a cetrain cloth with a diameter of 20 inches is placed in a

[Guest]

source : GMAT prep 2

a cetrain cloth with a diameter of 20 inches is placed in a circular tray with a diameter of 24 inches . What fraction of trays surface is not covered by a cloth?
1) 1/6
2)1/5
3)11/36
4)25/36
5)5/6

Ans- 11/36

Please explain

[Guest]

area of a circle formula: pi*r^2
tray area: pi*12^2 = 144*pi
cloth area: pi*10^2 = 100*pi

area of tray not covered by the cloth: 144*pi - 100*pi = 44*pi
ratio of tray's uncovered area to tray's total area: 44*pi/144*pi = 11/36

[RonPurewal]

area of a circle formula: pi*r^2
tray area: pi*12^2 = 144*pi
cloth area: pi*10^2 = 100*pi

area of tray not covered by the cloth: 144*pi - 100*pi = 44*pi
ratio of tray's uncovered area to tray's total area: 44*pi/144*pi = 11/36

well played.

--

you can also use a certain general result about area ratios. here's that result:
if two SIMILAR (same shape, different sizes) figures have lengths in the ratio A:B, then their areas are in the ratio A^2:B^2.

since all circles are similar to all other circles, you can certainly apply this here.
the ratio of the circles' diameters is 20:24 = 5:6.
therefore, the ratio of their areas is 5^2:6^2 = 25:36.
by the definition of a ratio, this means that the cloth covers 25/36 of the tray.
therefore, 1 - 25/36 = 11/36 of the tray is uncovered.

[Guest]

How do you know the cloth is circular? Is it not possible for the cloth to be a different shape such as oval?

[RonPurewal]

How do you know the cloth is circular? Is it not possible for the cloth to be a different shape such as oval?

i've seen this problem before; it's supposed to say "a circular cloth".
having recognized it on sight, i didn't even read its text carefully enough to notice the mistranscription here.
thanks.

by the way, in gmat world, a circle is the only thing that has a "diameter" or a "radius". so if you see either of those two terms and don't think you're necessarily talking about circles, read the problem again, more carefully, and you should notice a direct reference to circles somewhere.

[manjitzing]

By the word SURFACE how do you infer that it will be AREA=PixR^2, It might be Surface AREA=2xPixR??
When to consider Surface area and when to consider area?

[RonPurewal]

By the word SURFACE how do you infer that it will be AREA=PixR^2, It might be Surface AREA=2xPixR??

that is not a surface-area formula; it's the formula for the circumference of a circle, which is irrelevant to this problem.

in fact, that can't be an area formula, since it wouldn't have the right units. (it would have the same units as 'r', which would be linear units -- feet, inches, etc. -- while an area would have to be in square feet, or square inches, etc.)

When to consider Surface area and when to consider area?

they say that the tray is "circular". that means that it's two-dimensional and in the shape of a circle.
in general, if they give you a word that describes something two-dimensional -- "circular", "triangular", "square", etc. -- then the object is two-dimensional; if they give you a word that describes something three-dimensional -- "spherical", "cubical", etc., -- then the object is three-dimensional.

also, you should just use your intuition. in this problem, they're talking about a cloth and a tray, so it should be pretty clear that we're talking about two-dimensional measures.