1. **State the problem:** Simplify the expression $\frac{3}{4} - \frac{1}{4} + \frac{5}{4} \times \frac{2}{5}$.\n\n2. **Recall the order of operations:** Multiplication comes before addition and subtraction. So first, calculate $\frac{5}{4} \times \frac{2}{5}$.\n\n3. **Multiply the fractions:** $$\frac{5}{4} \times \frac{2}{5} = \frac{5 \times 2}{4 \times 5} = \frac{10}{20}.$$\n\n4. **Simplify the fraction:** $$\frac{10}{20} = \frac{\cancel{10}}{\cancel{20}} = \frac{1}{2}.$$\n\n5. **Rewrite the expression:** Now the expression is $\frac{3}{4} - \frac{1}{4} + \frac{1}{2}$.\n\n6. **Combine the first two fractions:** $$\frac{3}{4} - \frac{1}{4} = \frac{3 - 1}{4} = \frac{2}{4} = \frac{1}{2}.$$\n\n7. **Add the remaining fractions:** $$\frac{1}{2} + \frac{1}{2} = \frac{1 + 1}{2} = \frac{2}{2} = 1.$$\n\n**Final answer:** $1$