2/7*10 4/5:1/7:30:1 11/25*8:1/6

To simplify the given expression \( \frac{2}{7} \times 10 \, \frac{4}{5} : \frac{1}{7} : 30 : 1 \, \frac{11}{25} \times 8 : \frac{1}{6} \), let's break it down step by step.

First, convert the mixed numbers into improper fractions:

1. \( 10 \, \frac{4}{5} = 10 + \frac{4}{5} = \frac{50}{5} + \frac{4}{5} = \frac{54}{5} \)
2. \( 1 \, \frac{11}{25} = 1 + \frac{11}{25} = \frac{25}{25} + \frac{11}{25} = \frac{36}{25} \)

Now we can rewrite the full expression using improper fractions:

\[ \frac{2}{7} \times \frac{54}{5} : \frac{1}{7} : 30 : \frac{36}{25} \times 8 : \frac{1}{6} \]

Next, let's calculate each part individually.

Step 1: Calculate \( \frac{2}{7} \times \frac{54}{5} \)

\[ \frac{2 \times 54}{7 \times 5} = \frac{108}{35} \]

Step 2: The rest of the parts

Now let's deal with the division of parts marked by the ":"

The expression can be converted into multiplication by flipping the denominators:

\[ \frac{108}{35} \div \frac{1}{7} \div 30 \div \frac{36}{25} \div \frac{1}{6} \]

This is equivalent to:

\[ \frac{108}{35} \times 7 \times \frac{1}{30} \times \frac{25}{36} \times 6 \]

Step 3: Combine all the multiplications together

Now perform the multiplications step by step:

1. \( \frac{108}{35} \times 7 = \frac{756}{35} \)
2. \( \frac{756}{35} \div 30 = \frac{756}{35 \times 30} = \frac{756}{1050} \)
3. \( \frac{756}{1050} \times \frac{25}{36} = \frac{756 \times 25}{1050 \times 36} \)
4. \( \frac{756 \times 25}{1050 \times 36} \times 6 = \frac{756 \times 25 \times 6}{1050 \times 36} \)

Final Step: Calculate the final fraction

1. Calculate the numerator: \( 756 \times 25 \times 6 \)
2. Calculate the denominator: \( 1050 \times 36 \)

Now we can compute those products for the final answer:

• Numerator: \( 756 \times 25 = 18900 \) and \( 18900 \times 6 = 113400 \)
• Denominator: \( 1050 \times 36 = 37800 \)

Now we have:

\[ \frac{113400}{37800} \]

This fraction can be simplified by finding the GCD of 113400 and 37800, which gives us:

\[ \frac{113400 \div 37800}{37800 \div 37800} = \frac{3}{1} = 3 \]

Therefore, the simplified expression is:

\[ \boxed{3} \]