Understanding Diamond Cost and Weight Relationship

The problem states that the cost of a piece of diamond varies with the square of its weight. This means if the weight is \(W\), the cost \(C\) can be expressed as \(C = k W^2\), where \(k\) is a constant value.

We are given an original diamond with a value of Rs. 6,084. Let its original weight be \(W_{\text{original}}\). So, the initial condition is:

\(6084 = k (W_{\text{original}})^2\)

Analyzing the Diamond Cutting into Pieces

The original diamond is cut into three pieces. The weights of these three pieces are in the ratio 3 ∶ 2 ∶ 1. Let the common ratio factor be \(w\). Then the weights of the three pieces are \(3w\), \(2w\), and \(w\).

The total weight of the three pieces must be equal to the original weight of the diamond before cutting. So,

\(W_{\text{original}} = 3w + 2w + w = 6w\)

Calculating the Constant 'k' and the Value of \(kw^2\)

Now we can substitute the total weight \(W_{\text{original}} = 6w\) back into the original cost equation:

\(6084 = k (6w)^2\)

\(6084 = k \times 36w^2\)

From this equation, we can find the value of \(kw^2\), which will be useful in calculating the cost of the individual pieces:

\(kw^2 = \frac{6084}{36}\)

Performing the division:

\(6084 \div 36 = 169\)

So, \(kw^2 = 169\).

Determining the Value of Each Piece After Cutting

Now, let's calculate the value of each of the three pieces using the cost-weight relationship \(C = k W^2\) and the fact that \(kw^2 = 169\).

• The first piece has a weight of \(3w\). Its value is \(C_1 = k (3w)^2 = k \times 9w^2 = 9 (kw^2)\).
• The second piece has a weight of \(2w\). Its value is \(C_2 = k (2w)^2 = k \times 4w^2 = 4 (kw^2)\).
• The third piece has a weight of \(w\). Its value is \(C_3 = k (w)^2 = kw^2\).

Calculating the Total Value After Cutting

The total value of the diamond after it has been cut into three pieces is the sum of the values of the individual pieces:

Total value after cutting = \(C_1 + C_2 + C_3\)

Total value after cutting = \(9 (kw^2) + 4 (kw^2) + (kw^2)\)

Total value after cutting = \((9 + 4 + 1) (kw^2)\)

Total value after cutting = \(14 (kw^2)\)

Substitute the value \(kw^2 = 169\):

Total value after cutting = \(14 \times 169\)

Let's calculate \(14 \times 169\):

So, the total value of the diamond pieces after cutting is Rs. 2,366.

Finding the Loss Involved in Cutting

The loss involved in cutting the diamond is the difference between the original value of the diamond and the total value of the pieces after cutting.

Loss = Original value - Total value after cutting

Loss = Rs. 6,084 - Rs. 2,366

Let's calculate the difference:

The loss involved in the cutting is Rs. 3,718.

Summary of Loss Calculation

By understanding the relationship between the diamond's cost and the square of its weight, we calculated the value of the individual pieces after cutting and found the total value is significantly less than the original. The difference represents the loss.

• Original Value: Rs. 6084
• Original Weight Proportion: \(6w\)
• Cost relation: \(6084 = k (6w)^2 \implies kw^2 = 169\)
• Piece Weights: \(3w, 2w, w\)
• Piece Values: \(9(kw^2), 4(kw^2), kw^2 \implies 9(169), 4(169), 169\)
• Total Value After Cutting: \(14(kw^2) = 14(169) = 2366\)
• Loss: \(6084 - 2366 = 3718\)

Revision Table: Diamond Cost and Weight Problem

Additional Information: Proportional Relationships

This diamond cost problem is a good example of a concept called direct proportionality, specifically varying with the square of a quantity. Here's a bit more about proportional relationships:

• Direct Proportionality: If a quantity A is directly proportional to a quantity B, it means that as B increases, A increases at the same rate relative to B. Mathematically, \(A = kB\), where \(k\) is a constant. For example, the cost of apples might be directly proportional to the number of apples.
• Varying with Square: When a quantity varies directly with the square of another, like in this problem \(C = k W^2\), the first quantity changes much faster. If the weight doubles, the cost becomes four times. If the weight triples, the cost becomes nine times. This is why cutting the diamond results in a significant loss; the total value based on summing squares of smaller weights is less than the value based on the square of the total weight.
• Inverse Proportionality: If a quantity A is inversely proportional to a quantity B, it means that as B increases, A decreases. Mathematically, \(A = k/B\) or \(AB = k\). For example, the time taken to travel a distance might be inversely proportional to the speed.

Understanding how quantities relate through proportionality helps solve many problems in physics, economics, and other areas, including quantitative aptitude questions like this one.