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A project requires a rectangular sheet of cardboard satisfyi

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A project requires a rectangular sheet of cardboard satisfyi [#permalink] New post 30 Jan 2019, 09:28
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23% (00:00) correct 76% (03:03) wrong based on 13 sessions
A project requires a rectangular sheet of cardboard satisfying the following requirement: When the sheet is cut into identical rectangular halves, each of the resulting rectangles has the same ratio of length to width as the original sheet. Which of the following sheets comes closest to satisfying the requirement?

(A) A sheet measuring 7 inches by 10 inches
(B) A sheet measuring 8 inches by 14 inches
(C) A sheet measuring 10 inches by 13 inches
(D) A sheet measuring 3 feet by 5 feet
(E) A sheet measuring 5 feet by 8 feet
[Reveal] Spoiler: OA

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Re: A project requires a rectangular sheet of cardboard satisfyi [#permalink] New post 30 Jan 2019, 20:31
Expert's post
GreenlightTestPrep wrote:
A project requires a rectangular sheet of cardboard satisfying the following requirement: When the sheet is cut into identical rectangular halves, each of the resulting rectangles has the same ratio of length to width as the original sheet. Which of the following sheets comes closest to satisfying the requirement?

(A) A sheet measuring 7 inches by 10 inches
(B) A sheet measuring 8 inches by 14 inches
(C) A sheet measuring 10 inches by 13 inches
(D) A sheet measuring 3 feet by 5 feet
(E) A sheet measuring 5 feet by 8 feet



So, when you half a rectangle, one side will get halved and other will remain the same.
Thus the ratio of B:L should be same as L/2:B or L:2B..
This means B:L=L:2B or \(L^2=2B^2\)..
Let us check the options with this.


(A) A sheet measuring 7 inches by 10 inches
B=7 and L=10, so \(L^2=2B^2...10^2=2*7^2....100=98\)..YES almost equal

(B) A sheet measuring 8 inches by 14 inches
B=8 and L=14, so \(L^2=2B^2...14^2=2*8^2....196=128\)..NO

(C) A sheet measuring 10 inches by 13 inches
B=10 and L=13, so \(L^2=2B^2...13^2=2*10^2....169=200\)..NO

(D) A sheet measuring 3 feet by 5 feet
B=3 and L=5, so \(L^2=2B^2...5^2=2*3^2....25=18\)..NO

(E) A sheet measuring 5 feet by 8 feet
B=5 and L=8, so \(L^2=2B^2...8^2=2*5^2....64=50\)..NO

A
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: A project requires a rectangular sheet of cardboard satisfyi [#permalink] New post 31 Jan 2019, 10:46
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GreenlightTestPrep wrote:
A project requires a rectangular sheet of cardboard satisfying the following requirement: When the sheet is cut into identical rectangular halves, each of the resulting rectangles has the same ratio of length to width as the original sheet. Which of the following sheets comes closest to satisfying the requirement?

(A) A sheet measuring 7 inches by 10 inches
(B) A sheet measuring 8 inches by 14 inches
(C) A sheet measuring 10 inches by 13 inches
(D) A sheet measuring 3 feet by 5 feet
(E) A sheet measuring 5 feet by 8 feet


Here's an algebraic solution:

Let x be length of the LONG side of the original rectangle
Let y be length of the SHORT side of the original rectangle
Then cut the rectangle into two pieces
Image

We want the resulting rectangles to have the same ratio of length to width as the original sheet.
In other words, we want x/y = y/(x/2)
Cross multiply to get: x²/2 = y²
Multiply both sides by 2 to get: x² = 2y²
Divide both sides by y² to get: x²/y² = 2
Take square root of both sides to get: x/y = √2

IMPORTANT: For the GRE, it's often useful to know the following APPROXIMATIONS: √2 ≈ 1.4, √3 ≈ 1.7, √5 ≈ 2.2

So, we know that x/y ≈ 1.4
In other words, the ratio (LONG side)/(SHORT side) ≈ 1.4

Now check the answer choices:

(A) 10/7 = 1 3/7 ≈ 1.4 LOOKS GOOD!
(B) 14/8 = 1 6/8 = 1.75 ELIMINATE
(C) 13/10 = 1.3 ELIMINATE
(D) 5/3 = 1 2/3 ≈ 1.66 ELIMINATE
(E) 8/5 = 1.6 ELIMINATE

Answer: A

Cheers,
Brent
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Re: A project requires a rectangular sheet of cardboard satisfyi [#permalink] New post 02 Aug 2019, 16:33
GreenlightTestPrep wrote:

In other words, we want x/y = y/(x/2)


Wouldn't this mean we are comparing the new length/width ratio to the old width/length ratio? Shouldn't we keep it as x/y=x/(y/2)??
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Re: A project requires a rectangular sheet of cardboard satisfyi [#permalink] New post 03 Aug 2019, 06:08
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chillMike wrote:
GreenlightTestPrep wrote:

In other words, we want x/y = y/(x/2)


Wouldn't this mean we are comparing the new length/width ratio to the old width/length ratio? Shouldn't we keep it as x/y=x/(y/2)??


I believe the intent here is to compare the longer sides to the shorter sides.
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Re: A project requires a rectangular sheet of cardboard satisfyi   [#permalink] 03 Aug 2019, 06:08
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