A circular table has a glass center with a diameter of 4x inches,

I set x=5 so that the diameter is 20 inches. The radius of the table is 10 inches and the radius of the table+ring is 11 inches.

I am having trouble getting the pie symbol to work so I am using "n" to represent pie.

area of table - area of table + ring = answer

area of table = n10^2
area of table+ring = n11^2

Factor out n (pie) to get 10^2 / 11^2 or 100/121 or roughly 5/6. This is the fraction we want to get when we plug in x=5 for each of the answer choices.

A. Plug in 5 to get 1/6. This is far from 100/121
B. 1/5, still way off
C. 25/36 or roughly 5/7, very close
D. 11/36 - far off
E. 5/6 - bingo

Answer E!

Answer = D

Area of glass surface \(= \pi (2x)^2 = \pi 4x^2\) ................... (1)

Total Area\(= \pi (2x+2)^2\) .............. (2)

Area of surrounding metal ring = (2) - (1)

Required fraction \(= \frac{\pi (2x+2)^2 - \pi 4x^2}{\pi (2x+2)^2} = \frac{x^2 + 2x + 1 - x^2}{(x+1)^2} = \frac{2x+1}{(x+1)^2}\)

Radius of the table without the metal ring = 2x, therefore area of the table without the metal ring = 4pi * x^2

Radius of the table with the metal ring = 2x+2, therefore area of the table with the metal ring = 4pi * (x+1)^2

Fraction of the table's surface is made up by the metal ring = {4pi * (x+1)^2 - 4pi * x^2}/4pi * (x+1)^2 = {(x+1)^2 - x^2}/(x+1)^2 = 2x+1/(x+1)^2

Therefore D) is the answer

Official Solution:

A circular table has a glass center with a diameter of \(4x\) inches, which is surrounded by a metal ring with a width of 2 inches. In terms of \(x\), what fraction of the table's surface is made up by the metal ring?

A. \(\frac{1}{x+1}\)
B. \(\frac{1}{x}\)
C. \(\frac{x^{2}}{(x+1)^{2}}\)
D. \(\frac{2x +1}{(x+1)^{2}}\)
E. \(\frac{x}{x + 1}\)


We must determine what fraction of the table's total area (glass and metal) is made up by the metal ring. The expression we are interested in, then, is \(\frac{Area_{whole} - Area_{glass}}{Area_{whole}}\).

The formula for the area of a circle is \(A = \pi r^2\), where \(r\) is the radius of the circle. Here, we are given the diameter of the glass center. Since the radius of a circle is half its diameter, the radius of the glass is \(r_{g} = \frac{4x}{2} = 2x\) inches.

The metal ring extends 2 inches in every direction, so the diameter of the whole table is \(4x + 4\), as in the diagram below:



Thus, the radius of the whole table is \(r_{w} = \frac{4x + 4}{2} = 2x + 2\) inches.

Combine the area formula with the fraction determined above: \(\frac{Area_{whole} - Area_{glass}}{Area_{whole}} = \frac{\pi (r_{w})^2 - \pi (r_{g})^2}{\pi (r_{w})^2}\).

Plug in the values for the radii: \(\frac{\pi (2x +2)^2 - \pi (2x)^2}{\pi (2x + 2)^2}\).

Use FOIL to multiply the polynomials: \(\frac{(4x^{2} + 8x + 4)\pi - (4x^{2})\pi}{(4x^{2} + 8x + 4)\pi}\).

Factor out \(4 \pi\) from the top and bottom of the fraction and reduce: \(\frac{x^{2} + 2x + 1- x^{2}}{x^{2} + 2x + 1}\).

Simplify the numerator and factor the denominator: \(\frac{2x +1}{(x+1)^{2}}\).

Answer choice D is correct.

Given that this approach requires some complex algebraic manipulations and that the answer choices all contain variables, we may instead choose to solve this problem by plugging in a value for \(x\). We will say that \(x = 2\). Then the diameter of the glass is \(4(2) = 8\) inches, and the radius of the glass is 4 inches. The radius of the whole table is 2 inches more, or 6 inches. Computing the fraction, we find: \(\frac{Area_{whole} - Area_{glass}}{Area_{whole}} = \frac{\pi(6)^2 - \pi(4)^2}{\pi(6)^2}\). Simplified, this is: \(\frac{36\pi - 16\pi}{36\pi} = \frac{20\pi}{36\pi}\), or \(\frac{5}{9}\).

We now substitute 2 for \(x\) in all the answer choices and see which expressions produce \(\frac{5}{9}\). Remember that we must always check all five answer choices, in case the value we picked for \(x\) produces the "right" output for two or more answer choices.

Choice A: \(\frac{1}{x+1} = \frac{1}{3}\). This is not equal to \(\frac{5}{9}\).

Choice B: \(\frac{1}{x} = \frac{1}{2}\). This is not equal to \(\frac{5}{9}\).

Choice C: \(\frac{x^{2}}{(x+1)^{2}} = \frac{4}{9}\). This is not equal to \(\frac{5}{9}\).

Choice D: \(\frac{2x +1}{(x+1)^{2}} = \frac{5}{9}\). So answer choice D works.

Choice E: \(\frac{x}{x + 1} = \frac{2}{3}\). This is not equal to \(\frac{5}{9}\).

Answer: D.


_________________

Hi All,

While this is an old series of posts, here's an example of why it's really important to take the proper notes (and sometimes draw pictures (especially when dealing with Geometry questions). The above approach to this question is perfect - TESTing VALUES will work really well here. However, the original poster made a mistake - we're told that the ring has a WIDTH of 2, so that width appears all the way 'around' the central circle of the table. In the above calculation, the area of 'everything' is (12^2)pi --> NOT (11^2)pi. Answer choice E is actually there on purpose - to punish people who do this part of the math incorrectly. Without the necessary drawing/notes, the original poster didn't even realize that the mistake occurred... and chose one of the wrong answers as a result.

On many GMAT questions (but especially the tougher ones), the wrong answers are NOT going to be random - the GMAT writers can anticipate the mistakes that you MIGHT make - and create the 'end result' that you would get to. These types of answers are NOT traps - they're punishments (when you make mistakes, the GMAT is really good at punishing you for those mistakes). This is all meant to say that you need to do the necessary work to eliminate mistakes from your process; when you do that work, you'll be rewarded with a higher score.

GMAT assassins aren't born, they're made,
Rich

but how do we know that circular glass+ring make up the table. we only know that circular glass is in the center of the table?

Glass table Radius = 2x
Total table Radius = 2x+2

Metal Area = pi * (2x+2)^2 - pi * (2x)^2
Total Area = pi * (2x+2)^2

Metal / Total = (2x+2 + 2x) (2x+2 - 2x) / (2x+2)^2
= 2(2x+1)(2) / 2^2(x+1)^2
= 2x+1 / (x+1)^2
= D