1. **State the problem:** Calculate the value of the expression $$\frac{13}{25} + \frac{7}{70} + \frac{2}{5} \times \frac{8}{70}$$. 2. **Recall the order of operations:** Multiplication must be done before addition. 3. **Calculate the multiplication part:** $$\frac{2}{5} \times \frac{8}{70} = \frac{2 \times 8}{5 \times 70} = \frac{16}{350}$$ 4. **Simplify the fraction $$\frac{16}{350}$$:** Divide numerator and denominator by 2: $$\frac{16}{350} = \frac{8}{175}$$ 5. **Rewrite the expression with the simplified multiplication result:** $$\frac{13}{25} + \frac{7}{70} + \frac{8}{175}$$ 6. **Find a common denominator for all fractions:** Denominators are 25, 70, and 175. Prime factors: - 25 = $5^2$ - 70 = $2 \times 5 \times 7$ - 175 = $5^2 \times 7$ The least common denominator (LCD) is $2 \times 5^2 \times 7 = 350$. 7. **Convert each fraction to have denominator 350:** $$\frac{13}{25} = \frac{13 \times 14}{25 \times 14} = \frac{182}{350}$$ $$\frac{7}{70} = \frac{7 \times 5}{70 \times 5} = \frac{35}{350}$$ $$\frac{8}{175} = \frac{8 \times 2}{175 \times 2} = \frac{16}{350}$$ 8. **Add the fractions:** $$\frac{182}{350} + \frac{35}{350} + \frac{16}{350} = \frac{182 + 35 + 16}{350} = \frac{233}{350}$$ 9. **Final answer:** $$\boxed{\frac{233}{350}}$$