\frac{\left(\frac{18}{5}-\frac{2}{38}\right)}{\frac{3}{28}\cdot \frac{18}{9}}

To simplify the expression:

\[
\frac{\left(\frac{18}{5}-\frac{2}{38}\right)}{\frac{3}{28}\cdot \frac{18}{9}}
\]

### Step 1: Simplify the numerator

First, simplify the subtraction:

\[
\frac{18}{5} - \frac{2}{38}
\]

Find a common denominator for \(\frac{18}{5}\) and \(\frac{2}{38}\). The least common denominator (LCD) of 5 and 38 is \(190\).

Convert each fraction:

\[
\frac{18}{5} = \frac{18 \times 38}{5 \times 38} = \frac{684}{190}, \quad \frac{2}{38} = \frac{2 \times 5}{38 \times 5} = \frac{10}{190}
\]

Now subtract the fractions:

\[
\frac{684}{190} - \frac{10}{190} = \frac{684 - 10}{190} = \frac{674}{190}
\]

### Step 2: Simplify the denominator

Now simplify the denominator:

\[
\frac{3}{28} \cdot \frac{18}{9}
\]

First, simplify \(\frac{18}{9}\):

\[
\frac{18}{9} = 2
\]

So the expression becomes:

\[
\frac{3}{28} \cdot 2 = \frac{6}{28} = \frac{3}{14}
\]

### Step 3: Combine the simplified numerator and denominator

Now the entire expression becomes:

\[
\frac{\frac{674}{190}}{\frac{3}{14}} = \frac{674}{190} \times \frac{14}{3} = \frac{674 \times 14}{190 \times 3} = \frac{9436}{570}
\]

### Step 4: Simplify the result

Now simplify \(\frac{9436}{570}\). The greatest common divisor (GCD) of 9436 and 570 is 2. So divide both the numerator and the denominator by 2:

\[
\frac{9436}{570} = \frac{9436 \div 2}{570 \div 2} = \frac{4718}{285}
\]

Thus, the simplified result is:

\[
\frac{4718}{285}
\]

Let me know if you'd like more details or explanations. Here are some related questions you might want to explore:

1. How do you find the least common denominator between fractions?
2. What are the steps for multiplying fractions?
3. How do you simplify complex fractions?
4. What is the method for reducing fractions to their simplest form?
5. How do you factor the greatest common divisor?

**Tip:** Always check for common factors when simplifying fractions to make calculations easier.

Learn how to simplify the expression \frac{\left(\frac{18}{5}-\frac{2}{38}\right)}{\frac{3}{28}\cdot \frac{18}{9}} using the common denominator method for subtraction and multiplying fractions. Follow a step-by-step guide to obtain the simplified result \frac{4718}{285}, suitable for students in Grades 8-10.

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