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

Video Solution

Solution Steps

00:00 Solve

00:03 Division is also multiplication by the reciprocal

00:22 Move the product to the numerator

00:28 Use the distributive property and split 414 into 360 plus 54

00:33 Simplify what's possible

00:38 Division is also multiplication by the reciprocal

00:52 Multiply the outer factor by each term in parentheses

00:56 Negative times positive always equals negative

01:00 Negative times negative always equals positive

01:06 Multiply numerator by numerator and denominator by denominator

01:11 Convert fraction to number

01:14 Find common denominator

01:23 Combine like terms

01:31 And this is the solution to the question

Step-by-Step Solution

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 $69x+\frac{3-z^2}{yz}$ .