help

I would be happy if someone could show the working on this here. Thanks

The new sign for a local business contains two different sections. One of the sections will be
produced from wood, while the other will be metal. Metal is three times as expensive as wood.
The cost of metal needed for each sign is proportional to the diameter of the sign, while the cost
of wood needed is proportional to the square of the diameter. If the diameter of the sign is
doubled, then the total cost of the materials will be tripled.

What percentage (to the nearest 1%) of the sign is metal?

Ans: 1/4

Assume the sign is of unit volume.
Assume that wood costs w per unit volume and therefore that metal costs 3w per unit volume. Assume that the fraction of the sign that is metal is a and therefore the fraction that is wood is (1-a).

Cost of metal in first sign therefore is (3w x a) and cost of wood is w(1-a). Thus total cost of sign = 3aw + w - aw = 2aw +w.

In sign with the double diameter the cost of the metal has gone up by a factor of 2 (proportional) and the cost of the wood has gone up by a factor of 4 (proportional to square of the diameter)
Now new metal cost is 2 x 3aw and new wood cost is 4 x w(1-a). Thus new total cost is 6aw + 4w - 4aw = 2aw + 4w. This new total cost is 3 times the original total cost. Therefore 2aw + 4w = 3(2aw +w) =6aw + 3w. Rearranging gives w =4aw. Dividing through by w and then by 4 gives a = 1/4

Honestly this explanation is 5 star.
Thank you for the simplicity.