We need to solve the following expression: $2 \frac{6}{7} \times (\frac{3}{5} - \frac{1}{2}) + 2 \frac{3}{10} - \frac{4}{5}$

1. Problem Description

We need to solve the following expression:

$2 \frac{6}{7} \times (\frac{3}{5} - \frac{1}{2}) + 2 \frac{3}{10} - \frac{4}{5}$

2. Solution Steps

First, convert the mixed numbers to improper fractions:

$2 \frac{6}{7} = \frac{2 \times 7 + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7}$

$2 \frac{3}{10} = \frac{2 \times 10 + 3}{10} = \frac{20 + 3}{10} = \frac{23}{10}$

Now, substitute these into the expression:

$\frac{20}{7} \times (\frac{3}{5} - \frac{1}{2}) + \frac{23}{10} - \frac{4}{5}$

Next, we need to evaluate the expression in parentheses. The least common denominator of 5 and 2 is

1

0. $\frac{3}{5} - \frac{1}{2} = \frac{3 \times 2}{5 \times 2} - \frac{1 \times 5}{2 \times 5} = \frac{6}{10} - \frac{5}{10} = \frac{6-5}{10} = \frac{1}{10}$

Substitute this result back into the expression:

$\frac{20}{7} \times \frac{1}{10} + \frac{23}{10} - \frac{4}{5}$

Multiply the first two fractions:

$\frac{20}{7} \times \frac{1}{10} = \frac{20 \times 1}{7 \times 10} = \frac{20}{70} = \frac{2}{7}$

Substitute this back into the expression:

$\frac{2}{7} + \frac{23}{10} - \frac{4}{5}$

Now we need to find a common denominator for the three fractions. The least common denominator of 7, 10, and 5 is

7

0. $\frac{2}{7} = \frac{2 \times 10}{7 \times 10} = \frac{20}{70}$

$\frac{23}{10} = \frac{23 \times 7}{10 \times 7} = \frac{161}{70}$

$\frac{4}{5} = \frac{4 \times 14}{5 \times 14} = \frac{56}{70}$

Now substitute these into the expression:

$\frac{20}{70} + \frac{161}{70} - \frac{56}{70} = \frac{20 + 161 - 56}{70} = \frac{181 - 56}{70} = \frac{125}{70}$

Simplify the fraction:

$\frac{125}{70} = \frac{5 \times 25}{5 \times 14} = \frac{25}{14}$

Convert the improper fraction to a mixed number:

$\frac{25}{14} = 1 \frac{11}{14}$

3. Final Answer

$1 \frac{11}{14}$