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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.