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S95-21

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S95-21 [#permalink]

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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. \(\frac{x}{x + 1}\)
[Reveal] Spoiler: OA

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Re S95-21 [#permalink]

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New post 16 Sep 2014, 01:49
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:

Image

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
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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: S95-21 [#permalink]

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New post 18 Aug 2017, 04:48
question says "what fraction of the table's surface is made up by the metal ring?"
I thought it is asking for a circumference not area of a circle ? any help to comprehend this question

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Re: S95-21 [#permalink]

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New post 20 Aug 2017, 13:26
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By "surface", it meant the AREA.

Circumference would mean the "outline" of the circle.

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Re: S95-21   [#permalink] 20 Aug 2017, 13:26
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