Ordering Fractions And Mixed Numbers Worksheet - Free Printable

Let's solve each problem step by step, explaining the reasoning and methods used to order fractions and mixed numbers correctly.

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A) Arrange in Increasing Order

We will convert all numbers to improper fractions or decimals for easier comparison. We'll use common denominators where needed, or convert to decimals for clarity.

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#### 1) $\frac{10}{9}, \frac{4}{9}, 7\frac{1}{9}, \frac{13}{9}$

Convert all to improper fractions:
- $\frac{10}{9}$
- $\frac{4}{9}$
- $7\frac{1}{9} = \frac{64}{9}$
- $\frac{13}{9}$

Now compare numerators (same denominator):
- $\frac{4}{9} < \frac{10}{9} < \frac{13}{9} < \frac{64}{9}$

So increasing order:
$$
\boxed{\frac{4}{9},\ \frac{10}{9},\ \frac{13}{9},\ 7\frac{1}{9}}
$$

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#### 2) $2\frac{4}{5}, \frac{11}{4}, 2\frac{1}{3}, \frac{6}{7}$

Convert to improper fractions:
- $2\frac{4}{5} = \frac{14}{5}$
- $\frac{11}{4}$
- $2\frac{1}{3} = \frac{7}{3}$
- $\frac{6}{7}$

Find a common denominator or convert to decimals:

| Fraction | Decimal |
|--------|--------|
| $\frac{6}{7}$ | ≈ 0.857 |
| $\frac{14}{5} = 2.8$ | 2.8 |
| $\frac{7}{3} ≈ 2.333$ | 2.333 |
| $\frac{11}{4} = 2.75$ | 2.75 |

Order:
$\frac{6}{7} < 2\frac{1}{3} < \frac{11}{4} < 2\frac{4}{5}$

So increasing order:
$$
\boxed{\frac{6}{7},\ 2\frac{1}{3},\ \frac{11}{4},\ 2\frac{4}{5}}
$$

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#### 3) $\frac{1}{13}, 4\frac{1}{6}, 3\frac{1}{2}, \frac{12}{10}$

Convert:
- $\frac{1}{13} ≈ 0.077$
- $4\frac{1}{6} = \frac{25}{6} ≈ 4.166$
- $3\frac{1}{2} = 3.5$
- $\frac{12}{10} = 1.2$

Order:
$\frac{1}{13} < \frac{12}{10} < 3\frac{1}{2} < 4\frac{1}{6}$

So increasing order:
$$
\boxed{\frac{1}{13},\ \frac{12}{10},\ 3\frac{1}{2},\ 4\frac{1}{6}}
$$

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#### 4) $\frac{7}{8}, 11\frac{3}{8}, \frac{5}{8}, 12\frac{1}{8}$

All have denominator 8:
- $\frac{7}{8} = 0.875$
- $11\frac{3}{8} = \frac{91}{8} = 11.375$
- $\frac{5}{8} = 0.625$
- $12\frac{1}{8} = \frac{97}{8} = 12.125$

Order:
$\frac{5}{8} < \frac{7}{8} < 11\frac{3}{8} < 12\frac{1}{8}$

So increasing order:
$$
\boxed{\frac{5}{8},\ \frac{7}{8},\ 11\frac{3}{8},\ 12\frac{1}{8}}
$$

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#### 5) $\frac{9}{5}, 1\frac{1}{5}, \frac{1}{5}, 1\frac{3}{5}$

Convert:
- $\frac{9}{5} = 1.8$
- $1\frac{1}{5} = 1.2$
- $\frac{1}{5} = 0.2$
- $1\frac{3}{5} = 1.6$

Order:
$\frac{1}{5} < 1\frac{1}{5} < 1\frac{3}{5} < \frac{9}{5}$

So increasing order:
$$
\boxed{\frac{1}{5},\ 1\frac{1}{5},\ 1\frac{3}{5},\ \frac{9}{5}}
$$

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#### 6) $\frac{3}{7}, \frac{17}{3}, \frac{1}{2}, 5\frac{1}{4}$

Convert to decimals:
- $\frac{3}{7} ≈ 0.4286$
- $\frac{17}{3} ≈ 5.666$
- $\frac{1}{2} = 0.5$
- $5\frac{1}{4} = 5.25$

Order:
$\frac{3}{7} < \frac{1}{2} < 5\frac{1}{4} < \frac{17}{3}$

So increasing order:
$$
\boxed{\frac{3}{7},\ \frac{1}{2},\ 5\frac{1}{4},\ \frac{17}{3}}
$$

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B) Arrange in Decreasing Order

We now go from largest to smallest.

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#### 1) $\frac{4}{9}, 4\frac{3}{4}, \frac{8}{3}, 4\frac{1}{5}$

Convert:
- $\frac{4}{9} ≈ 0.444$
- $4\frac{3}{4} = 4.75$
- $\frac{8}{3} ≈ 2.666$
- $4\frac{1}{5} = 4.2$

Order from largest:
$4\frac{3}{4} > 4\frac{1}{5} > \frac{8}{3} > \frac{4}{9}$

So decreasing order:
$$
\boxed{4\frac{3}{4},\ 4\frac{1}{5},\ \frac{8}{3},\ \frac{4}{9}}
$$

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#### 2) $\frac{5}{6}, 10\frac{1}{6}, \frac{10}{6}, 9\frac{5}{6}$

Convert:
- $\frac{5}{6} ≈ 0.833$
- $10\frac{1}{6} ≈ 10.166$
- $\frac{10}{6} ≈ 1.666$
- $9\frac{5}{6} ≈ 9.833$

Order:
$10\frac{1}{6} > 9\frac{5}{6} > \frac{10}{6} > \frac{5}{6}$

So decreasing order:
$$
\boxed{10\frac{1}{6},\ 9\frac{5}{6},\ \frac{10}{6},\ \frac{5}{6}}
$$

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#### 3) $1\frac{2}{3}, \frac{7}{3}, \frac{2}{3}, 8\frac{1}{3}$

Convert:
- $1\frac{2}{3} = \frac{5}{3} ≈ 1.666$
- $\frac{7}{3} ≈ 2.333$
- $\frac{2}{3} ≈ 0.666$
- $8\frac{1}{3} = \frac{25}{3} ≈ 8.333$

Order:
$8\frac{1}{3} > \frac{7}{3} > 1\frac{2}{3} > \frac{2}{3}$

So decreasing order:
$$
\boxed{8\frac{1}{3},\ \frac{7}{3},\ 1\frac{2}{3},\ \frac{2}{3}}
$$

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#### 4) $\frac{15}{12}, \frac{14}{5}, \frac{11}{15}, 7\frac{3}{8}$

Simplify and convert:
- $\frac{15}{12} = \frac{5}{4} = 1.25$
- $\frac{14}{5} = 2.8$
- $\frac{11}{15} ≈ 0.733$
- $7\frac{3}{8} = 7.375$

Order:
$7\frac{3}{8} > \frac{14}{5} > \frac{15}{12} > \frac{11}{15}$

So decreasing order:
$$
\boxed{7\frac{3}{8},\ \frac{14}{5},\ \frac{15}{12},\ \frac{11}{15}}
$$

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#### 5) $4\frac{1}{6}, 1\frac{7}{9}, 5\frac{1}{5}, \frac{13}{7}$

Convert:
- $4\frac{1}{6} ≈ 4.166$
- $1\frac{7}{9} ≈ 1.777$
- $5\frac{1}{5} = 5.2$
- $\frac{13}{7} ≈ 1.857$

Order:
$5\frac{1}{5} > 4\frac{1}{6} > \frac{13}{7} > 1\frac{7}{9}$

So decreasing order:
$$
\boxed{5\frac{1}{5},\ 4\frac{1}{6},\ \frac{13}{7},\ 1\frac{7}{9}}
$$

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#### 6) $\frac{9}{10}, 3\frac{9}{10}, 5\frac{5}{10}, \frac{5}{10}$

Convert:
- $\frac{9}{10} = 0.9$
- $3\frac{9}{10} = 3.9$
- $5\frac{5}{10} = 5.5$
- $\frac{5}{10} = 0.5$

Order:
$5\frac{5}{10} > 3\frac{9}{10} > \frac{9}{10} > \frac{5}{10}$

So decreasing order:
$$
\boxed{5\frac{5}{10},\ 3\frac{9}{10},\ \frac{9}{10},\ \frac{5}{10}}
$$

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✔ Final Answers Summary:

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A) Increasing Order

1) $\frac{4}{9},\ \frac{10}{9},\ \frac{13}{9},\ 7\frac{1}{9}$
2) $\frac{6}{7},\ 2\frac{1}{3},\ \frac{11}{4},\ 2\frac{4}{5}$
3) $\frac{1}{13},\ \frac{12}{10},\ 3\frac{1}{2},\ 4\frac{1}{6}$
4) $\frac{5}{8},\ \frac{7}{8},\ 11\frac{3}{8},\ 12\frac{1}{8}$
5) $\frac{1}{5},\ 1\frac{1}{5},\ 1\frac{3}{5},\ \frac{9}{5}$
6) $\frac{3}{7},\ \frac{1}{2},\ 5\frac{1}{4},\ \frac{17}{3}$

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B) Decreasing Order

1) $4\frac{3}{4},\ 4\frac{1}{5},\ \frac{8}{3},\ \frac{4}{9}$
2) $10\frac{1}{6},\ 9\frac{5}{6},\ \frac{10}{6},\ \frac{5}{6}$
3) $8\frac{1}{3},\ \frac{7}{3},\ 1\frac{2}{3},\ \frac{2}{3}$
4) $7\frac{3}{8},\ \frac{14}{5},\ \frac{15}{12},\ \frac{11}{15}$
5) $5\frac{1}{5},\ 4\frac{1}{6},\ \frac{13}{7},\ 1\frac{7}{9}$
6) $5\frac{5}{10},\ 3\frac{9}{10},\ \frac{9}{10},\ \frac{5}{10}$

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✔ All answers are arranged correctly using decimal conversion or common denominators as needed.

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