Understanding Diamond Cost and Breakage Loss

This problem deals with the concept of direct proportionality and calculating percentage loss when a valuable item like a diamond breaks. The key information is that the cost of the diamond is directly proportional to the square of its weight.

Establishing the Proportionality Relation

We are given that the cost (C) of the diamond is directly proportional to the square of its weight (W). Mathematically, this relationship can be written as:

$$ C \propto W^2 $$

To turn this proportionality into an equation, we introduce a constant of proportionality, let's call it 'k'.

$$ C = k \times W^2 $$

Here, k is a constant value that relates the cost and the square of the weight for a diamond of a specific quality.

Calculating the Proportionality Constant (k)

We are given that a 14 gm diamond costs Rs. 2560. We can use this information to find the value of 'k'.

• Original Weight (W) = 14 gm
• Original Cost (C) = Rs. 2560

Substitute these values into the equation $C = k \times W^2$:

$$ 2560 = k \times (14)^2 $$

$$ 2560 = k \times 196 $$

Now, solve for 'k':

$$ k = \frac{2560}{196} $$

We can simplify this fraction by dividing both the numerator and the denominator by their common factors. Both are divisible by 4:

$$ k = \frac{2560 \div 4}{196 \div 4} = \frac{640}{49} $$

So, the proportionality constant $k = \frac{640}{49}$.

Analyzing Diamond Breakage and Piece Weights

The original 14 gm diamond broke into two pieces with weights in the ratio 5 : 9. To find the weight of each piece, we first find the total number of parts in the ratio:

Total parts = 5 + 9 = 14 parts.

Since the total weight is 14 gm, each part of the ratio corresponds to 1 gm ($14 \text{ gm} / 14 \text{ parts} = 1 \text{ gm/part}$).

• Weight of the first piece = 5 parts × 1 gm/part = 5 gm
• Weight of the second piece = 9 parts × 1 gm/part = 9 gm

Let's verify: 5 gm + 9 gm = 14 gm, which is the original weight.

Calculating Cost After Breakage

After the breakage, the value of the diamond is the sum of the values of the two pieces. We use the same proportionality relation $C = k \times W^2$ for each piece.

• Cost of the first piece ($C_1$) with weight $W_1 = 5$ gm:

  $$ C_1 = k \times W_1^2 = \frac{640}{49} \times (5)^2 = \frac{640}{49} \times 25 $$

• Cost of the second piece ($C_2$) with weight $W_2 = 9$ gm:

  $$ C_2 = k \times W_2^2 = \frac{640}{49} \times (9)^2 = \frac{640}{49} \times 81 $$

The total cost of the broken diamond ($C_{\text{broken}}$) is the sum of the costs of the two pieces:

$$ C_{\text{broken}} = C_1 + C_2 = \left(\frac{640}{49} \times 25\right) + \left(\frac{640}{49} \times 81\right) $$

We can factor out the common term $\frac{640}{49}$:

$$ C_{\text{broken}} = \frac{640}{49} \times (25 + 81) $$

$$ C_{\text{broken}} = \frac{640}{49} \times 106 $$

Now, calculate the value:

$$ C_{\text{broken}} = \frac{67840}{49} $$

Let's calculate the approximate value:

$$ C_{\text{broken}} \approx 1384.4898 $$

So, the total cost after breakage is approximately Rs. 1384.49.

Determining the Loss Due to Breakage

The loss incurred is the difference between the original cost and the total cost after breakage.

Original Cost ($C_{\text{original}}$) = Rs. 2560

Cost After Breakage ($C_{\text{broken}}$) = Rs. $\frac{67840}{49}$

Loss = $C_{\text{original}} - C_{\text{broken}}$

$$ \text{Loss} = 2560 - \frac{67840}{49} $$

To subtract, find a common denominator (49):

$$ \text{Loss} = \frac{2560 \times 49 - 67840}{49} $$

$$ \text{Loss} = \frac{125440 - 67840}{49} $$

$$ \text{Loss} = \frac{57600}{49} $$

Let's calculate the approximate value:

$$ \text{Loss} \approx 1175.5102 $$

The loss incurred is approximately Rs. 1175.51.

Calculating the Loss Percentage

The loss percentage is calculated with respect to the original cost.

$$ \text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{Original Cost}} \right) \times 100 $$

$$ \text{Loss Percentage} = \left( \frac{57600/49}{2560} \right) \times 100 $$

$$ \text{Loss Percentage} = \left( \frac{57600}{49 \times 2560} \right) \times 100 $$

$$ \text{Loss Percentage} = \left( \frac{57600}{125440} \right) \times 100 $$

Now, perform the calculation:

$$ \text{Loss Percentage} \approx 0.45918 \times 100 $$

$$ \text{Loss Percentage} \approx 45.918 \text{ percent} $$

Rounding to two decimal places, the loss percentage is 45.92 percent.

Revision Table: Diamond Cost and Loss Concepts

Additional Information: Proportionality in Problems

Proportionality is a fundamental concept in mathematics and physics used to describe how quantities relate to each other. In direct proportionality, as one quantity increases, the other increases at a constant rate. In inverse proportionality, as one quantity increases, the other decreases.

• Direct Proportionality: If $y$ is directly proportional to $x$, we write $y \propto x$, which means $y = kx$ for some constant $k$. The graph is a straight line through the origin.
• Direct Proportionality to a Power: As seen in this diamond problem, a quantity can be directly proportional to a power of another quantity, like $C \propto W^2$, meaning $C = kW^2$.
• Inverse Proportionality: If $y$ is inversely proportional to $x$, we write $y \propto \frac{1}{x}$, which means $y = \frac{k}{x}$ or $xy = k$ for some constant $k$. The graph is a hyperbola.

Understanding proportionality helps in solving many problems involving scaling of quantities based on given relationships.