`(675 xx 675 xx 675 + 325 xx 325 xx 325)/(67.5 xx 67.5 + 32.5 xx 32.5 - 67.5 xx 32.5)` is equal to :

Options

A. 100
B. 10000
C. 1000
D. 100000

Answer

To solve the expression \((675 \times 675 \times 675 + 325 \times 325 \times 325)/(67.5 \times 67.5 + 32.5 \times 32.5 - 67.5 \times 32.5)\), we can follow these steps:

### Step 1: Recognize the cubes in the numerator
The numerator can be rewritten as:
\[ 675^3 + 325^3 \]

### Step 2: Recognize the identity for cubes
We can use the algebraic identity:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
Here, let \(A = 675\) and \(B = 325\). Thus:
\[ 675^3 + 325^3 = (675 + 325)(675^2 - 675 \times 325 + 325^2) \]

### Step 3: Calculate \(A + B\)
Calculating \(675 + 325\):
\[ 675 + 325 = 1000 \]

### Step 4: Calculate \(A^2 - AB + B^2\)
Now we need to calculate \(675^2 - 675 \times 325 + 325^2\):
- Calculate \(675^2\):
\[ 675^2 = 455625 \]
- Calculate \(325^2\):
\[ 325^2 = 105625 \]
- Calculate \(675 \times 325\):
\[ 675 \times 325 = 219375 \]
Now substituting these values:
\[ 675^2 - 675 \times 325 + 325^2 = 455625 - 219375 + 105625 \]
Calculating this gives:
\[ 455625 - 219375 + 105625 = 341875 \]

### Step 5: Substitute back into the numerator
So, the numerator becomes:
\[ 1000 \times 341875 \]

### Step 6: Simplify the denominator
Now, let's simplify the denominator:
\[ 67.5^2 + 32.5^2 - 67.5 \times 32.5 \]
Calculating each term:
- Calculate \(67.5^2\):
\[ 67.5^2 = 4556.25 \]
- Calculate \(32.5^2\):
\[ 32.5^2 = 1056.25 \]
- Calculate \(67.5 \times 32.5\):
\[ 67.5 \times 32.5 = 2193.75 \]
Now substituting these values:
\[ 67.5^2 + 32.5^2 - 67.5 \times 32.5 = 4556.25 + 1056.25 - 2193.75 \]
Calculating this gives:
\[ 4556.25 + 1056.25 - 2193.75 = 3419.75 \]

### Step 7: Final expression
Now we can write the entire expression:
\[ \frac{1000 \times 341875}{3419.75} \]

### Step 8: Simplify the fraction
To simplify:
\[ \frac{1000 \times 341875}{3419.75} = \frac{341875000}{3419.75} \]
Calculating this gives:
\[ 100000 \]

### Conclusion
Thus, the final answer is:
\[ \boxed{100000} \]