Is the perimeter of square S greater than the perimeter of

Note: The perimeter of the two figures if fixed. For a square, we can calculate this if we know any one measurement (side, diagonal). However, the shape of the triangle can vary for a fixed perimeter. But as a rule : For a given perimeter, an equilateral triangle will have the maximum area. We will use this info to solve the question.

From 1: s/t =4/5 (here s= side of the square & t= side of the triangle) => s=4x & t=5x (using ratios)
Now, we can evaluate and compare their area. (but we don't need to calculate, just know that we can)
{SUFFICIENT option A or D possible}
From 2: s+t=18. (2 unknowns and 1 equation) => cannot be solved {INSUFFICIENT}

Answer A

Hi All,

My answer to the above question is A too.

However as per Manhattan Guide, the answer is D. I am pasting the solution printed in Manhattan here. Can anyone explain to me why is D right?

Solution as per Manhattan: n either a square or an equilateral triangle, you can …nd out just
about any measurement from any other measurement. In a square, for instance,
the length of a side can give you area, perimeter, or the length of the diagonal.
In an equilateral triangle, the base and height are always related in the same
ratio, so knowing the area gives you perimeter, the length of the height, or the
length of any side.
Statement (1), then, is sufficient. If you know the ratio of the areas, you
can determine the ratio of the perimeters. To do so, you'd have to find the
ratio between the perimeter of a square and the area of a square, and also
the perimeter of an equilateral triangle and the area of an equilateral triangle.
While you won’t spend the time to do so on this question, the fact that you
could means that the given ratio is sufficient.
Statement (2) is also sufficient. Again, since sides of squares and equilateral
triangles always have the same ratios to the perimeters of the same figures,
knowing the ratio between the sides of these two figures is enough to determine
the relationship between the perimeters of the figures. Choice (D) is correct.


Thanks

Target question: Is the perimeter of square S greater than the perimeter of equilateral triangle T?

Statement 1: The ratio of the length of a side of S to the length of a side of T is 4:5.
Let x = the length of EACH side of the square
Let y = the length of EACH side of the equilateral triangle

So, we can write: x/y = 4/5
Cross multiply to get: 5x = 4y
Divide both sides by 5 to get x =4y/5
We can also write: x = 0.8y

The perimeter of the equilateral triangle = y + y + y = 3y
The perimeter of the square = x + x + x + x = 4x

Since we now know x = 0.8y, we can replace x with 0.8y to get:
The perimeter of the square = 0.8y + 0.8y + 0.8y + 0.8y = 3.2y

Since the perimeter of the equilateral triangle = 3y, and the perimeter of the square = 3.2y, the answer to the target question is YES, the perimeter of square S is greater than the perimeter of equilateral triangle T
Statement 1 is SUFFICIENT

Statement 2: The sum of the lengths of a side of S and a side of T is 18.
There are several scenarios that satisfy statement 2. Here are two:
Case a: Each side of the square has length 17, and each side of the equilateral triangle has length 1. So the perimeter of the square = 17 + 17 + 17 + 17 = 68, and the perimeter of the triangle = 1 + 1 + 1 = 3. In this case, the answer to the target question is YES, the perimeter of square S is greater than the perimeter of equilateral triangle T
Case b: Each side of the square has length 1, and each side of the equilateral triangle has length 18. So the perimeter of the square = 1 + 1 + 1 + 1 = 4, and the perimeter of the triangle = 17 + 17 + 17 = 51. In this case, the answer to the target question is NOT, the perimeter of square S is not greater than the perimeter of equilateral triangle T
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Solution:

We need to determine whether the perimeter of square S is greater than the perimeter of equilateral triangle T. If we let s be a side of square S, then the perimeter of square S is 4s. Similarly, if we let t be a side of equilateral triangle T, then the perimeter of equilateral triangle T is 3t. Therefore, we need to determine whether 4s > 3t.

Statement One Alone:

Since the ratio of the length of a side of S to the length of a side of T is 4:5, we can let s = 4x and t = 5x where x is some positive number. So we have:

4s > 3t ?

4(4x) > 3(5x) ?

16x > 15x ?

Since x is positive, 16x is indeed greater than 15x. That is, the perimeter of square S is indeed greater than the perimeter of equilateral triangle T. Statement one alone is sufficient.

Statement Two Alone:

Knowing the sum of the lengths of a side of S and a side of T is 18 is not sufficient. For example, if s = 10 and t = 8, the 4s = 40 is greater than 3t = 24. However, if s = 7 and t = 11, then 4s = 28 is not greater than 3t = 33. Statement two alone is not sufficient.

Answer: A

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