1. **Problem 1:** Insert two rational numbers between (i) $\frac{2}{7}$ and $\frac{3}{4}$, (ii) $\frac{5}{6}$ and $\frac{3}{8}$. **Step 1:** Understand that rational numbers between two numbers can be found by averaging or finding numbers between their decimal or fractional values. **Step 2:** Convert fractions to decimals for easier insertion. (i) $\frac{2}{7} \approx 0.2857$, $\frac{3}{4} = 0.75$. (ii) $\frac{5}{6} \approx 0.8333$, $\frac{3}{8} = 0.375$. **Step 3:** For (i), find two numbers between 0.2857 and 0.75. For example, take averages: First number: $\frac{0.2857 + 0.75}{2} = 0.51785$ Second number: $\frac{0.2857 + 0.51785}{2} = 0.401775$ Convert back to fractions approximately: $0.401775 \approx \frac{2}{5}$, $0.51785 \approx \frac{1}{2}$. **Step 4:** For (ii), since $\frac{3}{8} < \frac{5}{6}$, reorder to find numbers between 0.375 and 0.8333. First number: $\frac{0.375 + 0.8333}{2} = 0.60415$ Second number: $\frac{0.375 + 0.60415}{2} = 0.489575$ Approximate fractions: $0.489575 \approx \frac{1}{2}$, $0.60415 \approx \frac{3}{5}$. **Answer 1:** (i) Two rational numbers between $\frac{2}{7}$ and $\frac{3}{4}$ are $\frac{2}{5}$ and $\frac{1}{2}$. (ii) Two rational numbers between $\frac{3}{8}$ and $\frac{5}{6}$ are $\frac{1}{2}$ and $\frac{3}{5}$. --- 4. **Problem 4:** Insert 6 rational numbers between 4.6 and 8.4. **Step 1:** Convert decimals to fractions or work with decimals directly. 4.6 and 8.4 are decimals; difference is $8.4 - 4.6 = 3.8$. **Step 2:** To insert 6 rational numbers, divide the interval into 7 equal parts: Step size $= \frac{3.8}{7} \approx 0.542857$. **Step 3:** Calculate the 6 numbers by adding multiples of step size to 4.6: 1st: $4.6 + 0.542857 = 5.142857$ 2nd: $4.6 + 2 \times 0.542857 = 5.685714$ 3rd: $4.6 + 3 \times 0.542857 = 6.228571$ 4th: $4.6 + 4 \times 0.542857 = 6.771428$ 5th: $4.6 + 5 \times 0.542857 = 7.314285$ 6th: $4.6 + 6 \times 0.542857 = 7.857142$ **Step 4:** Express these as fractions if desired: $5.142857 = \frac{36}{7}$, $5.685714 = \frac{40}{7}$, $6.228571 = \frac{44}{7}$, $6.771428 = \frac{47}{7}$, $7.314285 = \frac{51}{7}$, $7.857142 = \frac{55}{7}$. **Answer 4:** Six rational numbers between 4.6 and 8.4 are $\frac{36}{7}, \frac{40}{7}, \frac{44}{7}, \frac{47}{7}, \frac{51}{7}, \frac{55}{7}$.