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S95-21

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Math Expert
Joined: 02 Sep 2009
Posts: 43322

Kudos [?]: 139395 [0], given: 12789

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16 Sep 2014, 00:49
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Difficulty:

55% (hard)

Question Stats:

66% (03:07) correct 34% (02:34) wrong based on 38 sessions

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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

_________________

Kudos [?]: 139395 [0], given: 12789

Math Expert
Joined: 02 Sep 2009
Posts: 43322

Kudos [?]: 139395 [0], given: 12789

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16 Sep 2014, 00:49
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:

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: $$\frac{2x +1}{(x+1)^{2}}$$.

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 $$\frac{5}{9}$$.

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Joined: 18 Mar 2015
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Location: India
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18 Aug 2017, 03:48
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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Intern
Joined: 24 Jul 2017
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20 Aug 2017, 12:26
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By "surface", it meant the AREA.

Circumference would mean the "outline" of the circle.

Kudos [?]: 1 [1], given: 34

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