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

Statement 1. Area=15k+(15-k)k=189 Hence 30k-(k^2)=189 ==> (-k^2)+30k-189=0 D=b^2 - 4ac= 900 - 4(-1)(-189)=144 Hence k=(-30 +12)/(-2)=9 or k= (-30-12)(-2)=21 But k<15 hence k=9 Sufficient
Statement 2. Perimeter = 15+15+2k+(15-k) + (15-k)=60 ==> 60=60 k is eliminated Hence statement 2 is not sufficient

Answer A

Can someone please explain me this question in detail step by step. How could we assume the missing the piece as square ?

Santro,

The lengths of the two complete sides are 15 while the lengths of the two smaller sides are both k. If you imagine a square with sides of 15, then you can (hopefully) see that the missing piece has two sides that are 15 - k in length. Now, looking at the measure of the angles, we can also see that this missing piece has all 90 degree angles. Thus, it is either a rectangle or square. But, since the top piece is 15- k in length and the left side piece is 15 - k in length AND they are both parallel to the sides opposite them, we can conclude that the missing piece is also a square.

I reached the second option through algebraic approach, which returns the exact value of k=7.5
Sum of all sides = 3(15-k)+k+30= 60. Which reduces to k=7.5.
Could anyone please point out where I'm making mistake???

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The perimeter = 15 + 15 + k + k + (15 - k) + (15 - k) = 60. k's would cancel and you'd get 60 = 60.

Hi,

Can we solve this problem if we consider two rectangle areas: 15 x K (first) + K (15-K)(second) =189 ? , I tried with this but can´t get the solution.

Hi Omar31Peru,

Yes, you can deal with the information in Fact 1 by setting up a Quadratic equation (although it comes with a 'tweak' to the logic).

You can start with your equation and simplify as follows:

(15)(K) + (K)(15-K) = 189
15K + 15K - K^2 = 189
30K - K^2 = 189
0 = K^2 - 30K + 189
0 = (K-9)(K-21)
K = 9 or 21

Based on what we know about the garden, K CANNOT be 21, so the only possible value would be K=9. Thus, Fact 1 is SUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

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

Answer: A

Cheers,
Brent

How can we conclude that the missing piece is a square?

I used the following equation=>
=(15-k)*k=36=> k=3 or 12
Both are viable answers so I couldnt prove this was sufficient. What did i get wrong?

Hi livfcind,

To start, there's an error in your equation. The two dimensions of the "missing upper-left corner" are BOTH (15 - K) and since we are clearly dealing with 90-degree angles, that 'corner' would have to be a square.

Based on the information in Fact 1, if you want to create an equation that will account for the "missing upper-left corner" of the picture, then the equation would be...

(15 - K)(15 - K) = 36
(15 - K)^2 = 36

So, what numbers, when 'squared', equal 36? The answer is +6 or -6 (but since we're dealing with a geometric shape, we can't have a "negative length", so that corner is a 6x6 square and the value of K must be 9.

GMAT assassins aren't born, they're made,
Rich

Video solution from Quant Reasoning:
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Solution:

Question Stem Analysis:

We need to determine the value of k. Notice that each of the two unlabeled sides has a length of 15 - k ft.

Statement One Alone:

Knowing that the area of the garden is 189 square feet, we can create the equation:

15^2 - (15 - k)^2 = 189

225 - 189 = (15 - k)^2

36 = (15 - k)^2

15 - k = 6 or 15 - k = -6

k = 9 or k = 21

Since k can’t be greater than 15 (otherwise 15 - k would be negative), k = 9. Statement one alone is sufficient.

Statement Two Alone:

Knowing that the perimeter of the garden is 60 feet, we can create the equation:

15 + 15 + k + (15 - k) + (15 - k) + k = 60

Simplifying this equation, we have:

60 = 60

That is, the equation always holds as long as k is any positive value less than 15. Statement two alone is not sufficient.

Answer: A

Answer: Option A

Video solution by GMATinsight



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Beauty of this question is not in the reasoning or calculation part, but the three traps it sets up for the people solving. Seldom have I seen so many traps in the same question.

Text solution below, however for those interested in detailed step by step solution (video) along with clear call out of the TRAPS/MISTAKES here is a video solution:



SOLUTION
I am going to start with statement 2
Statement 2: Perimeter = 60
The perimeter of a polygon is sum of all sides of the polygon. In our case, starting from the bottom and going clockwise:

15 + k + (15-k) + (15-k) + k + 15 = 60

This is TRAP 1: With the constraint of time, some would look at this and say "One equation one variable" and mark this as "Sufficient". But thats the trap if you do not simplify this equation. When you simplify you realize that k cancels off and you only get
60 = 60 !! No value of k possible form the given information.
INSUFF

Statement 1: Area = 189
Pretty easy to come to the quadratic
k^2 - 30k + 189 = 0

TRAP 2: If you stop here and say - obviously k is not unique since 2 possible values of k possible. Hence insufficient. You will be falling into the trap 2. Remember - one of the values of k could be negative or discarded, so you must keep on going here.

k=9 or k=21

TRAP 3: If you stop here and say - well I expected 2 values and I got two "+ve" values of k. No unique answer hence INSUFF! You will be falling into the trap 3 after all the hard work! Just take a step further and check the "feasibility of the two values"

k=9 : possible
k=21: impossible because the max side in the polygon is clearly 15. k cannot be more than that . So we must discard

So only k=9 possible. Hence (A) is sufficient to answer the question

(A)

Check out the video for clear takeaways! Also An EXPERT TIP, chalk this up as one of the most important questions you have to review few days before the exam so you don't fall for these traps.

Stmt 1.
Area= 189
15k+ k(15-k)=189
\(k^{2}-30k+189=0\)
k=21, 9
k < 15 and hence only one solution
Sufficient

Stmt 2:
2*15 + 2*k + 2*(15-k)=60
2k gets cancelled and we cant calculate k
Not Sufficient

Ans A.

This problem becomes way easier if you extend the bottom left side and top right to make a square and then you see we have a square with a square taken out.
this is a hidden difference of squares problem that shows up a lot in geometry

so the dimensions of our omitted square are 15-k our larger square is 15 x 15

(1) area =189 --> 15^2 - (15-k)^2 = 189
225 - (225-30k+k^2) = 189
and k^2-30k-189 = 0 no need to do any more math we know we can get a solution

suff

(2) perimiter = 60 so 2k + 30 + 2*(15-k) = 60 notice the variable disappears so this is not sufficient.

oa A