Divide the reciprocal of the difference of the numbers 2/7 and 4/5 by 1 2/7 times.

Solution:

1. Calculate the difference between \frac{2}{7} and \frac{4}{5}:
* Convert to common denominator: 35
* \frac{2}{7} = \frac{10}{35}
* \frac{4}{5} = \frac{28}{35}
* Difference: \frac{10}{35} - \frac{28}{35} = \frac{-18}{35}

2. Find the reciprocal of the difference:
* Reciprocal of \frac{-18}{35} is \frac{35}{-18}

3. Convert 1 \frac{2}{7} to improper fraction:
* 1 \frac{2}{7} = \frac{9}{7}

4. Divide the reciprocal by 1 \frac{2}{7}:
* \frac{35}{-18} \div \frac{9}{7} = \frac{35}{-18} \times \frac{7}{9}

5. Simplify:
* = \frac{35 \times 7}{-18 \times 9} = \frac{245}{-162}

6. Simplify further by dividing by the GCD of 245 and 162:
* GCD = 1
* = \frac{245}{-162}

7. Simplified, the expression becomes:
* \frac{245}{-162} = -\frac{245}{162}

8. The result is:
* -\frac{245}{162}