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The figure shows a square patio surrounded by a walkway of w

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The figure shows a square patio surrounded by a walkway of w

by mitzwillrockgmat » Sat Jun 12, 2010 1:29 pm
The figure below 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

Image

I understand this question up to a point but then i get stuck when it comes to the quadratic equation. Can someone show how to solve it? I prefer not to use the quadratic eq formula because it consumes a lot of time.

This is what i did:

Walkway are = large square area - patio area
132 = (3x+5)^2 - (x+5)^2
132 = (9x^2 +30x + 25) - (x^2 +10x +25)
132 = 9x^2 + 30x + 25 - x^2 - 10x - 25
132 = 8^2 + 20x
8^2 +20x -132 =0
this divided by 4 becomes
2x^2 +5x - 33 = 0

i can solve this using quadratic equation formula (-b +- sqrt b^2 - 4ac ) /2a but takes time, also it doesn't always strike you in the heat of the moment that calculating the negative value is of no point when considering lengths or distances.

So is there a faster method?!

Btw the 2 as the coefficient of x^2 indicate that the sum & product should be 5/2 & 33/2 respectively correct?

Thanks!
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by rparvathaneni » Sat Jun 12, 2010 1:56 pm
It would be easier if we approach these kind of problems in reverse manner.

So let us consider we start with 68

68 is not a perfect square which will make the things a little complex for us, so let us consider 64

so we get x+5=8=>x=3

so total area is (3(3) + 5)^2 = 14^2 => 196

196-64=132 which complies with the given data.
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by mitzwillrockgmat » Sat Jun 12, 2010 2:49 pm
rparvathaneni wrote:It would be easier if we approach these kind of problems in reverse manner.

So let us consider we start with 68

68 is not a perfect square which will make the things a little complex for us, so let us consider 64

so we get x+5=8=>x=3

so total area is (3(3) + 5)^2 = 14^2 => 196

196-64=132 which complies with the given data.
hi!

i get your point on back solving would really help if one is truly stuck on this problem!

but don't you think given 3 perfect squares in the answer choices, one will have to be really lucky to pick 64 on the first try & get it right?

if one started from 100 the problem would take over 2 minutes to reach 64 no?

or is there a reason why you picked 64 first & not 81 or 100? pls advise thanks!
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by GMATGuruNY » Sun Jun 13, 2010 4:07 am
When you're trying out the answer choices, you can streamline the process by:

-- avoiding answers choices that are unlikely to be correct
-- of the remaining answer choices, starting with the one right in the middle (normally answer choice C)

If the answer right in the middle works, then it's the correct answer.
If the answer right in the middle is too big, you can eliminate both it and the answers that are even larger, and try those that are smaller.
If the answer right in the middle is too small, you can eliminate both it and the answers that are even smaller, and try those that are larger.

Starting with middle answer choice can help you to eliminate three answer choices at once. Remember this rule:

Start with C and cancel three.

The correct answer to this problem is likely to be a perfect square. So the most viable candidates are 64, 81, and 100.

Let's start with 81, the answer right in the middle.

If the area of the patio is 81 sq. meters, each side of the patio is 9 meters.
Since the width of the patio is 5 meters greater than the width of the walkway, the width of the walkway would be 9 - 5 = 4 meters.
Then each side of the entire figure (patio + walkway) would have to be 9 + 4 + 4 = 17 meters, and the area of the entire figure would have to be 17 * 17 = 289 sq. meters.
But if the area of the patio is 81 sq. meters and the area of the walkway is 132 meters (as given in the problem), the area of the entire figure should be only 81 + 132 = 213 sq. meters, not the 289 sq. meters we got above.

So answer choice D is too big and the correct answer must be answer choice B.

Trying only one answer choice led us to the correct answer. See how efficient trying out the answers can be?
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by mitzwillrockgmat » Sun Jun 13, 2010 5:12 am
GMATGuruNY wrote:When you're trying out the answer choices, you can streamline the process by:

-- avoiding answers choices that are unlikely to be correct
-- of the remaining answer choices, starting with the one right in the middle (normally answer choice C)

If the answer right in the middle works, then it's the correct answer.
If the answer right in the middle is too big, you can eliminate both it and the answers that are even larger, and try those that are smaller.
If the answer right in the middle is too small, you can eliminate both it and the answers that are even smaller, and try those that are larger.

Starting with middle answer choice can help you to eliminate three answer choices at once. Remember this rule:

Start with C and cancel three.

The correct answer to this problem is likely to be a perfect square. So the most viable candidates are 64, 81, and 100.

Let's start with 81, the answer right in the middle.

If the area of the patio is 81 sq. meters, each side of the patio is 9 meters.
Since the width of the patio is 5 meters greater than the width of the walkway, the width of the walkway would be 9 - 5 = 4 meters.
Then each side of the entire figure (patio + walkway) would have to be 9 + 4 + 4 = 17 meters, and the area of the entire figure would have to be 17 * 17 = 289 sq. meters.
But if the area of the patio is 81 sq. meters and the area of the walkway is 132 meters (as given in the problem), the area of the entire figure should be only 81 + 132 = 213 sq. meters, not the 289 sq. meters we got above.

So answer choice D is too big and the correct answer must be answer choice B.

Trying only one answer choice led us to the correct answer. See how efficient trying out the answers can be?
ahh that is a neat trick! im going to try this on questions now & see how it works! thank you! :)
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