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S95-22

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Note: Figure not drawn to scale.

A rectangular yard (white area) is surrounded by a hedge (shaded region) with a length of 15 meters and a width of 12 meters, as shown in the figure above. The area of the yard is equal to the area of the hedge. If the ratio of the length of the hedge to the width of the hedge is the same as the ratio of the length of the yard to the width of the yard, what is the width, in meters, of the yard?


A. \(6\sqrt{2}\)
B. \(\frac{5}{4}\)
C. \(\frac{6}{\sqrt{2}}\)
D. \(7\frac{\sqrt{2}}{2}\)
E. \(\frac{6}{2}\)
[Reveal] Spoiler: OA

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Image

Note: Figure not drawn to scale.

A rectangular yard (white area) is surrounded by a hedge (shaded region) with a length of 15 meters and a width of 12 meters, as shown in the figure above. The area of the yard is equal to the area of the hedge. If the ratio of the length of the hedge to the width of the hedge is the same as the ratio of the length of the yard to the width of the yard, what is the width, in meters, of the yard?


A. \(6\sqrt{2}\)
B. \(\frac{5}{4}\)
C. \(\frac{6}{\sqrt{2}}\)
D. \(7\frac{\sqrt{2}}{2}\)
E. \(\frac{6}{2}\)


Let the length of the yard be \(x\), and let the width of the yard be \(y\).

We are told that the length of the hedge is 15 meters and that the width of the hedge is 12 meters, so we can find the area of the yard and hedge combined using the formula for area of a rectangle: \(A_{\text{yard and hedge}} = \text{length} \times \text{width} = 15 \times 12 = 180\). Similarly, we can find the area of the yard: \(A_{yard} = xy\).

We are told that the area of the yard and the hedge are equal, so the area of the hedge must also be \(A_{hedge} = A_{yard} = xy\). We can use this information to write \(xy + xy = 2xy = 180\), or \(xy = 90\).

We know that the length and width of the yard are in the same ratio as the length and width of the hedge. We express this with a proportion: \(\frac{15}{12} = \frac{x}{y}\). Cross-multiply to get \(15y = 12x\). Simplify to get \(5y = 4x\).

Now we have two equations for \(x\) and \(y\). We will solve for these unknowns using substitution. Since we want to find the width, \(y\), we first solve for \(x\) in terms of \(y\). From the equation \(xy = 90\), we get \(x = \frac{90}{y}\). We plug this into \(5y = 4x\) and get \(5y = 4 \times \frac{90}{y} = \frac{360}{y}\). Dividing both sides by 5 and multiplying by \(y\), we find \(y^2 = 72\). We now take the square root of both sides to get \(y = \sqrt{72}\).

To simplify this square root, we look for a perfect square that is a factor of 72. We note that \(72 = 36 \times 2\), and so \(y = \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).


Answer: A
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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New post 25 Oct 2016, 09:01
any easy explanation....

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Re S95-22 [#permalink]

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New post 03 Jun 2017, 04:47
I don't agree with the explanation. you have assumed 15/12 = x/y, which means the ratio of length to width for whole yard and hedge = length to width ratio of the yard. To my mind this is not what the problem reads. Please correct me if my interpretation is not correct.

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New post 03 Jun 2017, 17:58
Ganeshsrinivasan wrote:
I don't agree with the explanation. you have assumed 15/12 = x/y, which means the ratio of length to width for whole yard and hedge = length to width ratio of the yard. To my mind this is not what the problem reads. Please correct me if my interpretation is not correct.


Discussion here might help: http://gmatclub.com/forum/a-rectangular ... 88513.html

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: S95-22 [#permalink]

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New post 18 Aug 2017, 05:03
I have used expression for area of hedge = area of yard = (15-2x)(12-2x)=15*12
can anyone point out the mistake ? I feel this equation would be valid had square was given in the question. Am I correct?

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Re: S95-22   [#permalink] 18 Aug 2017, 05:03
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