Question

An iron beam made with rare materials has its market price dependent on the square of its length. The beam broke into two pieces in the ratio of 4 : 9. If it is sold as two separate pieces, what would be the percentage profit or loss compared to its original value?

• A. 44.44% loss
• B. 50% loss
• C. 55.55% loss
• D. 60% loss

Hint:

For pricing based on the square of length, breaking an item into smaller parts always results in a loss because the sum of the squares of smaller lengths is less than the square of the total length.

Answer 1

The Correct Option is A

Pricing Model:

The beam's price is determined by the square of its length. Let the original length be \(L\). The original price is \(L^2\).

Broken Piece Lengths:

The beam fractures into two segments with lengths in a 4:9 ratio. The lengths of these pieces are:

Piece 1: \(\frac{4}{13}L\), Piece 2: \(\frac{9}{13}L\).

Broken Piece Pricing:

The price of each segment is proportional to the square of its length:

Price of Piece 1: \(\left(\frac{4}{13}L\right)^2 = \frac{16}{169}L^2\)

Price of Piece 2: \(\left(\frac{9}{13}L\right)^2 = \frac{81}{169}L^2\)

Total Price of Broken Pieces:

\[ \text{Total Price} = \frac{16}{169}L^2 + \frac{81}{169}L^2 = \frac{97}{169}L^2 \]

Loss Calculation:

The original price was \(L^2\). The loss is calculated as Original Price - Price of Broken Pieces:

\[ \text{Loss} = L^2 - \frac{97}{169}L^2 = \frac{169}{169}L^2 - \frac{97}{169}L^2 = \frac{72}{169}L^2 \]

Percentage Loss:

\[ \text{Percentage Loss} = \frac{\text{Loss}}{\text{Original Price}} \times 100 = \frac{\frac{72}{169}L^2}{L^2} \times 100 = \frac{72}{169} \times 100 = 44.44\%. \]

The calculated percentage loss is 44.44%.