1. **State the problem:** Calculate the value of the expression $\frac{2}{5} + \frac{7}{8} \times \frac{17}{19} \div 6 \div 5$ step by step. 2. **Recall the order of operations:** Multiplication and division are performed before addition, from left to right. 3. **Rewrite the expression:** $$\frac{2}{5} + \frac{7}{8} \times \frac{17}{19} \div 6 \div 5$$ 4. **Calculate the multiplication and divisions first:** - Multiply $\frac{7}{8}$ and $\frac{17}{19}$: $$\frac{7}{8} \times \frac{17}{19} = \frac{7 \times 17}{8 \times 19} = \frac{119}{152}$$ - Divide by 6: $$\frac{119}{152} \div 6 = \frac{119}{152} \times \frac{1}{6} = \frac{119}{912}$$ - Divide by 5: $$\frac{119}{912} \div 5 = \frac{119}{912} \times \frac{1}{5} = \frac{119}{4560}$$ 5. **Now add $\frac{2}{5}$ and $\frac{119}{4560}$:** - Find common denominator: $$5 = \frac{4560}{912} \Rightarrow \text{common denominator} = 4560$$ - Convert $\frac{2}{5}$ to denominator 4560: $$\frac{2}{5} = \frac{2 \times 912}{5 \times 912} = \frac{1824}{4560}$$ - Add the fractions: $$\frac{1824}{4560} + \frac{119}{4560} = \frac{1824 + 119}{4560} = \frac{1943}{4560}$$ 6. **Simplify the fraction if possible:** - Check for common factors between 1943 and 4560. Since 1943 is prime relative to 4560, fraction is in simplest form. **Final answer:** $$\frac{1943}{4560}$$