1. **Problem statement:** Arrange the given rivet or drill sizes in order from smallest to largest. 2. **Key idea:** To compare fractions or mixed numbers, convert them to decimals or find a common denominator. 3. **Step-by-step for each problem:** **Problem 6:** Sizes: $\frac{3}{4}$, $\frac{7}{8}$, $\frac{13}{16}$, $\frac{49}{64}$, $\frac{27}{32}$ - Convert to decimals: - $\frac{3}{4} = 0.75$ - $\frac{7}{8} = 0.875$ - $\frac{13}{16} = 0.8125$ - $\frac{49}{64} = 0.765625$ - $\frac{27}{32} = 0.84375$ - Order from smallest to largest: $$0.75 < 0.765625 < 0.8125 < 0.84375 < 0.875$$ - So, order is $\frac{3}{4} < \frac{49}{64} < \frac{13}{16} < \frac{27}{32} < \frac{7}{8}$ **Problem 7:** Sizes: $\frac{5}{16}$, $\frac{3}{8}$, $\frac{9}{32}$, $\frac{21}{64}$, $\frac{1}{4}$ - Convert to decimals: - $\frac{5}{16} = 0.3125$ - $\frac{3}{8} = 0.375$ - $\frac{9}{32} = 0.28125$ - $\frac{21}{64} = 0.328125$ - $\frac{1}{4} = 0.25$ - Order: $$0.25 < 0.28125 < 0.3125 < 0.328125 < 0.375$$ - So, order is $\frac{1}{4} < \frac{9}{32} < \frac{5}{16} < \frac{21}{64} < \frac{3}{8}$ **Problem 8:** Sizes: $\frac{5}{64}$, $\frac{7}{8}$, $\frac{3}{4}$, $\frac{29}{32}$, $\frac{13}{16}$ - Convert to decimals: - $\frac{5}{64} = 0.078125$ - $\frac{7}{8} = 0.875$ - $\frac{3}{4} = 0.75$ - $\frac{29}{32} = 0.90625$ - $\frac{13}{16} = 0.8125$ - Order: $$0.078125 < 0.75 < 0.8125 < 0.875 < 0.90625$$ - So, order is $\frac{5}{64} < \frac{3}{4} < \frac{13}{16} < \frac{7}{8} < \frac{29}{32}$ **Problem 9:** Sizes: $\frac{45}{64}$, $\frac{5}{6}$, $\frac{3}{4}$, $\frac{11}{16}$, $\frac{23}{32}$ - Convert to decimals: - $\frac{45}{64} = 0.703125$ - $\frac{5}{6} \approx 0.8333$ - $\frac{3}{4} = 0.75$ - $\frac{11}{16} = 0.6875$ - $\frac{23}{32} = 0.71875$ - Order: $$0.6875 < 0.703125 < 0.71875 < 0.75 < 0.8333$$ - So, order is $\frac{11}{16} < \frac{45}{64} < \frac{23}{32} < \frac{3}{4} < \frac{5}{6}$ **Problem 10:** Sizes: $\frac{1}{4}$, $\frac{5}{16}$, $\frac{1}{16}$, $\frac{17}{64}$, $\frac{9}{2}$, $\frac{3}{4}$, $64$, $332$, $\frac{7}{8}$, $72$, $\frac{1}{2}$, $\frac{23}{64}$ - Convert to decimals: - $\frac{1}{4} = 0.25$ - $\frac{5}{16} = 0.3125$ - $\frac{1}{16} = 0.0625$ - $\frac{17}{64} = 0.265625$ - $\frac{9}{2} = 4.5$ - $\frac{3}{4} = 0.75$ - $64 = 64$ - $332 = 332$ - $\frac{7}{8} = 0.875$ - $72 = 72$ - $\frac{1}{2} = 0.5$ - $\frac{23}{64} = 0.359375$ - Order: $$0.0625 < 0.25 < 0.265625 < 0.3125 < 0.359375 < 0.5 < 0.75 < 0.875 < 4.5 < 64 < 72 < 332$$ - So, order is $\frac{1}{16} < \frac{1}{4} < \frac{17}{64} < \frac{5}{16} < \frac{23}{64} < \frac{1}{2} < \frac{3}{4} < \frac{7}{8} < \frac{9}{2} < 64 < 72 < 332$ **Final answers:** - 6: $\frac{3}{4} < \frac{49}{64} < \frac{13}{16} < \frac{27}{32} < \frac{7}{8}$ - 7: $\frac{1}{4} < \frac{9}{32} < \frac{5}{16} < \frac{21}{64} < \frac{3}{8}$ - 8: $\frac{5}{64} < \frac{3}{4} < \frac{13}{16} < \frac{7}{8} < \frac{29}{32}$ - 9: $\frac{11}{16} < \frac{45}{64} < \frac{23}{32} < \frac{3}{4} < \frac{5}{6}$ - 10: $\frac{1}{16} < \frac{1}{4} < \frac{17}{64} < \frac{5}{16} < \frac{23}{64} < \frac{1}{2} < \frac{3}{4} < \frac{7}{8} < \frac{9}{2} < 64 < 72 < 332$