A precious stone breaks into four pieces having weights in the proportion 1 ∶ 2 ∶ 3 ∶ 4. The value of such a stone is proportional to the square of its weight. What is the percent loss in the value incurred due to breaking?
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Precious Stone Breaking Problem
This problem involves calculating the loss in value of a precious stone when it breaks. The value of the stone is directly proportional to the square of its weight. We are given the proportions of the weights of the broken pieces.
Stone Weight Proportions
Let the weights of the four pieces be in the ratio $1 : 2 : 3 : 4$.
We can represent the actual weights as $1k$, $2k$, $3k$, and $4k$, where $k$ is a constant.
The total weight of the original stone before breaking was the sum of the weights of the pieces:
Original Weight $= 1k + 2k + 3k + 4k = 10k$
Value Calculation
The value of the stone is proportional to the square of its weight. Let $c$ be the constant of proportionality.
Original Value:
Original Value $= c \times (\text{Original Weight})^2 = c \times (10k)^2 = c \times 100k^2$
Value of Broken Pieces:
The value of each piece is calculated separately:
- Value of piece 1 (weight $1k$) $= c \times (1k)^2 = c \times 1k^2$
- Value of piece 2 (weight $2k$) $= c \times (2k)^2 = c \times 4k^2$
- Value of piece 3 (weight $3k$) $= c \times (3k)^2 = c \times 9k^2$
- Value of piece 4 (weight $4k$) $= c \times (4k)^2 = c \times 16k^2$
The total value of the stone after breaking is the sum of the values of the individual pieces:
Total Value (Broken) $= c \times 1k^2 + c \times 4k^2 + c \times 9k^2 + c \times 16k^2$
Total Value (Broken) $= c \times (1 + 4 + 9 + 16)k^2 = c \times 30k^2$
Percent Loss in Value
The loss in value due to breaking is the difference between the original value and the total value of the broken pieces.
Loss in Value $= \text{Original Value} - \text{Total Value (Broken)}$
Loss in Value $= c \times 100k^2 - c \times 30k^2 = c \times (100 - 30)k^2 = c \times 70k^2$
The percent loss in value is calculated as:
Percent Loss $= \left( \frac{\text{Loss in Value}}{\text{Original Value}} \right) \times 100\%$
Percent Loss $= \left( \frac{c \times 70k^2}{c \times 100k^2} \right) \times 100\%$
We can cancel out $c$ and $k^2$ from the numerator and denominator:
Percent Loss $= \left( \frac{70}{100} \right) \times 100\%$
Percent Loss $= 0.70 \times 100\% = 70\%$
Therefore, the percent loss in the value incurred due to breaking is 70%.
Important Questions from Direct or Indirect Proportion
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