Worksheet for ordering fractions in increasing order.

To solve the problem of ordering fractions in increasing order, we need to compare each set of fractions and arrange them from smallest to largest. Here's how we can approach each set step by step:

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General Steps:

1. Find a Common Denominator: Convert all fractions in a set to have the same denominator so they can be easily compared.
2. Compare Numerators: Once the denominators are the same, compare the numerators to determine the order.
3. Write the Fractions in Order: Arrange the fractions from smallest to largest based on the comparison.

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Solutions for Each Set:

#### 1) $\frac{3}{4}, \frac{1}{6}, \frac{5}{12}, \frac{4}{9}$
- Common Denominator: The least common multiple (LCM) of 4, 6, 12, and 9 is 36.
- Convert Fractions:
- $\frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36}$
- $\frac{1}{6} = \frac{1 \times 6}{6 \times 6} = \frac{6}{36}$
- $\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}$
- $\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$
- Order by Numerator: $\frac{6}{36}, \frac{15}{36}, \frac{16}{36}, \frac{27}{36}$
- Final Order: $\frac{1}{6}, \frac{5}{12}, \frac{4}{9}, \frac{3}{4}$

#### 2) $\frac{4}{8}, \frac{7}{16}, \frac{5}{12}, \frac{1}{8}, \frac{3}{12}$
- Common Denominator: The LCM of 8, 16, 12 is 48.
- Convert Fractions:
- $\frac{4}{8} = \frac{4 \times 6}{8 \times 6} = \frac{24}{48}$
- $\frac{7}{16} = \frac{7 \times 3}{16 \times 3} = \frac{21}{48}$
- $\frac{5}{12} = \frac{5 \times 4}{12 \times 4} = \frac{20}{48}$
- $\frac{1}{8} = \frac{1 \times 6}{8 \times 6} = \frac{6}{48}$
- $\frac{3}{12} = \frac{3 \times 4}{12 \times 4} = \frac{12}{48}$
- Order by Numerator: $\frac{6}{48}, \frac{12}{48}, \frac{20}{48}, \frac{21}{48}, \frac{24}{48}$
- Final Order: $\frac{1}{8}, \frac{3}{12}, \frac{5}{12}, \frac{7}{16}, \frac{4}{8}$

#### 3) $\frac{3}{8}, \frac{1}{4}, \frac{4}{10}, \frac{3}{6}, \frac{5}{8}$
- Common Denominator: The LCM of 8, 4, 10, 6 is 120.
- Convert Fractions:
- $\frac{3}{8} = \frac{3 \times 15}{8 \times 15} = \frac{45}{120}$
- $\frac{1}{4} = \frac{1 \times 30}{4 \times 30} = \frac{30}{120}$
- $\frac{4}{10} = \frac{4 \times 12}{10 \times 12} = \frac{48}{120}$
- $\frac{3}{6} = \frac{3 \times 20}{6 \times 20} = \frac{60}{120}$
- $\frac{5}{8} = \frac{5 \times 15}{8 \times 15} = \frac{75}{120}$
- Order by Numerator: $\frac{30}{120}, \frac{45}{120}, \frac{48}{120}, \frac{60}{120}, \frac{75}{120}$
- Final Order: $\frac{1}{4}, \frac{3}{8}, \frac{4}{10}, \frac{3}{6}, \frac{5}{8}$

#### 4) $\frac{2}{5}, \frac{9}{15}, \frac{8}{20}, \frac{1}{5}$
- Common Denominator: The LCM of 5, 15, 20 is 60.
- Convert Fractions:
- $\frac{2}{5} = \frac{2 \times 12}{5 \times 12} = \frac{24}{60}$
- $\frac{9}{15} = \frac{9 \times 4}{15 \times 4} = \frac{36}{60}$
- $\frac{8}{20} = \frac{8 \times 3}{20 \times 3} = \frac{24}{60}$
- $\frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60}$
- Order by Numerator: $\frac{12}{60}, \frac{24}{60}, \frac{24}{60}, \frac{36}{60}$
- Final Order: $\frac{1}{5}, \frac{2}{5}, \frac{8}{20}, \frac{9}{15}$

#### 5) $\frac{4}{9}, \frac{9}{12}, \frac{5}{18}, \frac{3}{18}, \frac{1}{3}$
- Common Denominator: The LCM of 9, 12, 18, 3 is 36.
- Convert Fractions:
- $\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$
- $\frac{9}{12} = \frac{9 \times 3}{12 \times 3} = \frac{27}{36}$
- $\frac{5}{18} = \frac{5 \times 2}{18 \times 2} = \frac{10}{36}$
- $\frac{3}{18} = \frac{3 \times 2}{18 \times 2} = \frac{6}{36}$
- $\frac{1}{3} = \frac{1 \times 12}{3 \times 12} = \frac{12}{36}$
- Order by Numerator: $\frac{6}{36}, \frac{10}{36}, \frac{12}{36}, \frac{16}{36}, \frac{27}{36}$
- Final Order: $\frac{3}{18}, \frac{5}{18}, \frac{1}{3}, \frac{4}{9}, \frac{9}{12}$

#### 6) $\frac{5}{8}, \frac{7}{16}, \frac{3}{4}, \frac{1}{4}, \frac{12}{24}$
- Common Denominator: The LCM of 8, 16, 4, 24 is 48.
- Convert Fractions:
- $\frac{5}{8} = \frac{5 \times 6}{8 \times 6} = \frac{30}{48}$
- $\frac{7}{16} = \frac{7 \times 3}{16 \times 3} = \frac{21}{48}$
- $\frac{3}{4} = \frac{3 \times 12}{4 \times 12} = \frac{36}{48}$
- $\frac{1}{4} = \frac{1 \times 12}{4 \times 12} = \frac{12}{48}$
- $\frac{12}{24} = \frac{12 \times 2}{24 \times 2} = \frac{24}{48}$
- Order by Numerator: $\frac{12}{48}, \frac{21}{48}, \frac{24}{48}, \frac{30}{48}, \frac{36}{48}$
- Final Order: $\frac{1}{4}, \frac{7}{16}, \frac{12}{24}, \frac{5}{8}, \frac{3}{4}$

#### 7) $\frac{7}{10}, \frac{2}{5}, \frac{9}{20}, \frac{5}{10}, \frac{10}{20}$
- Common Denominator: The LCM of 10, 5, 20 is 20.
- Convert Fractions:
- $\frac{7}{10} = \frac{7 \times 2}{10 \times 2} = \frac{14}{20}$
- $\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$
- $\frac{9}{20} = \frac{9}{20}$
- $\frac{5}{10} = \frac{5 \times 2}{10 \times 2} = \frac{10}{20}$
- $\frac{10}{20} = \frac{10}{20}$
- Order by Numerator: $\frac{8}{20}, \frac{9}{20}, \frac{10}{20}, \frac{10}{20}, \frac{14}{20}$
- Final Order: $\frac{2}{5}, \frac{9}{20}, \frac{5}{10}, \frac{10}{20}, \frac{7}{10}$

#### 8) $\frac{9}{15}, \frac{5}{12}, \frac{7}{18}, \frac{4}{9}, \frac{1}{6}$
- Common Denominator: The LCM of 15, 12, 18, 9, 6 is 180.
- Convert Fractions:
- $\frac{9}{15} = \frac{9 \times 12}{15 \times 12} = \frac{108}{180}$
- $\frac{5}{12} = \frac{5 \times 15}{12 \times 15} = \frac{75}{180}$
- $\frac{7}{18} = \frac{7 \times 10}{18 \times 10} = \frac{70}{180}$
- $\frac{4}{9} = \frac{4 \times 20}{9 \times 20} = \frac{80}{180}$
- $\frac{1}{6} = \frac{1 \times 30}{6 \times 30} = \frac{30}{180}$
- Order by Numerator: $\frac{30}{180}, \frac{70}{180}, \frac{75}{180}, \frac{80}{180}, \frac{108}{180}$
- Final Order: $\frac{1}{6}, \frac{7}{18}, \frac{5}{12}, \frac{4}{9}, \frac{9}{15}$

#### 9) $\frac{4}{9}, \frac{2}{6}, \frac{15}{21}, \frac{7}{12}, \frac{9}{18}$
- Common Denominator: The LCM of 9, 6, 21, 12, 18 is 252.
- Convert Fractions:
- $\frac{4}{9} = \frac{4 \times 28}{9 \times 28} = \frac{112}{252}$
- $\frac{2}{6} = \frac{2 \times 42}{6 \times 42} = \frac{84}{252}$
- $\frac{15}{21} = \frac{15 \times 12}{21 \times 12} = \frac{180}{252}$
- $\frac{7}{12} = \frac{7 \times 21}{12 \times 21} = \frac{147}{252}$
- $\frac{9}{18} = \frac{9 \times 14}{18 \times 14} = \frac{126}{252}$
- Order by Numerator: $\frac{84}{252}, \frac{112}{252}, \frac{126}{252}, \frac{147}{252}, \frac{180}{252}$
- Final Order: $\frac{2}{6}, \frac{4}{9}, \frac{9}{18}, \frac{7}{12}, \frac{15}{21}$

#### 10) $\frac{1}{25}, \frac{7}{15}, \frac{9}{25}, \frac{3}{10}, \frac{2}{5}$
- Common Denominator: The LCM of 25, 15, 10, 5 is 150.
- Convert Fractions:
- $\frac{1}{25} = \frac{1 \times 6}{25 \times 6} = \frac{6}{150}$
- $\frac{7}{15} = \frac{7 \times 10}{15 \times 10} = \frac{70}{150}$
- $\frac{9}{25} = \frac{9 \times 6}{25 \times 6} = \frac{54}{150}$
- $\frac{3}{10} = \frac{3 \times 15}{10 \times 15} = \frac{45}{150}$
- $\frac{2}{5} = \frac{2 \times 30}{5 \times 30} = \frac{60}{150}$
- Order by Numerator: $\frac{6}{150}, \frac{45}{150}, \frac{54}{150}, \frac{60}{150}, \frac{70}{150}$
- Final Order: $\frac{1}{25}, \frac{3}{10}, \frac{9}{25}, \frac{2}{5}, \frac{7}{15}$

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Final Answer:

\[
\boxed{
\begin{array}{ll}
1) & \frac{1}{6}, \frac{5}{12}, \frac{4}{9}, \frac{3}{4} \\
2) & \frac{1}{8}, \frac{3}{12}, \frac{5}{12}, \frac{7}{16}, \frac{4}{8} \\
3) & \frac{1}{4}, \frac{3}{8}, \frac{4}{10}, \frac{3}{6}, \frac{5}{8} \\
4) & \frac{1}{5}, \frac{2}{5}, \frac{8}{20}, \frac{9}{15} \\
5) & \frac{3}{18}, \frac{5}{18}, \frac{1}{3}, \frac{4}{9}, \frac{9}{12} \\
6) & \frac{1}{4}, \frac{7}{16}, \frac{12}{24}, \frac{5}{8}, \frac{3}{4} \\
7) & \frac{2}{5}, \frac{9}{20}, \frac{5}{10}, \frac{10}{20}, \frac{7}{10} \\
8) & \frac{1}{6}, \frac{7}{18}, \frac{5}{12}, \frac{4}{9}, \frac{9}{15} \\
9) & \frac{2}{6}, \frac{4}{9}, \frac{9}{18}, \frac{7}{12}, \frac{15}{21} \\
10) & \frac{1}{25}, \frac{3}{10}, \frac{9}{25}, \frac{2}{5}, \frac{7}{15}
\end{array}
}
\]