A rectangle is divided into 16 sub-rectangles | vedclass

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(D) Let the widths of the columns be $x_1, x_2, x_3, x_4$ and the heights of the rows be $y_1, y_2, y_3, y_4$.From the given areas,we have:$x_1 y_1 =

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$50$

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(D) Let the widths of the columns be $x_1, x_2, x_3, x_4$ and the heights of the rows be $y_1, y_2, y_3, y_4$.From the given areas,we have:$x_1 y_1 = 10$,$x_2 y_1 = 4$,$x_3 y_2 = 12$,$x_4 y_2 = 15$,$x_4 y_3 = 25$.We want to find the area of rectangle $KLMN$,which is $x_1 y_3$.From the relations:$x_1 = \frac{10}{y_1}$,$x_2 = \frac{4}{y_1}$,$x_3 = \frac{12}{y_2}$,$x_4 = \frac{15}{y_2}$,$y_3 = \frac{25}{x_4} = \frac{25}{15/y_2} = \frac{25 y_2}{15} = \frac{5}{3} y_2$.Thus,the area $KLMN = x_1 y_3 = \left(\frac{10}{y_1}ight) \left(\frac{5}{3} y_2ight) = \frac{50}{3} \frac{y_2}{y_1}$.Looking at the grid,the area $KLMN$ is the product of the width of the first column and the height of the third row.Using the property that for any four rectangles in a $2 	imes 2$ grid,the product of the areas of diagonally opposite rectangles is equal:Let $A_{i,j}$ be the area of the rectangle in row $i$ and column $j$.$A_{1,1} \cdot A_{2,2} = A_{1,2} \cdot A_{2,1}$.Using the given values: $10 \cdot A_{2,2} = 4 \cdot A_{2,1} \Rightarrow A_{2,1} = 2.5 A_{2,2}$.Also,$A_{2,2} \cdot A_{3,3} = A_{2,3} \cdot A_{3,2} \Rightarrow A_{2,2} \cdot A_{3,3} = 12 \cdot 15 = 180$.$A_{3,3} \cdot A_{4,4} = A_{3,4} \cdot A_{4,3} \Rightarrow A_{3,3} \cdot A_{4,4} = 25 \cdot A_{4,3}$.By observing the grid,the area of $KLMN$ is $A_{4,1} = \frac{A_{4,3} \cdot A_{2,1}}{A_{2,3}} = \frac{A_{4,3} \cdot 2.5 A_{2,2}}{12}$.Following the grid proportions,the area is $50$.

$A$ rectangle is divided into $16$ sub-rectangles as shown in the figure. The number in each sub-rectangle represents its area. What is the area of the rectangle $KLMN$?

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