Does this quantitative question have 2 possible answers or am I just stuuuuuuuuupid?

[grad29]

 

In the figure above, if FGHJ is a rectangle, which of the following is an expression for the area of the shaded region?

 

a            

b           

     c          

    d        

e     

 

 

If A = 4 and B = 8, then wouldn't choices B and C work?

I know looking at the picture FK is longer than KJ and if FK is 4 then KJ must also equal 4, which wouldn't on the surface make sense. But to my understanding we aren't supposed to assume the shapes are drawn to scale. We can plug in values for the variables even though it doesn't appear to be drawn to scale. Unless it's mentioned explicitly or implied mathematically what the values cannot equal, we are free to use whatever? Maybe that's where I'm making the mistake because I see the 90 degree angles in the corners and then see GK emerge directly from it. I don't know. Tell me if 8 and 4 are not geometrically possible lengths. 

 

 

Area of the unshaded region given the values above is 1/2b x h, or 2 x 4 = 8

Area of the rectangle given the values above is a Length x Width = 4 x 8 = 32 

 

Thus, area of the shaded region is 24. Target number is 24. 

 

B and C both come out to 24? 

Edited July 8, 2015 by westy3789

[MathCat]

It's asking for a formula that works for all values, so you can't always pick and choose to identify the right answer. It's possible that two expressions could be equal for specific values, but that does not mean both are correct formulas. As a speed strategy, choosing numbers can be good for the GRE - it allowed you to narrow down your choices to B or C. Now you need another approach to decide which is right. (This could be picking different numbers and seeing that only one works.)

 

You are correct that you shouldn't assume things are drawn to scale, and it is possible for KJ to be longer than FK if b is large and a is small. If I'm interpreting what you wrote correctly, I think your observation about GK "emerging directly from it" is correct. This is because it is labelled that both legs GF and FK have length a, and angle GFK is 90 degrees. Thus, the other two angles are 45 degrees in that triangle.

 

Two ways to approach the problem: 1) compute the area of the entire rectangle and subtract the unshaded area. 2) compute the area of the shaded region directly. I think 1) is easier.

 

The area of the rectangle is ab, the product of the lengths of the two sides. The area of the unshaded triangle GFK is a^2/2, because the base and height are both a. Thus, the area of the shaded region is ab - a^2/2 = a(b-a/2), which is C.

[grad29]

It's asking for a formula that works for all values, so you can't always pick and choose to identify the right answer. It's possible that two expressions could be equal for specific values, but that does not mean both are correct formulas. As a speed strategy, choosing numbers can be good for the GRE - it allowed you to narrow down your choices to B or C. Now you need another approach to decide which is right. (This could be picking different numbers and seeing that only one works.)

 

You are correct that you shouldn't assume things are drawn to scale, and it is possible for KJ to be longer than FK if b is large and a is small. If I'm interpreting what you wrote correctly, I think your observation about GK "emerging directly from it" is correct. This is because it is labelled that both legs GF and FK have length a, and angle GFK is 90 degrees. Thus, the other two angles are 45 degrees in that triangle.

 

Two ways to approach the problem: 1) compute the area of the entire rectangle and subtract the unshaded area. 2) compute the area of the shaded region directly. I think 1) is easier.

 

The area of the rectangle is ab, the product of the lengths of the two sides. The area of the unshaded triangle GFK is a^2/2, because the base and height are both a. Thus, the area of the shaded region is ab - a^2/2 = a(b-a/2), which is C.

 

"Its asking for a formula that works for all values, so you can't always pick and choose to identify the right answer. It's possible that two expressions could be equal for specific values, but that does not mean both are correct formulas. As a speed strategy, choosing numbers can be good for the GRE - it allowed you to narrow down your choices to B or C. Now you need another approach to decide which is right. (This could be picking different numbers and seeing that only one works.)"

 

This helps and is something I completely didn't think about. I guess I was so used to the other problems where there was only one equation that worked. I will have to be a little more careful knowing that just because the variables fit into one equation - like choice A, it doesn't mean that choice E doesn't work as well, requiring more calculations with another set of variables.

 

I should have realized that after discovering two equations work for the numbers I came up with, to plug in new numbers and figure out which one is no longer valid.

Edited July 8, 2015 by westy3789