The price of a diamond varies as the cube of its volume. A cubical variety of this diamond was worth ₹ 10,00,000. If this diamond accidentally broke into 8 equal cubical diamonds, then the total loss in value amounts to

### Concept & Proportionality When a quantity varies directly as the cube of another, we can set up a proportionality equation. Here, the price ($P$) varies as the cube of the volume ($V$). $$P = k \times V^3$$ When a large cube is cut into smaller equal cubes, the total volume is conserved, so the volume of each small cube is a fraction of the original volume. ### Step-by-Step Solution 1. **Initial State:** Let the original volume be $V$ and the initial price be $P$. We are given $P = 10,00,000$. 2. **Finding the Constant:** $P = k V^3$. 3. **Breaking the Diamond:** The diamond breaks into 8 equal cubical diamonds. Since total volume is conserved, the volume of each small diamond is $v = \frac{V}{8}$. 4. **New Value Calculation:** The price of one small diamond ($p$) is: $p = k \times v^3 = k \times (\frac{V}{8})^3 = k \times \frac{V^3}{512} = \frac{P}{512}$ 5. **Total Value of Broken Pieces:** Total value = $8 \times p = 8 \times \frac{P}{512} = \frac{P}{64}$. 6. **Calculating the Loss:** Loss = Original Price - Total New Value Loss = $P - \frac{P}{64} = \frac{63P}{64}$ 7. **Final Calculation:** Substituting $P = 10,00,000$: Loss = $\frac{63}{64} \times 10,00,000 = 63 \times 15,625 = 9,84,375$. ### Exam Strategy & Shortcut Realize that the new total value is simply $\frac{n}{n^3}$ of the original value where $n$ is the number of pieces (if broken evenly along 3 dimensions, but here it explicitly says 8 equal pieces, so volume is $1/8$, value of one is $1/8^3$, total is $8/8^3 = 1/64$). Loss fraction is $1 - \frac{1}{64} = \frac{63}{64}$. Simply calculate $\frac{63}{64} \times 1000000$. ### Common Pitfall Assuming the price is proportional to the volume itself or the side length squared, rather than the cube of the volume as explicitly stated in the problem. ### Final Answer Therefore, the correct answer is **₹ 9,84,375**.