Solve Complex Fraction Equation: 414÷(6/x) with Nested Operations

To solve this problem, we'll follow these detailed steps:

• Step 1: Simplify the first part $414 : \frac{6}{x}$.
  • Interpretation: This is equivalent to $414 \times \frac{x}{6}$.
  • Calculation: $414 \times \frac{x}{6} = 69x$.
• Step 2: Simplify the second part within the parentheses:
  • Start by simplifying $\frac{x}{y} : \frac{z}{x}$.
    • This is equivalent to $\frac{x}{y} \times \frac{x}{z} = \frac{x^2}{yz}$.
  • Next, simplify $\frac{x^2}{yz} - \frac{1}{z}$.
    • To subtract the fractions, use the common denominator $yz$.
    • $\frac{x^2}{yz} - \frac{1 \times y}{yz} = \frac{x^2 - y}{yz}$.
  • Finally, simplify the entire expression $\frac{x^2 - y}{yz} : (y \cdot \frac{1}{3})$.
    • $y \cdot \frac{1}{3} = \frac{y}{3}$.
    • The division of fractions becomes multiplication of reciprocals: $\frac{x^2 - y}{yz} \times \frac{3}{y}$.
    • Result: $\frac{3(x^2 - y)}{yz \cdot y} = \frac{3x^2 - 3}{yz^2}$.
• Step 3: Combine the results from Steps 1 and 2.
  • The expression becomes: $69x - \frac{3x^2 - 3}{yz^2} = 69x + \frac{3 - z^2}{yz}$.

Therefore, the solution to the problem is $69x+\frac{3-z^2}{yz}$.