The given expression is:

$\frac{10 + 9 \cdot 8 - 7}{6 \cdot 5 - 4 - 3 + 2}$

Step-by-Step Solution:

Numerator: $10 + 9 \cdot 8 - 7$

1. Follow the order of operations (PEMDAS/BODMAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
2. First, perform multiplication: $9 \cdot 8 = 72$.
3. Substitute back: $10 + 72 - 7$.
4. Perform addition and subtraction from left to right:
   • $10 + 72 = 82$,
   • $82 - 7 = 75$.

Thus, the numerator is $75$.

Denominator: $6 \cdot 5 - 4 - 3 + 2$

1. Again, follow the order of operations.
2. First, perform multiplication: $6 \cdot 5 = 30$.
3. Substitute back: $30 - 4 - 3 + 2$.
4. Perform subtraction and addition from left to right:
   • $30 - 4 = 26$,
   • $26 - 3 = 23$,
   • $23 + 2 = 25$.

Thus, the denominator is $25$.

Final Step: Divide the Numerator by the Denominator

$\frac{75}{25} = 3$

Final Answer:

$3$

Would you like further explanation or details?

Related Questions:

1. How does the order of operations affect solving expressions like this?
2. What are other examples where a similar approach is required for fractions?
3. How can parentheses change the solution in such problems?
4. What are practical applications of such calculations in real life?
5. Can such expressions be represented graphically?

Tip: Always simplify each component of the expression before combining results for clarity.