If the perimeter of square region S and the perimeter of rec

If the perimeter of square region S and the perimeter of rectangular region R are equal and the sides of R are in the ratio 2:3 then the ratio of the area of R to the area of S

A. 25:16
B. 24:25
C. 5:6
D. 4:5
E. 4:9

Let the side of square be \(s\) and the side of rectangle \(a\) and \(b\)

Given:

The sides of R are in the ratio 2:3 --> \(\frac{a}{b}=\frac{2}{3}\) --> \(b=\frac{3a}{2}\).
The perimeter of square region S and the perimeter of rectangular region R are equal --> \(P=4s=2(a+b)\) --> \(2s=a+b=\frac{5a}{2}\) --> \(s=\frac{5a}{4}\).

Question: \(\frac{area \ of \ R}{area \ of \ S}=\frac{ab}{s^2}=?\)

Substitute b = 3a/2 and s = 5a/4 into the question: \(\frac{ab}{s^2}=\frac{3a^2}{2}*\frac{16}{25a^2}=\frac{24}{25}\).

Answer: B.


OR: you can pick numbers: let the sides of rectangle be 4 and 6 (ratio 2:3) then the perimeter of the rectangle will be 2(4+6)=20, thus the side of the square will be 20/4=5. Next, the area of the rectangle will be 4*6=24 and the area of the square will be 5^2=25, so the ratio of the areas will be 24/25.

Answer: B.


OR: if we take the side of rectangle to be 2x and 3x (for some positive multiple x), then the perimeter of the rectangle will be 2(2x+3x)=10x, thus the side of the square will be 10x/4=5x/2. Next, the area of the rectangle will be 2x*3x=6x^2 and the area of the square will be (5x/2)^2=25x^2/4, so the ratio of the areas will be 24/25.

Answer: B.

Similar question: https://gmatclub.com/forum/if-the-perim ... 96132.html
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In figure R -

Length = 3 ; Breadth = 2

Area of R = 6 and Perimeter of R = 2 ( 3 + 2 ) => 10

In figure S -

Sides = S (Say)

So, 4s = 10

Or, s = 5/2

Area of S = 25/4

Hence area of R to the area of S = 6 : 25/4 => 24 : 25

Hence, Correct answer will be (B) 24: 25
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GOT IT, 2ND APPROACH IS EASIER...THAAAAAANKS B.

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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We know Perimeter of a square (Ps) = 4*side
Perimeter of a rectangle (Pr) = 2(length+breath)

Let us assume 40 to be the perimeter of the square (since we know each side of a square is equal and the perimeter is divisible by 4, also take in to account the length and breadth of the rectangle is in the ration 2k:3k = 5k; we can assume such a number)

Therefore,
Ps = Pr = 40
Area of the square = 100 sq. units
We know 2(length+breadth) = 40
i.e. length + breadth = 20 (or 5k = 20 given that l:b (or b:l) = 2:3)
Therefore length = 8, breath = 12

Area of the rectangle = 8*12 = 96 sq. units

Question asked = Area of the rectangle : Area of the square = 96:100 ==> 24:25

Note : The explanation might be bigger, but it takes less than 15 seconds to solve this problem if you assume numbers and try the problem.

hi pushpitkc, can you explain how after this \(s=\frac{5a}{4}\). we got this --->

\(\frac{ab}{s^2}=\frac{3a^2}{2}*\frac{16}{25a^2}=\frac{24}{25}\).

thanks

Hi dave13

Rectangle: Length - a , Breadth - b | Square: size - s.

We know that \(b=\frac{3a}{2}\) and \(s=\frac{5a}{4}\)

Ratio of area = \(\frac{ab}{s^2}\) = \(\frac{\frac{3a^2}{2}}{\frac{25a^2}{16}}\) = \({\frac{3a^2}{2}}*{\frac{16}{25a^2}} = \frac{24}{25}\)

Hope this clears your confusion!

Hi pushpitkc,

appreciate your explanation just one question just one question we need to find ratio of area of rectangle to the area of square

but we know only breadth of rectangle \(b=\frac{3a}{2}\) i mean it doesnt represent the area of rectangle whereas area of square is aexprresed correctly \(s=\frac{5a}{4}\)

my question is: why do we divide only breadth of rectangle by area of square AND NOT area of rectangle by area of square

Hi dave13

The area of the rectangle is length*breadth = a*b. We know that the breadth \(b=\frac{3a}{2}\).

The area of the rectangle will be \(a*b = a*\frac{3a}{2} = \frac{3a^2}{2}\)

Hope this helps you!

We are given that the perimeters of square region S and rectangular region R are equal and that the sides of R are in the ratio 2 : 3. Let’s label the sides of our figures:

Width of rectangle R = 2x

Length of rectangle R = 3x

Side of square S = s

The perimeter of rectangular region R is 2(2x) + 2(3x) = 4x + 6x = 10x.

The perimeter of square region S is 4s.

Since the two perimeters are equal we can create the following equation:

4s = 10x

2s = 5x

s = (5/2)x

Lastly, we need to determine the areas of both rectangle R and square S.

Area of rectangle R = length * width

A = (3x)(2x) = 6x^2

Since s = (5/2)x, we can use (5/2)x for the side of S.

Area of square S = side^2

A = [(5/2)x]^2

A = (25x^2)/4

We must determine the ratio of the area of region R to the area of region S.

Area of R/Area of S

6x^2/[(25x^2)/4]

6/(25/4)

24/25

Alternate Solution:

We know the sides of the rectangle have a ratio of 2:3; thus we can express the sides of this rectangle as 2x and 3x for some number x. The perimeter of the rectangle, in terms of x, is then 3x + 2x + 3x + 2x = 10x. This is also the perimeter of the square, so taking x = 2 will give us easy numbers to work with.

When x = 2, the sides of the rectangle are 4 and 6; thus the area of the rectangle is 4 x 6 = 24.

Also, when x = 2, the perimeter of the square is 10x = 20; thus a side of the square will be 5. The area of the square will then be 5 x 5 = 25.

So, the ratio of the area of the rectangle to the area of the square is 24:25.

Answer: B
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given
4s=2*(2+3)
s=5/2
so area of R = 6
area of S = 25/4
ratio R/S ; 24/25
IMO B

I like to approach such questions by picking smart numbers.
Given that the sides of the rectangular are in the ratio of 2:3 we know that the perimeter of that rectangular would be (2x2)+(2x3)=10
Because the perimeter of the square should be the same, the sides have to be 4x2,5=10
I do want to work with integers so I don't like 2,5.
Let's just double the sides of the rectangular (2x4)+(2x6)=20
Now, each side of the square has to be 5
Area of rectangular R -> 4x6=24
Area of square S -> 5x5=25
Ratio 24:25
Hence, B

Let's PLUG IN some values that meet the given conditions.

The sides of R are in the ratio 2:3
So, let the two sides have lengths 2 and 3.
This means the area of Region R = (2)(3) = 6
This means the ENTIRE perimeter of Region R is 2 + 2 + 3 + 3 = 10


The perimeters of square region S and rectangular region R are equal.
This means the perimeter of square region S is also 10
Since all 4 sides in a square are of equal length, each side must have length 2.5
So, the area of Region S = (2.5)(2.5) = 6.25

What is the ratio of the area of region R to the are of region S ?
We get: 6 : 6.25
Check the answer choices .... no matches. So, we need to take 6 : 6.25 and find an equivalent ratio.
If we multiply both parts by 4 we get: 24 : 25
So, the correct answer is B

Cheers,
Brent

Hi GMATters,

Here is my video solution to this question:

NB: You can always plug numbers in here if you want to as Jeff suggested above. It's really up to you, but I personally lean into Algebra because thinking about the correct numbers seems like just about as much effort as setting up the variables and letting it rip. This is my way; do what works best for you.

Enjoy!

I'll never understand why would solve this question any way aside from just picking numbers that make sense.

We could make the rectangle 2x3, but then the sides of the square would be 2.5. That doesn't sound all that awesome. How about we make the rectangle 4x6 and the square 5x5.
Area of the rectangle is 24. Area of the square is 25.
AreaR:AreaS = 24:25

Answer choice B.


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