1. **State the problem:** Multiply the whole number 6 by the mixed number $6 \frac{5}{7}$. 2. **Convert the mixed number to an improper fraction:** A mixed number $a \frac{b}{c}$ can be converted to an improper fraction using the formula: $$a \frac{b}{c} = \frac{ac + b}{c}$$ For $6 \frac{5}{7}$: $$6 \frac{5}{7} = \frac{6 \times 7 + 5}{7} = \frac{42 + 5}{7} = \frac{47}{7}$$ 3. **Rewrite the multiplication:** $$6 \times 6 \frac{5}{7} = 6 \times \frac{47}{7}$$ 4. **Express the whole number as a fraction:** $$6 = \frac{6}{1}$$ So the multiplication becomes: $$\frac{6}{1} \times \frac{47}{7}$$ 5. **Multiply the numerators and denominators:** $$\frac{6 \times 47}{1 \times 7} = \frac{282}{7}$$ 6. **Convert the improper fraction back to a mixed number:** Divide 282 by 7: $$282 \div 7 = 40 \text{ remainder } 2$$ So: $$\frac{282}{7} = 40 \frac{2}{7}$$ **Final answer:** $$6 \times 6 \frac{5}{7} = 40 \frac{2}{7}$$