Solve the Logic Puzzle | Find area of the Square ABCD | All rectangles in the square have same area 

Dear teacher, you make all exercises seems so easy, thanks!

I worked it out by using the fact that the brown rectangle is 1/5 the area of ABCD, and as it spans the width of the square, the short side must be 1/5 of the long side. Call the short side length x, therefore ABCD has sides of 5x. S = 5x^2. The blue rectangle’s long side is 5x - x = 4x, so S = 6 . 4x = 24x. Therefore 5x^2 = 24x. Divide by x (since x > 0) gives us 5x = 24. Since ABCD has sides = 5x, each side is 24. Therefore the area of the square is 24^2 = 576 units^2

A bit of a shorter method: As shown in video, blue rectangle has width = 6, and height = x, so S = 6x Now consider rectangle formed by yellow, purple and green rectangles Area of this composite rectangle = S + S + S = 3S = 3(6x) = 18x This rectangle has height x, so its width = 18x/x = 18 Therefore, AB = 18+6 = 24 Since ABCD is a square, then Area(ABCD) = 24^2 = 576

The rectangle formed by the yellow, pink and green rectangles has an area three times the area of the blue rectangle but has the same length x. Then the width should be three times the width of the blue rectangle: 3(6)= 18. And 18 + 6 = 24. Area of ABCD = (24)(24)= 576.

Super! This can also be done in a simple way. Pink, Green and Yellow rectangles combine to form a rectangle with an area 3S with the same height as the blue rectangle of area S. Hence, its width is 3*6 = 18. Length of the square is 18+6 = 24.

A la izquierda del rectángulo azul azul se puede colocar otros 3 azules, sustituyendo al amarillo, al morado y al verde → AB=(3+1)6 =24 → Área cuadrado ABCD = 24² =576 Gracias y saludos.

If l is the side of the square: 1) 6x=S; 2) lx=4S --> lx=4*6x --> l=24. Area=24²=576.

I solve a different way. I said the one rectangle was 6 by A/6 and the side of the square is a. The rectangle on the top is a-6 by A/(a-6). That would mean those 2 rectangle beneath would be (2A)/(a-6) by (a-6)/2. If you set those vertical sides equivalent to each other you get (A/a-6) + 2A/(a-6) = A/6. The A's cancel out and you are left with a - 6 = 18. Solve for a which is 24. I took a little extra time to solve for A which is 115.2 square units which is just 576 divided by 5 of course. I am a 61 year old Aerospace Engineer. These little exercises keep my mind sharp. Thanks Premath.

Sir! After finding the length of the rectangle of which the breadth is given you should have taken the rectangle of which you have founded the length and then have find the breadth of the whole rectangle since the area of whole rectangle will be S+S+S+S, then then breadth turns out to be 24 units and this is also the side of the square ABCD and thus the area turns out to be 576 sq.units that is the answer

Just another method: 1) Let x = length of the square side; 2) height of the blue rectangle = s/6; 3) height of the orange rectangle is s/x; 4) x= s/6 + s/x ; x= s(1/6+1/x); 5) s = x/(1/6+1/x) = 6x^2/(x+6) 6) x^2 = 5*s 7) x^2 = 5 *6*x^2/(x+6) => 1 = 30/(x+6) => x+6 = 30=> x= 24 8) Asquare = x^2 = 24^2 =576 cm^2

The four rectangles on top can be rearranged all in the same way as the one on the right. Then, we can see four identical rectangles with a width of 6 (and the same length). One side of the square ABCD, therefore, is 4 * 6 = 24 units. The area of the square ABCD is A = 24² = 576 square units (cm²).

Very easy idea , we enjoy with it . Thanks Sir. Thanks PreMath .

Another extremely elegant solution. Once again, many thanks from me.

Best method to solve this,Sir.Great job❤

Fantastic, great explanation.

The ratio of the horizontal length of the yellow rectangle to the horizontal length of the light blue rectangle is 3S:S=3:1 from the area ratio of yellow + pink + green and blue. Therefore, if the horizontal length of the yellow rectangle is x, then x:6=3:1 ∴x=18 Since the length of one side of a square is 18+6=24, the area of a square is 24²=576

Nice done! Another simple solution: Let's call X=AB=BC=CD=AD. So X^2=5S, which results in S=X^2/5. The area of the "orange" rectangle at the base of the square ABCD=S and the square ABCD=5S: therefore its width is X/5, as well as the length of the "blue" rectangle=4X/5. The area of the blue rectangle of width 6=S, which results in 6*4X/5=S=24X/5. Conclusion: S=X^2/5=24X/5 --> X^2/x=24 --> X=24. So Area of square ABCD=24^2=576.

Amazing 👍 Thanks for sharing😊

Yay! I solved it.

Let a be the length of square side and x the other side of orange rectangle. Then a^2=5S S=ax (a-x)(a-6)=3S From that a^2=5ax (a-x)(a-6)=3ax and x=a/5 (a-a/5)(a-6)=3a^2/5 4(a-6)=3a a=24 Area of square=24^2

Your solution is wonderful!

By observation each rectangle is 1/5 the full area, since there are five rectangles. This means each rectangle has the area s²/5, where s is the length of one side of ABCD. As the orange rectangle takes up the entire bottom edge of ABCD, two of its sides, DC and its opposite, are length s. This means the other two sides, along AD and CB, are (s²/5)/s = s/5. This means that the length of the blue rectangle along CB is 4s/5. Aʙ = 6(4s/5) s²/5 = 24s/5 s(s/5) = 24(s/5) s = 24 A = 24(24) = 576

6L area of given rectangle Other 4 rectangle having same area means 6L So whole square area is L^2 = 6L * 5 L^2 = 30L divide each side by L We will get L = 30 of that rectangle Now area of that rectangle is 30*6 =180 Total area is 180* 5 =900

Thanks for video.Good luck sir!!!!!!!!!

Each side of the square = x cm The square area = (x^2) cm^2 = 5S cm^2 The width of the orange square = (x/5) cm The length of the blue square = (4/5)x cm 6 (4/5)x = (1/5)(x^2) 24x = x^2 x = 24 area of the square = 24^2 cm^2 = 576 cm^2

Great!

let the longer side of the purple and green rectangles be a and their shorter sides will be the same since they have the same area be b. the length of the yellow rectangle will be 2b and it's breadth wil be a/2. the length of the blue rectangle will be 3/2a the area of the small rectangles are equal hence : ab = 3/2a x 6 b = 9 hence the side of the large square = 9 + 9+ 6 = 24 area of square = 576 units sq

There is a simpler method that only needs to consider the right and bottom sides. You don’t even need to introduce a new variable. We know the total area of the square is 5S, and thus the length of each side SQRT(5S). The blue square has one side of length 6; the other side must then be S/6. This makes the length of the vertical segment of the bottom (red) rectangle length SQRT(5S)-S/6. We can now calculate the area of S in terms of the red rectangle only, and as this is one equation in one variable we can solve it directly: S = SQRT(5S) * (SQRT(5S) - S/6); S = 5S - (S*SQRT(5S))/6; 4S = (S*SQRT(5S))/6; 24S = S*SQRT(5S); SQRT(5S)=24. Square both sides to find that 5S=576, and as we know the total area of the square is 5S, its area is thus 576. QED. Note we didn’t even have to look at any of the other squares in this method. They are completely superfluous to the solution.

Very interesting puzzle, it suffices to determine the value of S, then then the answer is 5S, let l be the length of the square, thus the height of the bottom rectangle is S/l, as the height of left most rectangle is S/6, then the length of the square is also equal to S(1/l+1/6)=l, as 5S=l^2, l^2(1/l+1/6)=5l, then l+l^2/6=5l, 6l+l^2=30l, l^2=24l thus l=24, the answer is 24^2=576.🙂

Very nice

6•18=108 108=S S•5=540 Area of the square =540

I solved it using just the blue and orange rectangles. Let the total side length of the square be g. The total area we are looking for is g². The total area is also 5 rectangles of equal area S, so 5S = g². It is also true that S = g²/5. (1) Look at the blue and orange rectangles. We need two spare variables, x and y. Let's define the blue to have side lengths of 6 and x, while orange has side lengths of g and y. The areas of those two are equal, so 6x = S (2), and gy = S (3). We also know that x + y = g (4), because the whole shape is a square. Per equations (1) and (3), we can conclude that gy = g²/5. Since we know g cannot be zero, we can safely divide both sides by g, giving y = g/5. Per equation (4), we can now express x in terms of g: x + g/5 = g, therefore x = 4g/5. Plugging that into equation (2) gives us 24g/5 = S. Equation (1) makes this an equation in terms of the side length g: 24g/5 = g²/5. Again, since we know g cannot be zero, we can safely divide both sides by g/5, so we get g = 24. Since the total area is g², That means the total area is 576.

Did it a bit differently. The base of the blue rectangle is 6 so its height is S/6. I have labelled each side of the square as « a ». The sum of the areas of the yellow, purple and green rectangles equals to 3*S. Combining these three rectangles gives a bigger rectangle with a base equal to a-6 and a height equal to S/6. Therefore, 3*S=(S/6)*(a-6) => 18=a-6 => a=24. The area of the square equals to a^2, equals to 24^2.

Tricky at first but easy at the end

Mostly by chance I have found a path which arguably is the shortest possible(a sheer luck), consider the blue rectangle , its height is h2 = S/6. Consider the composite rectangle on the left,it shares the same height and its area is 3 S , so he have 3 S = (W -6) * S / 6. Divide by S and you get a single equation in a single variable which is total width(and square side as well). Due to good luck and my endless laziness too🤣.

Nice❤

It wasn't required, but you can also find the value of x in your diagram, which is 19.2

It is a lot easier in this way. let the height of yellow rect is a and the width of pink rect is b. As the area of pink and green rect of same height are equal, the width of green rect and pink rect should be identical. Therefore, the width of green rect is also b. The Area of yellow rect is equal to a x (b+b)=2ab. It means the area of any rect is also equal to 2ab. The area of pink rect is b x height =2ab. So, the height of pink rect is 2a. The area of blue rect is equal to 6 x (a+2a)=18a. Area of blue rect is alos equal to 2ab. So, 2ab=18a => b=9. The width of the squae therefore is equal to 9+9+6=24. ==> the area of the square is 24^2.

24^2 = 576 Let the length of square = x; hence the area =x^2; hence the area of each rectangle =x^2/5 {since each rectangle has the SAME area} Let the length of the blue rectangle = L, then its area = 6L; hence 6L = x^2/5 ; hence L = x^2/30 Since the length of the yellow rectangle = x-6 [recall the length of the square =x, and the width of the blue is 6; hence the yellow is the difference between the two), then the area of the Yellow + Green + Purple = (x-6)* L since the Green and Purple are just below the Yellow, which all equal the length of the Blue). ; hence (x-6)*L = (x^2/5)*3 [ recall that the area of each rectangle =x^2/5; hence 3 of them would = 3(x^2/5] substituting the value for L = x^2/30 into the equation (x-6)* L = 3(x^2/5, would give the value for or "x" (x-6)* x^2/30 = 3x^2/5 (x-6) = 3x^2/x^2 * 30/5 x-6 = 3*6 x-6 = 18 x = 6+18 x =24 x^2 = 24*24 = 576 Answer

forget the brown rectangle! since all rectangles have equal area which is 6x! combined area of 4 rectangles yellow+blue+purple+green should be 24x.. so side AB should be 24! area of square 24×24

Similar method but I somehow came out with 576 furlongs squared. :)

576

But Sir, we are just taking it granted that the widths of the green & the purple rectangles are same. Though that is not a given condition.

There is 6, not 6cm. It should be written as 6[u], we don't have given that unit is centimeter, also don't need it.

Now, solve for _x._

I didn't use centimeters. Used inches and came up with the same results. 🙂

why 5.6.x = 5.6.9 < 24.24?

asnwer=105 cm isit

576