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The problem asks to calculate the exact value of the expression:

\[
\frac{4}{7} \div \frac{2}{3} \times \frac{5}{7}
\]

To solve this, let's break it down step by step:

### Step 1: Division of Fractions
When dividing fractions, multiply the first fraction by the reciprocal of the second fraction. So:

\[
\frac{4}{7} \div \frac{2}{3} = \frac{4}{7} \times \frac{3}{2} = \frac{4 \times 3}{7 \times 2} = \frac{12}{14}
\]

### Step 2: Simplify the Resulting Fraction
Simplify \(\frac{12}{14}\) by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 2:

\[
\frac{12}{14} = \frac{6}{7}
\]

### Step 3: Multiply the Result by \(\frac{5}{7}\)
Now multiply the result by \(\frac{5}{7}\):

\[
\frac{6}{7} \times \frac{5}{7} = \frac{6 \times 5}{7 \times 7} = \frac{30}{49}
\]

### Final Answer
The exact value of the expression is:

\[
\boxed{\frac{30}{49}}
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. What are the steps to multiply and divide fractions?
2. How do you find the reciprocal of a fraction?
3. How do you simplify fractions?
4. What is the significance of the GCD in fraction simplification?
5. Can you perform similar operations with mixed numbers?

### Tip:
Always remember to simplify fractions whenever possible to get the final answer in its simplest form.

Using a calculator, or otherwise, calculate the EXACT value of \(\frac{4}{7} \div \frac{2}{3} \times \frac{5}{7}\).

Learn how to calculate the exact value of the expression \(\frac{4}{7} \div \frac{2}{3} \times \frac{5}{7}\). This detailed solution covers key concepts such as the division and multiplication of fractions, as well as the simplification process. Suitable for students in Grades 5-7.

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