A six-sided mosaic contains 24 triangular pieces of tile of the same

Hi All,

We're told that a six-sided mosaic contains 24 triangular pieces of tile of the SAME size and shape, as shown in the figure above and that the sections of tile fit together perfectly. We're asked for the total square centimeters of tile in the mosaic. While this question might look a bit 'crazy', it's a great "concept question" meaning that if you know the concepts involved, then you won't have to do much math to actually get the correct answer. To answer this question, we'll need information to determine the area of any one of the triangles, then we can multiply that area by 24 to get the total area of the mosaic.

1) Each side of each triangular piece of tile is 9 centimeters long.

Fact 1 tells us that we're dealing with Equilateral triangles with sides of 9 cm. With this information, we can determine the area of each triangular tile (the best part is that we don't actually have to do that math here; we know that we CAN do that math - and that there will be just one value for the area - so we can stop working).
Fact 1 is SUFFICIENT

2) The mosaic can be put inside a rectangular frame that is 40 centimeters wide.

The information in Fact 2 puts a limit on what the maximum side of each triangle could be, but doesn't given us any data on the exact areas involved, so since the triangles don't have a set area, the answer to the question will change as the areas change.
Fact 2 is INSUFFICIENT

Final Answer:

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

ZoltanBP DavidTutorexamPAL GMATPrepNow

The question stem states "24 triangular pieces of tile of the same size and shape" so I am not sure why would we have different heights. Can we consider these small triangles to be 24 equilateral triangles and if the statement 2 would have stated that the mosaic shape is inscribed in the rectangular frame, then we would have had the height and subsequently we could get the side of the equilateral triangle and the area.

So the only missing info is whether the mosaic shape is inscribed or not. Is my thinking correct ?

Thanks
Saurabh Arjaria

(1) From the side of the equilateral triangle, we can get the value of the area of each triangle.
Area of equilateral triangle= root 3/4*(side)^2

Area of the mosaic = root 3/4*(side)^2 * 24( Total number of triangular pieces) . Sufficient.

(2) Nothing is known about the length, therefore. Not Sufficient.

A

Stmnt 1 gives us that the triangles are equilateral but think about what happens in case of isosceles triangles. Imagine that the horizontal base of each triangle is just 6 cm while other two sides are still 9 cm each. Will we face any problem in making the mosaic?

Stmtn 2 means that the mosaic fits into a 40 cms wide frame so its width is less than or equal to 40 cm. It is not necessary that its width is exactly 40 cm.

GMATBusters Bunuel chetan2u
Is it ok to say that STatement 2 doesn't tell us what kind these 24 triangles are. It only fixes the height of the triangles but unless we know something more about the triangles e.g equilateral, isosceles etc, we can determine the area of the individual triangle.

Hi altairahmad,

You are correct that Fact 2 does NOT tell us what 'type' of triangles we're dealing with nor the actual measurements of any of the sides (that information only gives us a 'maximum base length'; since we know that the length of 4 'bases' will fit inside a 40 cm width, thus each 'base' is something LESS than 10cm).

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

Solution:

Question Stem Analysis:

We need to determine the area of the mosaic, which is a hexagon comprised of 24 triangles of the same size and shape. Notice that if each triangle is an equilateral triangle and we are given a side of the triangle, then we can determine the area of one triangle and hence the areas of all 24 triangles, i.e., the area of the mosaic.

Statement One Alone:

Since we are given that each side of a triangular piece is 9 cm long, we know each triangle is an equilateral triangle. Therefore, we can determine the area of one triangle and hence the areas of all 24 triangles, i.e., the area of the mosaic. Statement one alone is sufficient.

Statement Two Alone:

Knowing the mosaic can be put inside a rectangular frame that is 40 centimeters wide is not sufficient to determine the area of the mosaic. The mosaic could be different sizes, as long it fits into the rectangular frame. However, since it can be different sizes, it can have different areas. Statement two alone is not sufficient.

Answer: A

Hi GMATinsight

I disagree with the last part of your explanation in the video i.e. you told: "for 2nd statement if it has been mentioned the mosaic exactly fits in the rectangle of width 40 cm then the statement would be sufficient"

Why I disagree:
Because then also it depends on how we are putting mosaic in the rectangle, there will be multiple possibilities for ex: 40== 4* length of each tile
OR
40 ==4* height of the equilateral triangle (if u have assumed tiles to be in the shape of equilateral triangle)
In both cases you will get different value of mosaic side.

I may be wrong hence requesting your reply on this please

Thank you

#1
hexagon is 6 times equilateral ∆ area
given side = 9 sufficeint to solve
#2
mosaic rectangluar size ; insufficeint
IMO A

Bunuel chetan2u I have the same doubt. other answers kept saying that this statement is insufficient because the length of a rectangle is not mentioned. But my understanding is its length is not required. If mosaic fits, then one of the diagonal of mosaic will be equal to 40cm and we can find the side of an equilateral triangle. The only missing info is whether the mosaics diagonal fits the width of the rectangle. could you please help with your explanation?

The question stem states "24 triangular pieces of tile of the same size and shape" so I am not sure why would we have different heights. Can we consider these small triangles to be 24 equilateral triangles and if the statement 2 would have stated that the mosaic shape is inscribed in the rectangular frame, then we would have had the height and subsequently we could get the side of the equilateral triangle and the area.

So the only missing info is whether the mosaic shape is inscribed or not. Is my thinking correct ?

Thanks
Saurabh Arjaria[/quote]

Bunuel chetan2u I have the same doubt. other answers kept saying that this statement is insufficient because the length of a rectangle is not mentioned. But my understanding is its length is not required. If mosaic fits, then one of the diagonal of mosaic will be equal to 40cm and we can find the side of an equilateral triangle. The only missing info is whether the mosaics diagonal fits the width of the rectangle. could you please help with your explanation?[/quote]


I agree with these statements and believe they are correct. Since the stem clearly states that we are dealing with a triangle which has the "same shape and size", it wouldn't be wrong to consider all the 24 triangles in the hexagon as equilateral triangles.

I believe the issue with statement II is that we cannot figure out if the rectangular frame is "circumscribing" the hexagon as stated in the first response of this question.

I must agree that this is a weird question.

I think, even if we assume "put inside" means "inscribed" (which would be wrong) and that all triangles are equilateral, we actually do not know how the mosaic actually fits in the rectangle. The 40 cms can be the BASE of 4 triangles or the HEIGHT of the 4 triangles.

Sarjaria84 Thanks for the explanation. Why are we dividing it by 2 twice? Thanks

Hi EMPOWERgmatRichC

Please could you explain how from (2) we get a limit on the sides of a triangle? We are told that the mosaic can fit in a rectangle of 40cm. So we can say that:

1) The two triangles at the base of the hexagon will cover at least 40 cm.
2) The four triangles (the height of the hexagon) will cover at least 40 cm.

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

@gmatinsight @empowergmatrichc I do not get how would we get the answer if we knew that the mosaic fits perfectly in the rectangle. Even if it did, we do not know what kind of a triangle is inside- for all we know they one triangle could have all different lengths or it could be isoceles or many such shapes and thus would have number of possible areas with respect to the hexagon so why would it be sufficient if we had info that hexagon fits into rectangle @scotttargettestprep


A six-sided mosaic contains 24 triangular pieces of tile of the same size and shape, as shown in the figure above. If the sections of tile fit together perfectly, how many square centimeters of tile are in the mosaic?

(1) Each side of each triangular piece of tile is 9 centimeters long.
(2) The mosaic can be put inside a rectangular frame that is 40 centimeters wide.


DS24602.01
OG2020 NEW QUESTION


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Answer: Option A

Video solution by GMATinsight



[[/quote]

Answer: Option A

Video solution by GMATinsight



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Hi Elite097,

The information in Fact 2 (about the hexagon fitting inside of a particular rectangle) is NOT enough to sufficiently answer the question (which is why the correct answer is neither B or D). From your post, I assume that you've worked through the prompt and read some of the explanations in this thread - and I think that you understand that Fact 2 is Insufficient. Are there any other aspects of this question that you'd like to discuss in more detail?

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

Contact Rich at: Rich.C@empowergmat.com