Did the ETS make a math error in the Official Guide to the GRE?

[Little Richard]

Hello, everyone. I am writing about a problem in the first practice test of the ETS Guide to the GRE, second edition. ON page 44 they give two overlapping circles and within the overlap a diamond. The diamond's top and bottom points are at the points of where the circles' sides intersect, and the left and right points rest on points O and P. The question asks, "In the figure above, O and P (left and right points of the diamond) are centers of the two circles. If each circle has radius R, what is the area of the shaded region?"

If you have solved this problem, let me know how. Because I will explain how I did it, and it looks like ETS made a typo.

I can reason that the diamond has all equal sides; each line originates from the center and goes to the circle's edge. So we need to find the area of the diamond whose perimeter is 4R, right? If we cut that diamond in half we get two equilateral triangles, each side being R.

To find the area of one half of the diamond, an equilateral triangle, we need 1/2 x base x height. The base is r. The height, using the proportions of a 30:60:90 triangle is r/2 x Sq.r. 3 So the area of one of the equilateral triangles is then:

(r x r/2[sq.r.3])/2 or, simplified, ([r^2]/2 x sq.r.3) / 2 We multiply that by 2, to get the area of the entire diamond, and it becomes (r^2)/2 x sq.r. 3, right?

The correct answer is, according to ETS, r^2 x (sq.r.3)/2 so the division by two is placed under the square root of 3 rather than r^2, which seems like a typo to me, and I cannot reason how to get that answer. However, math also is not my strength. Someone help?

[Dedi]

Let me give you an example on why both answers are correct:

 

1/2 * 2. Simple, yes? It can also be written as 1/2 * 2/1 = (1*2)/(2*1) = 2/2 = 1.

When dealing with fractions, think of whole numbers as being divided by one.

 

I'm not great at explaining the way I do math, so let me know if that didn't make sense.

[TakeruK]

Like Dedi said, both numbers are the same value. Just to put it another way, in case it helps:

 

You say that answer is r^2 /2 x sqrt(3); that is, in words "r squared divided by 2 times square root of 3"

 

The answer key says it is r^2 x sqrt(3)/2; that is, in words "r squared times square root of 3 divided by 2"

 

If you are only multiplying or dividing then the order does not matter. So these two answers are the same!!

 

If you also want a concrete example (try these on your calculator if you are not sure),

 

"3 times 4 divided by 2" is, in math, 3 x 4/2, which is equal to 6

 

but this is the same as 

 

"3 divided by 2 times 4", or in math, 3/2 x 4, which is also equal to 6.

[Little Richard]

Ohh, okay. I feel silly that it's so basic but the complexity must have thrown me off. Thanks to both of you!