The explanation to this question is a bit complicated (or confusing). If you have any question, feel free to comment down in the comments section.
It is important to use variables (such as ‘x’ or ‘y’ or ‘z’) when we are not given any values in the question.
Let cost of wood be = W
We know cost of metal is three times cost of wood.
Hence, cost of metal = 3W
Let amount of wood used = r
Therefore, total cost of wood = (W x t)
Let amount of metal used = u
Therefore, total cost of metal = (3W x u)
Total cost of sign = Wr + 3Wu
Let the diameter of sign = D
Wood is proportional to size of diameter: r = D²
Metal is proportional to the square of diameter: u = D
Since the diameter is doubled (2D), we get the following equations:
r = (2D)² = 4D² = 4r (Since D² = r)
u = 2D = 2u (Since D = u)
New total cost = (W x 4r) + (3W x 2u) = 4Wr + 6Wu
Since the diameter is doubled, the total cost will be tripled.
Therefore the new cost is three times the old cost.
4Wr + 6Wu = 3[Wr + 3Wu]
4Wr + 6Wu = 3Wr + 9Wu
Wr = 3Wu
r = 3u
Percentage of metal = Amount of metal ÷ Total amount of material
Total amount of material = amount of wood (r) + amount of metal (u)
Amount of metal = u
Percentage of metal = u ÷ (r + u)
Since r = 3u, we get:
Percentage of metal = u ÷ (3u + u) = u ÷ 4u = 1/4 = 25%