Areas!

Geometry Level 4

There are 7 squares with sides x , 3 x , 5 x x, 3x, 5x and 7 x 7x embedded on a rectangle. Such that one side of both squares with side x coincide with the rectangle and each of the square share on of it's side with the other square. Also two opposite sides of 7 x 7x square coincide with length of the rectangle.

If the ratio of blue color region to that of multicolored region is of the form a b \dfrac{a}{b} , where a and b are coprime positive integers, give your answer as the remainder when b b is divided by a a .


The answer is 1.

Viki Zeta by Viki Zeta , 19, India

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1 solution

area of whole rectangle = [ 2 ( x + 3 x + 5 x ) + 7 x ] ( 7 x ) = 175 x 2 \text{area of whole rectangle}=\left[2(x+3x+5x)+7x\right](7x)=175x^2

area of blue region = 4 ( 5 x ) ( x ) + 4 ( 3 x ) ( 2 x ) + 4 ( x ) ( 3 x ) = 56 x 2 \text{area of blue region}=4(5x)(x)+4(3x)(2x)+4(x)(3x)=56x^2

area of multi-colored = 175 x 2 − 56 x 2 = 119 x 2 \text{area of multi-colored}=175x^2-56x^2=119x^2

ratio = 56 x 2 119 x 2 = 56 119 = 8 17 \text{ratio}=\dfrac{56x^2}{119x^2}=\dfrac{56}{119}=\dfrac{8}{17}

Hence,

17 8 = 2 remainder 1 \dfrac{17}{8}=2~\text{remainder 1}

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