A circular mat with diameter 20 inches is placed on a square

Question

A circular mat with diameter 20 inches is placed on a square tabletop, each of whose sides is 24 inches long. Which of the following is closest to the fraction of the tabletop covered by the mat?

A. 5/12
B. 2/5
C. 1/2
D. 3/4
E. 5/6

Bunuel · Accepted Answer

area_{circle}=\pi{r^2}=\pi{10^2}\approx{314}  ( d=20  -->  r=10 );

area_{square}=24^2=576 ;

\frac{area_{circle}}{area_{square}}=\frac{314}{576}=\frac{157}{288} .

Now this fraction is obviously between 1/2 and 3/4 -->  \frac{1}{2}=\frac{144}{288}  and  \frac{3}{4}=\frac{216}{288}  --> 157 is closer to 144 than to 216.

Answer: C.

prashantbacchewar · Answer

Nice explanation Bunuel... Thanks

Swoosh617 · Answer

SO we are looking for the area of the cloth over the area of the table

Area of the Cloth = (pi)(r)^2 which is about (3)(10)(10)

Area of the Table = (24)(24)

So the quick way to estimate is looking at the fraction like this: (3/24)(100/24)

I hope this is easy to follow, so with some simplification i get (1/8)(4) = (1/2) Answer is C

prasannajeet · Answer

Hi Bunel
Please let me clarify how did you establish 1/2=144/288 and 1/3= 216/288???

Rgds
Prasannajeet

Bunuel · Answer

To compare 157/288, 1/2 ans 1/3 find their common denominator, which is 288 --> 1/2=144/288 (multiply by 144) and 1/3= 216/288 (multiply by 216).

Jaisri · Answer

Simplified the numbers as below:
Req fraction is (3.14 * 10 * 10)/(24 * 24)
Cancel 10 and 24 by 2: (3.14 * 5 * 5)/(12 * 12)
Cancel 5*5=25 in numerator with 12 in denominator: (3.14 * 2.something)/(1 * 12) = 6.something / 12 =~1/2. (Here it is important to note that 2.something and 6.something are in the lower end, i.e., much lesser than 2.5 and 6.5.)
Bunuel has given a standard way to deal with fraction comparison in the previous posts. In some cases, the above sort of simplifications might help in getting a fraction with realtively smaller numerator and denominators.

mvictor · Answer

if D = 20, then r = 10, and area of the circle is 100 pi (I converted pi to 22/7) and thus the area is 2200/7
area of the square is 24^2 = 576
now, to find the area that is covered by the circle, divide 2200/7 by 576. get 2200/4032, which is slightly more than 1/2.

other method, 2200/7 is slightly more than 300
300+/576 = aprox 1/2

DJ1986 · Answer

I don't get why everyone is over-complicating the math here since the question stem clearly asks to ESTIMATE.

Making pi = 3

Area of circle = 2*5*2*5*3
Area of sqr = 12*3*2*2

3s cancel. 4s cancel. left with 25/48 close to 25/50 = 1/2

SeregaP · Answer

let's do it in a few steps:
1) area of a mat: 10^2*3,14
2) area of a table: 24*24
then we can arrive at cca 77/144

Let's look at answer choices, they're in ascending order, 1/2 seems fine if tested

sealberg · Answer

Area of the Square = 24*24
Area of the Circle = 3*10*10 (approximating pi as 3)

Fraction of the square covered by the circle = 3*10*10 / 24*24
Cancel the 3 with one 24, reduce the two tens = 25/48, which is roughly 1/2. Answer C

JeffTargetTestPrep · Answer

Since the diameter of the mat is 20 inches, the radius is 10 inches. The area of the mat is:

area = πr^2 = π(10)^2 = 3.14 x 100 = 314 square inches

Since each side of the square tabletop is 24 inches long, the area is:

area = side^2 = 24 x 24 = 576 square inches

Thus, the fraction of the table covered by the mat is 314/576 = 157/288

157/288 is about 160/290 = 16/29, which is about 1/2.

Answer: C

abhishekdadarwal2009 · Answer

A circular mat with diameter 20 inches is placed on a square tabletop, each of whose sides is 24 inches long. Which of the following is closest to the fraction of the tabletop covered by the mat?

A. 5/12
B. 2/5
C. 1/2
D. 3/4
E. 5/6

so to solve this I worked like...

area of the mat/total area of the square tabletop
3.14*10*10/24*24=
314/576, which is closest to 1/2. so i picked C.

beenish12 · Answer

Why did you choose to compare only these 2 options out of the 5? Like if we are solving this in under 2 mins, how should we narrow down to comparing with these 2? Comparing with all 5 would be too time consuming.

Engineer1 · Answer

It is a GMAT prep question. Please update the tag.  Bunuel

Bunuel · Answer

________________________
Updated the tags. Thank you!

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A circular mat with diameter 20 inches is placed on a square

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