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WWW.C E M C .U WAT E R LO O.C A | T h e C E N T R E fo r E D U C AT I O N i n M AT H E M AT I C S a n d CO M P U T I N G Problem of the Week Problem C and Solution Wired Problem A piece of wire 60 cm in length is to be cut into two parts in the ratio 3 : 2. Each part is bent to form a square. Determine the ratio of the area of the larger square to the smaller square. Solution Let the length of the longer piece of wire be 3x cm and the length of the shorter piece of wire be 2x cm. Then 3x + 2x = 60 or 5x = 60 and x = 12 follows. Then the longer piece of wire is 3x = 3(12) = 36 cm and the smaller piece of wire is 2x = 2(12) = 24 cm. These two lengths correspond to the perimeters of the respective squares. Each of the wires is bent to form a square. The length of each side of the square is the perimeter of the square divided by 4. Therefore the side length of the larger square is 36 ÷ 4 = 9 cm and the side length of the smaller square is 24 ÷ 4 = 6 cm. The area of a square is calculated by squaring the side length. The area of the larger square is 92 = 81 cm2 and the area of the smaller square is 62 = 36 cm2 . The ratio of the area of the larger square to the area of the smaller square is 81 : 36. This ratio can be simplified by dividing each term by 9. The ratio in simplified form can then be written as 9 : 4. Therefore the ratio of the area of the larger square to the area of the smaller square is 9 : 4. An observation: The ratio of the area of the larger square to the area of the smaller square is 9 : 4 = 32 : 22 . Is it a coincidence that the ratio of the area of the larger square to the area of the smaller square is equal to the squares of each term in the given ratio? Also notice that the ratio of the area of the larger square to the area of the smaller square is equal to the ratio of the square of the perimeter of the larger square to the square of the perimeter of the smaller square. In this case, the perimeter of the larger square is 36 cm and the perimeter of the smaller square is 24 cm. Then 362 : 242 = 1296 : 576 = 1296 : 576 = 9 : 4. 144 144 It is left to the solver to see if these two results are true in general.