Question

A large rectangle is made up of nine identical rectangles whose longer side equals 10. What is the area of the large rectangle?

Answer 1

Answer:  360 square units

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Explanation:

Refer to the diagram below.

A small rectangle has its longer side equal to 10 and its shorter side is x, which is some real number on the interval 0 < x < 10.

Across the top, the two sides of length 10 add to 20. So the larger rectangle has a horizontal dimension of 20 units.

The vertical side of the larger rectangle is x+10+x = 2x+10 units

The area would be

area = length*width = 20(2x+10) = 40x+200

Each of the nine smaller rectangles has area of length*width = 10x. Since we have 9 identical copies, the smaller rectangle areas must add to 9*10x = 90x

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Equate those two expressions and solve for x

90x = 40x+200

90x-40x = 200

50x = 200

x = 200/50

x = 4

So each smaller rectangle has area of 10*x = 10*4 = 40 square units. In total, the nine smaller rectangles combine to an area of 9*40 = 360 square units

Note how x = 4 leads 40x+200 to result in 360 as well.

Answer 2

• On the top row of rectangles, the length is 10 + 10 = 20
• The middle row has 5 rectangles with all of their short sides equaling the top row, 20
• So 5x = 20 where x is the short side of the rectangle
• x = 4
• So each rectangle’s area = 4 x 10 = 40
• There are 9 rectangles
• So 9 x 40 = 360 = area of the large rectangle
• We can check our work
• Large rectangle side 1 = 10 + 10 = 20
• Large rectangle side 2 = 4 + 10 + 4 = 18
• 18 x 20 = 360