The figure above represents an $$L$$-shaped garden. What is the value of $$k?$$

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The figure above represents an $$L$$-shaped garden. What is the value of $$k?$$

by Gmat_mission » Thu Jan 07, 2021 10:50 am

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The figure above represents an $$L$$-shaped garden. What is the value of $$k?$$

(1) The area of the garden is $$189$$ square feet.
(2) The perimeter of the garden is $$60$$ feet.

Source: Official Guide

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Re: The figure above represents an $$L$$-shaped garden. What is the value of $$k?$$

by [email protected] » Thu Jan 14, 2021 7:35 am

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Gmat_mission wrote:
Thu Jan 07, 2021 10:50 am
2015-10-26_2054.png

The figure above represents an $$L$$-shaped garden. What is the value of $$k?$$

(1) The area of the garden is $$189$$ square feet.
(2) The perimeter of the garden is $$60$$ feet.

Source: Official Guide
Target question: What is the value of k?

Statement 1: The area of the garden is 189 square feet.
Let's drawn an auxiliary line that divides the shape into two rectangular regions A and B.

Regions A and B have the following measurements.

So, the area of region A = k(15 - k) = 15k - k²
The area of region B = 15k
So, the TOTAL area = 15k - k² + 15k = 30k - k²

Since we're told the area is 189, we can write: 30k - k² = 189
Rearrange to get: k² - 30k + 189 = 0
Factor: (k - 21)(k - 9) = 0
So, EITHER k = 21 OR k = 9

HOLD ON!
k cannot be greater than 15 (since one entire side has length 15)
So, it MUST be the case that k = 9
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The perimeter of the garden is 60 feet.
This statement provides NO NEW information, because the perimeter will ALWAYS be 60, regardless of the value of k.
Here's why:
If k = the two given sides, then the remaining two sides must both have a length of 15 - k

So, when we add all lengths, we get: PERIMETER = k + (15 - k) + (15 - k) + k + 15 + 15 = 60

If you're not convinced, consider these two possible cases:

Case a:

Notice that the perimeter = 60
In this case, the answer to the target question is k = 6

Case b:

Notice that the perimeter = 60
In this case, the answer to the target question is k = 5

Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT