y? Note: Figure not drawn to scale

1. 2x + 2y - 180
2. 180 - x - y
3. 360 - 2x - 2y
4. 360 - 2x - 3y
5. 180 + x - 2y

Correct Answer: A

Solution and Explanation:
Approach Solution 1:

The problem statement asked to find out the value of the angle w in terms of x and y.
As we know that the sum of the angles of a triangle is 180°. We also know that the sum of the angles in a straight line is 180°.
Therefore, the smaller triangle holding angle w includes angles of w, 180-x-y, and 180-x-y.
Hence we can infer that:
180 = w + 180-x-y + 180-x-y

Therefore, the value of w = 2x+2y-180

Approach Solution 2:

The problem statement asked to get the value of w in terms of x and y.
As per the mathematical rule, we know that the sum of three angles in any triangle is 180°.
Let's assume the ∠E = k in the big triangle ACE. Then we know
x + y + k = 180,
Or, we can say, k = 180 – x – y

Now, let’s analyse the angles around point F.
Those three angles form a straight line.
We all know according to the mathematical rule, the sum of all angles in a straight line is 180°. Since one angle is x and one is y, the other angle has to be k, i.e. ∠DFE = k

 

Now, let’s analyse triangle DEF. Therefore, the sum of the three angles in the triangle must also be 180°.
Hence, w + k + k = 180°
w = 180 – 2k

Therefore, we get the expressionism of w in terms of k.
In order to find the value of w in terms of x & y, we must substitute the expression for k.
That is, as we get, k = 180 – x – y.

Therefore, w = 180 – 2(180 – x – y)
=> w = 180 – 360 + 2x + 2y
=> w = 2x + 2y – 180
Therefore, the value of w = 2x + 2y – 180.

Approach Solution 3:

Since we know the sum of all three angles of a triangle is 180°, then we can say,

\(w + \angle{DEF} + \angle{DFE} = 180°\) —--(i)

∴ \(\angle{DEF}\) = 180°- x - y (if we consider ΔCAE, where \(\angle{C}\)= x° and \(\angle{A}\)= y°, then \(\angle{E}\) = 180°- x-y)

∴ \(\angle{DFE}\) = 180°- x - y (if we consider ΔAFE, where \(\angle{A}\)= y° and \(\angle{BFA}\)= x°, then \(\angle{DFE}\) =180°- x-y)

Let’s put the expression of \(\angle{DEF}\) and \(\angle{DFE}\) in equation (i), we get:

w + 180°- x -y + 180°- x - y = 180 degree
w + 360 degree - 2x - 2y = 180 degree
w = 2x + 2y - 360 degree + 180 degree
w = 2x + 2y -180

Therefore, the value of w = 2x + 2y – 180.

“What is the value of w in terms of x and y''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book "GMAT Official Guide 2019". The GMAT Problem Solving questions enable the candidates to improve their skills in calculating numerical problems. GMAT Quant practice papers allow the candidates to practice several questions that will boost their confidence to score better in the quant exam.

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