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# If a rectangular region has perimeter P inches and area A sq

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Re: If a rectangular region has perimeter P inches and area A sq  [#permalink]

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16 Jul 2017, 10:47
olifurlong wrote:
If a rectangular region has perimeter P inches and area A square inches, is the region square?

(1) P = 4/3*A
(2) P = 4√A

The goal is to determine if the shape is in fact a square.

Statement 1) P = 4/3*(A)

If true, then P=4x and A = x^2

4x = 4/3*(x^2)
x = x^2/3
x = 3

Only true if the side is length 3. We don't know if the side of the shape has length 3, so the statement is insufficient.

Statement 2) P = 4sqrt(A)

4x = 4sqrt(x^2)
4x=4x
x = x

This statement is true, so the statement is sufficient.
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Re: If a rectangular region has perimeter P inches and area A sq  [#permalink]

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17 Jul 2017, 05:47
(B) in IMO. First assume the rectangle is a square. Then l=b. P=4l and Area=l^2. If you consider statement 1, LHS is not equal to RHS.
If you consider statement 2, LHS=RHS. So B is sufficient.
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Re: If a rectangular region has perimeter P inches and area A sq  [#permalink]

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29 Jan 2018, 03:13
This question raises very important point on DS questions.

First of all it is a YES/NO question.

When you plug in numbers statement (2) becomes YES. And when you plug numbers in statement (2) the answer is almost always no. Theoretically both statements give you concrete YES or NO, therefore each alone must be sufficient.

But two statements must both either say YES or NO. If one statement yields YES but the other NO, then the statement that you know for 100% is sufficient is the correct answer. And there must be at least on set of numbers that would give YES answer for statement one. But you should not bother to find that set of numbers.
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Re: If a rectangular region has perimeter P inches and area A sq  [#permalink]

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19 May 2018, 14:41
igotthis wrote:
olifurlong wrote:
If a rectangular region has perimeter P inches and area A square inches, is the region square?
1) P=4/3A
2)P=4√A

Lets try finding the relation between area and perimeter of a square.
We know the area of a square with length x (A) = x^2
The perimeter of the square (P) = 4x

Since P = 4x --> x = P/4

So, A = (P/4)^2
A =(P^2)/16
Therefore, P^2 = 16A
and P = 4*sqrtA

Therefore, B is suff to answer this question

I did the question in a similar way. Could someone please explain why A is not sufficient?

For a square, we know:
A = x^2
P = 4x
A/P = x/4 => P = 4A/x

So shouldn't any equation in that form (i.e. Statement A: P = 4A/3) be sufficient? According to A, our side x would be 3 but I don't think the side would matter since we just need the equation to be in the form P = 4A/x.
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If a rectangular region has perimeter P inches and area A sq  [#permalink]

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19 May 2018, 22:41
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dabaobao wrote:
I did the question in a similar way. Could someone please explain why A is not sufficient?

For a square, we know:
A = x^2
P = 4x
A/P = x/4 => P = 4A/x

So shouldn't any equation in that form (i.e. Statement A: P = 4A/3) be sufficient? According to A, our side x would be 3 but I don't think the side would matter since we just need the equation to be in the form P = 4A/x.

Hey dabaobao ,

You did a very BIG mistake of not reading the question properly.

Question says "If a rectangular region" but you took it a square.

For a rectangle, Perimeter = 2(l+b) and area = l * b.

Now, try substituting the values in statement A, you won't be able to determine whether l = b for the region to be a square. Hence, A is insufficient.

Does that make sense?
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Re: If a rectangular region has perimeter P inches and area A sq  [#permalink]

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31 Jul 2018, 04:17
olifurlong wrote:
If a rectangular region has perimeter P inches and area A square inches, is the region square?

(1) P = 4/3*A
(2) P = 4√A

Bunuel
chetan2u

I have a small doubt. Can we take the ratio of different units i.e (inches/sq. inches) ?

Perimeter is in inches and Area is in square inches. Per my understanding ratio is independent of units, but shouldn't it (ratio) be obtained for similar units?

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If a rectangular region has perimeter P inches and area A sq  [#permalink]

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07 Aug 2018, 18:06
I have a question:

Even without seeing the (1) or (2), we can prove that this equation needs to be true: $$P = 4\sqrt{A}$$.
Proof: If the shape is a square,
The length of one side should be: $$\sqrt{A}$$,
The perimeter should be: $$P = 4\sqrt{A}$$

We know that the shape can be a square only if the equation above holds. If we are given the information in (1), we could definitely conclude that the shape is NOT a square, since it contradicts the equation above.

So I would think the answer should be D. Can anyone illuminate me about what is wrong in my rationale?
(I know the official answer is not D.)
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Re: If a rectangular region has perimeter P inches and area A sq  [#permalink]

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22 Nov 2018, 08:45
Top Contributor
olifurlong wrote:
If a rectangular region has perimeter P inches and area A square inches, is the region square?

(1) P = (4/3)(A)
(2) P = 4√A

Given: A rectangular region has perimeter P inches and area A square inches

Target question: Is the region square?
This is a good candidate for rephrasing the target question.

So, what kind of relationship between P and A do we need in order for the rectangular region to be a SQUARE?
Well, if the region is a square, then all 4 sides will be the same length
So, if P = the perimeter (sum of all 4 sides), then P/4 = the length of each side
If P/4 = the length of each side, then the AREA = (P/4)(P/4)
In other words: A = (P/4)(P/4)
Simplify: A = P²/16
Multiply both sides by 16 to get: 16A = P²
Take square root of both sides to get: 4√A = P
So, if 4√A = P, then we can be certain that the region is a square.
REPHRASED target question: Does 4√A = P?

Below, you'll find a video with tips on rephrasing the target question

Statement 1: P = (4/3)(A)
Let's TEST some values.
There are several values of P and A that satisfy statement 1. Here are two:
Case a: P = 12 and A = 9. In this case, the answer to the REPHRASED target question is YES, 4√A DOES EQUAL P
Case b: P = 4 and A = 3. In this case, the answer to the REPHRASED target question is NO, 4√A does NOT equal P
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: P = 4√A
Perfect!!
The answer to the REPHRASED target question is YES, 4√A DOES EQUAL P
Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT

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Re: If a rectangular region has perimeter P inches and area A sq &nbs [#permalink] 22 Nov 2018, 08:45

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