The Problem:

The problem states that the value of a diamond varies with the square of its weight. We are given a diamond worth Rs. 14,40,000, which breaks into three pieces. The weights of these pieces are in the ratio 3:4:5. We need to determine the loss in value due to the breakage.

Understanding the Problem:

To solve this problem, we need to understand the relationship between the weight and value of a diamond. The problem states that the value varies with the square of the weight. This means that if the weight of the diamond doubles, its value will increase by a factor of four.

Calculating the Loss in Value:

To calculate the loss in value due to breakage, we need to compare the weights of the original diamond and the three broken pieces. Let's assume the original weight of the diamond is x.

According to the given information, the weights of the broken pieces are in the ratio 3:4:5. So, the weights of the three pieces can be represented as 3x, 4x, and 5x.

Now, let's calculate the loss in value for each piece individually.

The value of the original diamond can be calculated by squaring its weight:
Value of original diamond = x^2

Similarly, we can calculate the values of the three broken pieces:
Value of first piece = (3x)^2 = 9x^2
Value of second piece = (4x)^2 = 16x^2
Value of third piece = (5x)^2 = 25x^2

The total loss in value is given by the difference between the value of the original diamond and the sum of the values of the broken pieces:
Loss in value = (Value of original diamond) - (Value of first piece + Value of second piece + Value of third piece)

Substituting the values, we have:
Loss in value = x^2 - (9x^2 + 16x^2 + 25x^2)

Simplifying further:
Loss in value = x^2 - 50x^2
Loss in value = -49x^2

Calculating the Loss in Value in Lakhs:

To convert the loss in value from rupees to lakhs, we need to divide the result by 1,00,000 (since 1 lakh = 1,00,000).

Loss in value (in lakhs) = (Loss in value) / 1,00,000
Loss in value (in lakhs) = (-49x^2) / 1,00,000

Conclusion:

The loss in value due to breakage is given by the equation (-49x^2) / 1,00,000, where x represents the original weight of the diamond. To calculate the exact loss in value, we need to know the original weight of the diamond.