(x+2 3/7):3 2/7=4/5

To solve the equation \((x + 2 \frac{3}{7}) : 3 \frac{2}{7} = \frac{4}{5}\), we first need to convert the mixed numbers into improper fractions.

1. Convert \(2 \frac{3}{7}\) to an improper fraction: \[ 2 \frac{3}{7} = \frac{2 \times 7 + 3}{7} = \frac{14 + 3}{7} = \frac{17}{7} \]

2. Convert \(3 \frac{2}{7}\) to an improper fraction: \[ 3 \frac{2}{7} = \frac{3 \times 7 + 2}{7} = \frac{21 + 2}{7} = \frac{23}{7} \]

Now we can rewrite the original equation: \[ (x + \frac{17}{7}) : \frac{23}{7} = \frac{4}{5} \]

The division of fractions can be expressed as: \[ \frac{x + \frac{17}{7}}{\frac{23}{7}} = \frac{4}{5} \]

To eliminate the fraction in the denominator, we can multiply both sides by \(\frac{23}{7}\): \[ x + \frac{17}{7} = \frac{4}{5} \times \frac{23}{7} \]

Now, calculate \(\frac{4}{5} \times \frac{23}{7}\): \[ \frac{4 \times 23}{5 \times 7} = \frac{92}{35} \]

So now our equation is: \[ x + \frac{17}{7} = \frac{92}{35} \]

Next, we need to get a common denominator to combine the fractions. The least common denominator of 7 and 35 is 35. We convert \(\frac{17}{7}\) into a fraction with a denominator of 35: \[ \frac{17}{7} = \frac{17 \times 5}{7 \times 5} = \frac{85}{35} \]

Now we substitute this back into the equation: \[ x + \frac{85}{35} = \frac{92}{35} \]

To isolate \(x\), subtract \(\frac{85}{35}\) from both sides: \[ x = \frac{92}{35} - \frac{85}{35} = \frac{92 - 85}{35} = \frac{7}{35} \]

Simplifying \(\frac{7}{35}\) gives us: \[ x = \frac{1}{5} \]

Thus, the solution is: \[ \boxed{\frac{1}{5}} \]