A project requires a rectangular sheet of cardboard satisfying the following requirement: When the sheet is cut into ide

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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

Answer: A

Source: Manhattan GMAT

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M7MBA wrote:
Fri May 14, 2021 2:13 am
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

Answer: A

Source: Manhattan GMAT
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
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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 GMAT, everyone should 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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