The figure shows a square patio surrounded by a walkway of width x

Question

https://gmatclub.com/forum/download/file.php?id=27151 The figure shows a square patio surrounded by a walkway of width x meters. If the area of the walkway is 132 square meters and the width of the patio is 5 meters greater than the width of the walkway, what is the area of the patio, in square meters? A. 56 B. 64 C. 68 D. 81 E. 100 patio.JPG

AJB77 · Answer

Clearly the patio is a square, since all of its sides of off 2x from the bigger square.

Let the patio's side be a. => x=a-5

Side of bigger square = a + 2(a-5) = 3a - 10

The area of the walkway = Area of big square - Area of patio

132 = (3a-10)^2 - a^2

132 = 9a^2 -60a + 100 -a^2 

This reduces to:

8a^2 - 60a - 32 = 0
2a^2 -15a - 8 = 0
2a^2 + a -16a - 8 = 0
a(2a + 1) -8(2a+1) = 0
(a-8)*(2a+1) = 0
a = 8 or -1/2, But a is positive => a=8

Patio's area = a^2 = 64

B is my answer.

amitdgr · Answer

The figure shows a square patio surrounded by a walkway of width x meters. If the area of the walkway is 132 square meters and the width of the patio is 5 meters greater than the width of the walkway, what is the area of the patio, in square meters?

A. 56
B. 64
C. 68
D. 81
E. 100

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Attachment:

patio.JPG [ 8.03 KiB | Viewed 29594 times ]

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Official Answer

B

A. 56
B. 64
C. 68
D. 81
E. 100

LiveStronger · Answer

B

(5+3x)^2 - (5+x)^2 = 132
2x(10+4x) = 132
x(5+2x) = 33 = 3*11
So, x can be 3 and not 11

8^2 = 64

amitdgr · Answer

nice one

especially this step ---- x(5+2x) = 33 = 3*11 I was wondering how to reduce the calculation thanks

bigfernhead · Answer

So, x can be 3 and not 11

Why not 11?

(I did this the ultra long way)

amitdgr · Answer

x > 0 so 5+3x > x

x(5+2x) = 3*11

x has to be the smaller value and (5+2x) the larger.

so x = 3

even I did it the ULTRA lonnggg wayy

gurpreet07 · Answer

can someone please explain this step how did we get this (5+3x)^2 - (5+x)^2 = 132

LiveStronger · Answer

Width of the patio is 5m greater than the width of the walkway. So, width of the Patio = 5+x (since x is the width of the walkway)

To get the area of the walkway, we need to substract the area of the bigger square minus the patio.

Area of patio = (5+x)^2
Side of the bigger aquare = x+x+5+x = 5+3x
So, area of the bigger square is = (5+3x)^2

Hence, (5+3x)^2 - (5+x)^2 = 132

marcap · Answer

Hi LiveStronger,
nice way to solve it. could you please explain me how do you go from step 1 to step 2. It took me 3 lines of operations and you can do it in just one. I definitely need to be faster in the exam so this kind of tips will be really useful.
thks!!

Step 1. (5+3x)^2 - (5+x)^2 = 132
Step 2. 22x(10+4x) = 132

LiveStronger · Answer

I used a^2 - b^2 = (a+b)(a-b) formula

IanStewart · Answer

The solutions above subtract the inner square from the outer one, which is a good approach. One other way to get to the same answer: divide up the patio. You have the four corners, measuring x by x, and four rectangles lined up with each side of the inner square which measure x by x+5. So:

4x^2 + 4x(x+5) = 132
2x^2 + 5x = 33

etc.

4x^2 + 4x(x+5) = 132
2x^2 + 5x = 33

etc.

GMAT TIGER · Answer

width of the walkway = x
width of the patio = x+5 = x+5
width of the whole area = x+5+x = 3x+5

(3x+5)^2 - (x+5)^2 = 132
9x^2 + 30x + 25 - x^2 -10x -25 = 132
8x^2 + 20x = 132
4 (2x^2 + 5x) = 132
2x^2 + 5x - 33 = 0
2x^2 + 11x - 6x - 33 = 0
x (2x+11) -3 (3x +11) = 0
(x-3) (2x+11) = 0
x = 3 or -11/2 but -11/2 is not possible as length/width cannot be in -ve. so x = 30

so area of the patio = (3+5)^2 = 64

gurpreet07 · Answer

Thanks GMAT TIGER and LiveStronger for d explanation......
Actually i go numb when i see numbers.......

can u please suggest me how should i improve my quant

prasun84 · Answer

If its convenient, u can also attempt backsolving it:-
On first glance A,C are clearly out.
(E) 100 means side of patio is 10 so width of walkway =5. the walkway has 4 rectangles 2 among them are each 20*5=100 --impossible for total of 132.
(D) 81 means side of patio is 9 so width of walkway =4. the walkway has 4 rectangles 2 among them are each 17*4=68. 2 such rectangles account for 136--hence impossible for total of 132.
(C) 64 means side of patio is 8 so width of walkway =3. the walkway has 4 rectangles 2 among them are each 14*3=42 and rest are 8*24.total= 2(42+24)=132.

tt11234 · Answer

the answer is B. here's how i did it...

the width of the patio is X + 5 and the width of the big square is 3X +5, therefore the are of the big square is equal to the area of the walkway plus the area of the patio, the equation is
(3X +5)^2 -132 = (X + 5)^2, 
X = 3 or X = -11/2
since the width of the patio = X+5 = 3 + 5 =8
the area equals to 8^ 2 =64
hope it helps!!

fluke · Answer

Smaller square=Patio
Let the side of Patio be "s".
Width of walkway=x
width of the patio is 5 meters greater than the width of the walkway(Actually it should be "side of the patio is 5 meters greater than the width of the walkway" because patio is a square)

So, 
 s=x+5

If we see the figure properly, the outer quadrilateral is also a square with side  s+x+x  OR  x+5+x+x=3x+5

We know, the area of the walkway is 132 square meters:

Area of walkway=Area of outer square-Area of inner square
Area of walkway=(3x+5)^2-(x+5)^2=132

(3x+5)^2-(x+5)^2=132

(3x)^2+(5)^2+2*3x*5-(x^2+5^2+2*5x)=132

9x^2+25+30x-x^2-25-10x=132

9x^2+30x-x^2-10x=132

8x^2+20x=132

2x^2+5x=33

2x^2+5x-33=0

2x^2+11x-6x-33=0

x(2x+11)-3(2x+11)=0

(x-3)(2x+11)=0

x=3 \hspace{3} OR \hspace{3} x=-\frac{11}{2}

Width can't be -ve. So,  x=3 
 s=x+5=3+5=8

Area of the patio =s^2=8^2=64

Ans: "B"

sudish · Answer

The algebraic solution to the problem is given below:

Given, Width of walkway = x
Width of Patio = Width of Walkway + 5 meters = x+5 meters
Area of walkway = 132 square meters

From the given figure, we know that the width of the square figure i.e. Patio + Walkway will be:
2 times the width of walkway + width of Patio i.e 2(x) + (x+5) = 3x+5

Therefore, the area of this combined figure (square) will be (3x+5)^2

Now, the area of the walkway will be equal to the difference between the area of the combined figure and the square Patio i.e. (3x+5)^2 - (x+5)^2
This is given to be 132 square meters.

Therefore, expanding and simplifying the equation, (3x+5)^2 - (x+5)^2 = 132
We get,
8x^2 + 20x - 132 = 0
i.e. 2x^2 + 5x - 33 = 0
Solving the above quadratic equation will yield the value of x as -6.5 and 3. Since the width of the walkway can't be negative, the value of x is 3 meters.

Using this, we can calculate the area of the Patio, which is (x+5)^2 i.e. (3+5)^2 = 8^2 = 64 square meters.

This is option B.

Hope this helps

Cheers!

PathFinder007 · Answer

Hi bunuel,

Could you please provide your comments on how we are getting 5+3x.

i didnt get it.

Thanks.

ENGRTOMBA2018 · Answer

It is given that width of the patio is 5 more than width of the walkway. Now, walkway's 'extension on either side of the patio' is 'x' and thus width of the of the patio =  W_P  = 5+x

Now, total width of the walkway =  W_W  = width of the patio + 2*x = 5+x+2x = 5+3x

The figure shows a square patio surrounded by a walkway of width x

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The figure shows a square patio surrounded by a walkway of width x

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