Solve Mixed Number Expression: 10⅖ - 2⅗ + 7⅙

To solve the given problem $10\frac{2}{7} - 2\frac{3}{7} + 7\frac{1}{6}$, we'll proceed with the following steps:

• Step 1: Convert the mixed numbers to improper fractions.
  For $10\frac{2}{7}$, convert to $\frac{72}{7}$, since $10 \times 7 + 2 = 70 + 2 = 72$.
  For $2\frac{3}{7}$, convert to $\frac{17}{7}$, since $2 \times 7 + 3 = 14 + 3 = 17$.
  For $7\frac{1}{6}$, convert to $\frac{43}{6}$, since $7 \times 6 + 1 = 42 + 1 = 43$.
• Step 2: Find a common denominator.
  The denominators are 7 and 6. The least common denominator (LCD) is 42.
• Step 3: Convert fractions to the common denominator.
  Convert $\frac{72}{7}$ to $\frac{432}{42}$ (since $72 \times 6 = 432$).
  Convert $\frac{17}{7}$ to $\frac{102}{42}$ (since $17 \times 6 = 102$).
  Convert $\frac{43}{6}$ to $\frac{301}{42}$ (since $43 \times 7 = 301$).
• Step 4: Perform the operations.
  First, subtract: $\frac{432}{42} - \frac{102}{42} = \frac{330}{42}$.
  Then, add: $\frac{330}{42} + \frac{301}{42} = \frac{631}{42}$.
• Step 5: Simplify the result.
  Since 631 and 42 have no common factors other than 1, the fraction is already in its simplest form. Convert back to a mixed number: $631 \div 42 = 15$ remainder $1$, so $\frac{631}{42} = 15\frac{1}{42}$.

Therefore, the solution to the problem is $15\frac{1}{42}$.