A rectangular yard is 20 yards wide and 40 yards long.

A rectangular yard is 20 yards wide and 40 yards long. It is surrounded by a thick hedge that grows on the border of the property, but completely within the boundaries of the yard. If the hedge covers an area of 171 square yards, what is the width?

A. 160/120
B. 171/120
C. 180/120
D. 191/120
E. 800/120



The area of the whole yard is 20*40=800 square yards;
The area of the uncovered yard is 800-171=629 square yards (the whole area minus the area of the portion of the yard which is covered by the hedge);

Now, the width of the uncovered yard is 20-2x and the length is 40-2x, so its area is (20-2x)(40-2x)=629 --> (20-2x)(40-2x)=17*37. At this point it's better to substitute the values from answer choices rather than to solve for x. Answer choice C, 3/2 fits.

Answer: C.


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Let n be the width of the hedge and assuming it grows uniformly

the 40(20) - (40-2n)(20-2n) = 171

4n^2 - 120n +171 = 0

Solving for n gives 1.5 or 28.5

From the choices, (C) is 1.5.
No choices gave n as 28.5, so (C) is the answer.

Let me try and explain. If we cut at each vertex , the closed figure can turn into a big ,thin strip with length being equal to perimeter and width same as before. The area will remain same as before.

We are told that the rectangular yard is 20 yards wide and 40 yards long. Why are you changing the dimensions of the yard? If you mean something elese please upload the figure.
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My mind couldn't go into this question. each option(width) produced lenght that is longer than that of the field whcih is supposed to be larger than. Any mistake in this question? Some one even suggested that they mean 171 (sqaured) yards. thats even more preposterous as 171squared is larger than 800. Can someone explain what the question is talking about first?

Hi Nez,

the width is 160/120 which is nearly 1.5 yards.. so its OK..
what 171 square yards means is that 171 is the area of width in square yards..
same as if unit of length is in m, the area can be in square m...
171 sq yards means only 171 and not 171^2..
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Hi,
Another method where you can make use of choices given to home on to correct choice..

lets imagine the strip be x yards all along the perimeter..
so Area= perimeter*x - four corners of x*x
A= 2(20+40)x-4x^2=171..
120x-4x^2=171..

two ways hereafter:-

1) substitute ways and find answer

2)since all choices are in fraction with denominator 120, lets take x=y/120, where y is an integer..
now 120x will be an integer and 171 is an integer so 4x^2 will also be an integer (since Integer-Integer=Integer)..
4x^2=4*y^2/120^2= (y/60)^2 should be an integer ..
so y or the NUMERATOR should be a multiple of 60..
only 180 in C is multiple of 60..so ans is C

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Area of the rectangular yard=20*40=800 sq. yard
Given, area of the hedge=171 sq. yard
So, area of the yard that is not covered with hedge=Area of the rectangular yard-Area of the hedge=800-171=629 sq. yard

let 'w' be the width of the hedge, then length of the uncovered yard, L=40-2w and width of the uncovered yard. W=20-2w
Hence, area of the uncovered yard=L*W=(40-2w)(20-2w)=629
Or, \(4w^2-120w+800=629\)
Or, \(4w^2-120w+171=0\)
On solving, we have w=57/2(Not in option) ,3/2 or180/120

Ans. (C)
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Solution

Given:

• A rectangular yard is 20 yards wide and 40 yards long.
• The yard is surrounded by a thick hedge that grows on the border of the property, but completely within the boundaries of the yard.
• The hedge covers an area of 171 square yards.

To find:

• The width of the hedge.

Approach and Working:

• The area of the whole yard = 40 x 20 = 800
• The area of the hedge = 171

o Hence, the area of the yard without the hedge = 800 – 171 = 629

If we assume the width of the hedge to be w, then

• Length of yard without hedge = 40 – 2w
• Width of yard without hedge = 20 – 2w
• Therefore, area of yard without hedge = (40 – 2w) (20 – 2w) = 629

o Or, (40 – 2w) x (20 – 2w) = 37 x 17 = (40 – 2 x 1.5) x (20 – 2 x 1.5)

Comparing both sides of the equation, we can say w = 1.5 = 180/120

Hence, the correct answer is option C.

Answer: C


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