Understanding the Stone Value Problem

This question deals with a problem where the value of a stone is not directly proportional to its weight, but to the square of its weight. This is a common type of problem that tests understanding of proportionality and ratios.

We are given that the value ($V$) of the stone is proportional to the square of its weight ($W$). Mathematically, this can be written as:

\(V \propto W^2\)

This means \(V = k \cdot W^2\) for some constant of proportionality, \(k\).

Initial Stone Value and Weight

Let the original weight of the stone be \(W\). The original value of this stone is given as Rs. 1,20,000.

So, we have the equation for the original stone:

\(1,20,000 = k \cdot W^2\)

Breaking the Stone and Weight Ratio

The stone is broken into two pieces. Let the weights of these two pieces be \(W_1\) and \(W_2\). The problem states that the weights are in the ratio 2 : 3.

\(W_1 : W_2 = 2 : 3\)

The sum of the weights of the two pieces must equal the original weight:

\(W_1 + W_2 = W\)

Since the weights are in the ratio 2:3, we can think of the total weight \(W\) being divided into \(2 + 3 = 5\) parts. The first piece has 2 parts and the second piece has 3 parts.

Therefore, we can express \(W_1\) and \(W_2\) in terms of \(W\):

• \(W_1 = \frac{2}{5} W\)
• \(W_2 = \frac{3}{5} W\)

Calculating the Value of Each Small Stone

The value of each smaller stone is also proportional to the square of its weight with the same constant \(k\). Let the values of the two pieces be \(V_1\) and \(V_2\).

For the first piece (weight \(W_1\)):

\(V_1 = k \cdot W_1^2 = k \cdot \left(\frac{2}{5} W\right)^2 = k \cdot \frac{4}{25} W^2\)

For the second piece (weight \(W_2\)):

\(V_2 = k \cdot W_2^2 = k \cdot \left(\frac{3}{5} W\right)^2 = k \cdot \frac{9}{25} W^2\)

Finding the Total Price of the Two Small Stones

The total price (or value) of the two small stones is the sum of their individual values:

\(V_{total} = V_1 + V_2\)

\(V_{total} = k \cdot \frac{4}{25} W^2 + k \cdot \frac{9}{25} W^2\)

We can factor out \(k \cdot W^2\):

\(V_{total} = k \cdot W^2 \left(\frac{4}{25} + \frac{9}{25}\right)\)

\(V_{total} = k \cdot W^2 \left(\frac{4+9}{25}\right)\)

\(V_{total} = k \cdot W^2 \left(\frac{13}{25}\right)\)

Using the Original Value to Find the Total Price

We know from the original stone's value that \(k \cdot W^2 = 1,20,000\). We can substitute this into the expression for \(V_{total}\):

\(V_{total} = 1,20,000 \cdot \frac{13}{25}\)

Now, we calculate the final value:

\(V_{total} = \frac{1,20,000 \times 13}{25}\)

First, divide 1,20,000 by 25:

\(\frac{1,20,000}{25} = \frac{120000}{100} \times 4 = 1200 \times 4 = 4800\)

Now, multiply the result by 13:

\(V_{total} = 4800 \times 13\)

\(4800 \times 13 = 4800 \times (10 + 3) = (4800 \times 10) + (4800 \times 3)\)

\(48000 + 14400 = 62400\)

So, the total price of the two small stones is Rs. 62,400.

Let's quickly compare the original value to the total value of the broken pieces. The original value was Rs. 1,20,000. The total value of the broken pieces is Rs. 62,400. This shows that breaking the stone results in a significant loss of total value, which is characteristic of such proportionality problems.

Final Answer Check

The calculated total price of the two small stones is Rs. 62,400. This matches one of the provided options.

The options were:

• Rs. 68,400
• Rs. 62,400
• Rs. 66,500
• Rs. 65,400

Our result of Rs. 62,400 corresponds to the second option.

Revision Table: Stone Value Calculation

Additional Information: Proportionality Concepts

Proportionality describes how two quantities relate to each other. There are different types:

• Direct Proportionality: If \(A\) is directly proportional to \(B\), then \(A = cB\) for some constant \(c\). This means as \(B\) increases, \(A\) increases at the same rate.
• Inverse Proportionality: If \(A\) is inversely proportional to \(B\), then \(A = c/B\) for some constant \(c\). This means as \(B\) increases, \(A\) decreases.
• Direct Proportionality to a Power: As seen in this problem, \(V\) is proportional to the square of \(W\), i.e., \(V = cW^2\). This means as \(W\) increases, \(V\) increases much faster (quadratically). If it were proportional to the cube, \(V = cW^3\), it would increase even faster.

In problems involving proportionality, identifying the relationship (\(V \propto W^2\)), setting up the equation with a constant (\(V = kW^2\)), and using the initial condition to understand the constant's role is crucial. When quantities are broken down (like the weight of the stone), calculating the corresponding proportional value for each part and then summing them gives the new total value.