Evaluate the expression shown below and write your answer **as a fraction** in simplest form.

0, point, 1, divided by, start fraction, 5, divided by, 28, end fraction, dot, start fraction, 3, divided by, 2, end fraction

Let's break down the given expression step-by-step.

The expression is:

\[
0.1 \div \left( \frac{5}{28} \cdot \frac{3}{2} \right)
\]

### Step 1: Convert the decimal to a fraction

First, let's convert \(0.1\) to a fraction:

\[
0.1 = \frac{1}{10}
\]

### Step 2: Evaluate the multiplication inside the parentheses

Next, we need to calculate the multiplication inside the parentheses:

\[
\frac{5}{28} \cdot \frac{3}{2} = \frac{5 \cdot 3}{28 \cdot 2} = \frac{15}{56}
\]

### Step 3: Divide by the resulting fraction

Now, we need to divide \(\frac{1}{10}\) by \(\frac{15}{56}\). To divide by a fraction, we multiply by its reciprocal:

\[
\frac{1}{10} \div \frac{15}{56} = \frac{1}{10} \cdot \frac{56}{15} = \frac{1 \cdot 56}{10 \cdot 15} = \frac{56}{150}
\]

### Step 4: Simplify the fraction

Now, we simplify \(\frac{56}{150}\) by finding the greatest common divisor (GCD) of 56 and 150. The GCD of 56 and 150 is 2, so we divide both the numerator and the denominator by 2:

\[
\frac{56}{150} = \frac{56 \div 2}{150 \div 2} = \frac{28}{75}
\]

### Final Answer

The expression simplifies to:

\[
\frac{28}{75}
\]

So, the answer is \(\frac{28}{75}\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the greatest common divisor (GCD) of two numbers?
2. What are the rules for dividing fractions?
3. How do you convert a repeating decimal to a fraction?
4. How do you simplify complex fractions?
5. What are some applications of dividing fractions in real life?

**Tip:** When dividing by a fraction, always remember to multiply by the reciprocal to simplify the calculation.

Learn how to evaluate the expression 0.1 divided by the product of fractions (5/28 * 3/2) and simplify the result to its simplest form (28/75). This problem covers fraction operations and converting decimals to fractions. Suitable for students in Grades 6-8.

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