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

Question

https://gmatclub.com/forum/download/file.php?id=28198 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. 2015-10-26_2054.png

VeritasPrepDennis · Accepted Answer

Stmt (1) - The area of the garden, if it were a square, would be 225 sq ft. The area of the L-shaped garden is 189 sq. ft - which is 36 sq ft less than a 15 x 15 square. Additionally, the missing piece is also a square and we can conclude that its dimensions are 6 x 6. The value of k is therefore 15 - 6, or 9. SUFFICIENT

Stmt (2) The perimeter of the garden will be 60 regardless of the value of k. For example, if k =5, the garden would have dimensions of 15, 5, 10, 10, 5 and 15 (total of 60). If k = 6, the garden's dimensions would be 15, 6, 9, 9, 6 and 15 (again total of 60). NOT SUFFICIENT

Answer is A.

EMPOWERgmatRichC · Answer

Hi All,

We're asked for the value of K in the drawing. While we normally have to be careful about trusting a picture in a DS question, we can trust the data in the drawing. We know that all of the angles are right angles and we know that the L-shaped garden is essentially a big square with a small square-shaped corner removed (and the length of a side on that square is (15-K)).

1) The area of the garden is 189 square feet.

A 15ft x 15ft square would have an area of 225 sq. ft. Since the garden is 189 sq ft., we know the missing corner = 225 - 189 = 36 sq. ft. This means that each side of that little square would have to be 6. Thus, 15 - K = 6, so K = 9. 
Fact 1 is SUFFICIENT

2) The perimeter of the garden is 60 feet.

With this Fact, we can create the following equation (where X is one of the unlabeled sides): 
15 + 15 + K + K + X + X = 60 
2K + 2X = 30 
K + X = 15

This is NOT new information (we knew this from the prompt). Thus, K could be any value in the range of 0 < K < 15. 
Fact 2 is INSUFFICIENT

Final Answer:  A

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

Konstantin1983 · Answer

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

santro789 · Answer

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

dmcashion · Answer

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.

Blackishmamba · Answer

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???

Posted from my mobile device

Bunuel · Answer

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

Omar31Peru · Answer

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.

EMPOWERgmatRichC · Answer

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

BrentGMATPrepNow · Answer

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. 
 https://i.imgur.com/hJBOxsO.png

Regions A and B have the following measurements. 
 https://i.imgur.com/RI6GMD7.png 
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  
 https://i.imgur.com/7X0ijQS.png 
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 : 
 https://i.imgur.com/WMhTHDo.png 
Notice that the perimeter = 60
In this case, the answer to the target question is  k = 6

Case b : 
 https://i.imgur.com/tWRWwQ3.png 
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

livfcind · Answer

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?

EMPOWERgmatRichC · Answer

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

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=scBf6rT58bs

ScottTargetTestPrep · Answer

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

GMATinsight · Answer

Answer: Option A

Video solution by  GMATinsight

https://www.youtube.com/watch?v=sJImYoMZKpw

gmatbyexample · Answer

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:

https://www.youtube.com/watch?v=Wo8Fe2UdnEQ

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.

sacharya · Answer

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.

ocelot22 · Answer

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

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

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GMAT Data Sufficiency (DS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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My Rewards