Added the OA.

Please provide always the explanation. It helps the community.

Thank you

When slope is linear, it equals 1 and divides the plane into two 45 degree angles and x=y. However, here w > 45, which means slope is more than 1 because it is more steep. In such cases, y coordinates > x coordinates.
n>m. I.E. n+m>2m

If you are good at drawing or visualization, it usually helps in GRE with coordinate geometry especially because planes are to scale.

Refer the attached figure..

At 45, \(m=n\)..
but at \(w>45\), \(n>m....n+m>m+m...n+m>2m....A>B\)

To get a better idea of what's going on here, let's see what happens when the angle is EXACTLY 45 degrees

If we drop a line from a point ON the line, we see we get an ISOSCELES RIGHT triangle.


This means that, for any point ON the line, ....

....the x-coordinate and the y-coordinate will be EQUAL

So, some other points on this line will look like this:



KEY CONCEPT: If w > 45, then the line will pass through the red region


We already know that, for any point ON THE BLUE LINE, the x-coordinate and the y-coordinate will be EQUAL
If the red region lies ABOVE THE BLUE LINE, what important thing can we say about the coordinates of ANY point in the red region???

We can say that, for any point in the red region, the y-coordinate will be greater than the x-coordinate

So, if w > 45, then we know that n > m

We're now ready to answer the question...

Given:
Quantity A: m + n
Quantity B: 2m

Subtract m from both quantities to get:
Quantity A: n
Quantity B: m

Since we now know that n > m, we can see that the correct answer is A

Cheers,
Brent

Thank you for your explanation
Is it possible to illustrate this point?
W>45. For example, it can be more than 90.
As a result, when W=100, A cannot be considered as a correct answer?! Because, there is no limitation on the range of W.

Great question.
This illustrates an important point about diagrams on the GRE.
Since the line appears to be in the first quadrant, we can assume that it is in the first quadrant.
That is, the diagram is telling us that 0 degrees < w < 90 degrees
So, when we add the given information (w > 45), we can be certain that 45 degrees < w < 90 degrees

w>45 , means the slope is more steep , hence m>1 , and also if the slope if more steep then it will have y>x .
so, n>m
n+m> m+m ( Adding m both side of inequality)
n+m> 2m

A 45 degree would have the slope of the line equal to 1.
In this case, we are told that w>45, which implies that the slope>1.
Knowing this, we can infer that for each marginal movement of X, the change in y is greater than the change in X. This implies that, the function of the line will be such that for any value of x, the absolute value of y is greater than the absolute value of x. As we are in the first quadrant, both x and y will be positive, implying that y>x for all x.
Therefore, 2m (which is like saying 2x) must be less than m+n, as n>m.

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