Solve the Fraction Equation: Simplify 30x Divided by 4y/5z

To solve this problem, we'll simplify the expression step-by-step:

• Step 1: Simplify $30x : \frac{4y}{5z}$.
• Step 2: Simplify $z : \frac{9y}{10x}$.
• Step 3: Simplify $\frac{x}{y} : \frac{1}{z}$.
• Step 4: Combine the simplified forms according to the overall expression.

Let's work through these steps:

Step 1: Simplify $30x : \frac{4y}{5z}$. This is equivalent to $30x \times \frac{5z}{4y} = \frac{150xz}{4y} = \frac{75xz}{2y}$.

Step 2: Simplify $z : \frac{9y}{10x}$. This is equivalent to $z \times \frac{10x}{9y} = \frac{10xz}{9y}$.

Step 3: Simplify $\frac{x}{y} : \frac{1}{z}$. This is equivalent to $\frac{x}{y} \times z = \frac{xz}{y}$.

Step 4: Substitute these results back into the original expression:

We have $\frac{75xz}{2y} - \left( \frac{10xz}{9y} + \frac{xz}{y} \right)$.

Combine the terms in the parentheses:

$\frac{10xz}{9y} + \frac{xz}{y} = \frac{10xz}{9y} + \frac{9xz}{9y} = \frac{19xz}{9y}$.

Now, compute the final expression:

$\frac{75xz}{2y} - \frac{19xz}{9y} = \left(\frac{75 \times 9 - 19 \times 2}{18}\right)\frac{xz}{y} = \frac{675 - 38}{18} \frac{xz}{y} = \frac{637}{18} \frac{xz}{y} = 35\frac{7}{18} \frac{xz}{y}$.

Therefore, the solution to the problem is $35\frac{7}{18}\frac{xz}{y}$.