GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 23 May 2019, 04:51

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# S95-21

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 55265

### Show Tags

16 Sep 2014, 01:49
00:00

Difficulty:

55% (hard)

Question Stats:

66% (02:59) correct 34% (02:36) wrong based on 44 sessions

### HideShow timer Statistics

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. $$\frac{x}{x + 1}$$

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 55265

### Show Tags

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:

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

_________________
Manager
Joined: 18 Mar 2015
Posts: 115
Location: India
Schools: ISB '19
GMAT 1: 600 Q47 V26
GPA: 3.59

### Show Tags

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
Intern
Joined: 24 Jul 2017
Posts: 4

### Show Tags

20 Aug 2017, 13:26
1
By "surface", it meant the AREA.

Circumference would mean the "outline" of the circle.
Re: S95-21   [#permalink] 20 Aug 2017, 13:26
Display posts from previous: Sort by

# S95-21

Moderators: chetan2u, Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.