Fraction arithmetic practice problems on a worksheet.

Let's solve each of the nine fraction problems step by step, following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

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1. $\frac{1}{4} \times \frac{5}{6} - \frac{1}{6}$

Step 1: Multiply $\frac{1}{4} \times \frac{5}{6} = \frac{5}{24}$

Step 2: Subtract $\frac{1}{6}$:
Convert $\frac{1}{6}$ to 24ths: $\frac{1}{6} = \frac{4}{24}$

$\frac{5}{24} - \frac{4}{24} = \frac{1}{24}$

✔ Answer: $\boxed{\frac{1}{24}}$

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2. $\frac{3}{5} \div \frac{5}{6} + \frac{1}{5}$

Step 1: Divide $\frac{3}{5} \div \frac{5}{6} = \frac{3}{5} \times \frac{6}{5} = \frac{18}{25}$

Step 2: Add $\frac{1}{5}$:
Convert $\frac{1}{5} = \frac{5}{25}$

$\frac{18}{25} + \frac{5}{25} = \frac{23}{25}$

✔ Answer: $\boxed{\frac{23}{25}}$

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3. $\left(\frac{2}{5} + \frac{8}{9}\right) \times \frac{1}{2}$

Step 1: Add inside parentheses:
LCM of 5 and 9 is 45.

$\frac{2}{5} = \frac{18}{45},\quad \frac{8}{9} = \frac{40}{45}$

$\frac{18}{45} + \frac{40}{45} = \frac{58}{45}$

Step 2: Multiply by $\frac{1}{2}$:

$\frac{58}{45} \times \frac{1}{2} = \frac{58}{90} = \frac{29}{45}$ (simplify by dividing numerator and denominator by 2)

✔ Answer: $\boxed{\frac{29}{45}}$

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4. $\frac{7}{9} + \frac{4}{5} \times \frac{4}{9}$

Step 1: Multiply first: $\frac{4}{5} \times \frac{4}{9} = \frac{16}{45}$

Step 2: Add $\frac{7}{9}$: Convert to 45ths: $\frac{7}{9} = \frac{35}{45}$

$\frac{35}{45} + \frac{16}{45} = \frac{51}{45} = \frac{17}{15}$ (simplify by dividing by 3)

$\frac{17}{15} = 1\frac{2}{15}$

✔ Answer: $\boxed{\frac{17}{15}}$ or $1\frac{2}{15}$

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5. $\frac{4}{5} - \frac{3}{5} \div \frac{5}{6}$

Step 1: Division first: $\frac{3}{5} \div \frac{5}{6} = \frac{3}{5} \times \frac{6}{5} = \frac{18}{25}$

Step 2: Subtract from $\frac{4}{5}$: Convert $\frac{4}{5} = \frac{20}{25}$

$\frac{20}{25} - \frac{18}{25} = \frac{2}{25}$

✔ Answer: $\boxed{\frac{2}{25}}$

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6. $\frac{3}{5} \times \left(\frac{1}{2} + \frac{2}{9}\right)$

Step 1: Add inside parentheses: LCM of 2 and 9 is 18.

$\frac{1}{2} = \frac{9}{18},\quad \frac{2}{9} = \frac{4}{18}$

$\frac{9}{18} + \frac{4}{18} = \frac{13}{18}$

Step 2: Multiply: $\frac{3}{5} \times \frac{13}{18} = \frac{39}{90} = \frac{13}{30}$ (divide numerator and denominator by 3)

✔ Answer: $\boxed{\frac{13}{30}}$

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7. $\left(\frac{5}{6} + \frac{2}{5}\right) \div \frac{3}{4}$

Step 1: Add inside parentheses: LCM of 6 and 5 is 30.

$\frac{5}{6} = \frac{25}{30},\quad \frac{2}{5} = \frac{12}{30}$

$\frac{25}{30} + \frac{12}{30} = \frac{37}{30}$

Step 2: Divide by $\frac{3}{4}$:
$\frac{37}{30} \div \frac{3}{4} = \frac{37}{30} \times \frac{4}{3} = \frac{148}{90} = \frac{74}{45}$ (simplify by dividing by 2)

$\frac{74}{45} = 1\frac{29}{45}$

✔ Answer: $\boxed{\frac{74}{45}}$ or $1\frac{29}{45}$

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8. $\frac{5}{8} \times \frac{2}{9} + \frac{1}{2}$

Step 1: Multiply: $\frac{5}{8} \times \frac{2}{9} = \frac{10}{72} = \frac{5}{36}$

Step 2: Add $\frac{1}{2}$: Convert $\frac{1}{2} = \frac{18}{36}$

$\frac{5}{36} + \frac{18}{36} = \frac{23}{36}$

✔ Answer: $\boxed{\frac{23}{36}}$

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9. $\frac{3}{5} + \frac{3}{4} \div \frac{1}{6}$

Step 1: Division first: $\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = \frac{9}{2}$

Step 2: Add $\frac{3}{5}$: Convert both to common denominator (LCM of 5 and 2 is 10)

$\frac{3}{5} = \frac{6}{10},\quad \frac{9}{2} = \frac{45}{10}$

$\frac{6}{10} + \frac{45}{10} = \frac{51}{10} = 5\frac{1}{10}$

✔ Answer: $\boxed{\frac{51}{10}}$ or $5\frac{1}{10}$

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✔ Final Answers Summary:

1. $\frac{1}{24}$
2. $\frac{23}{25}$
3. $\frac{29}{45}$
4. $\frac{17}{15}$ or $1\frac{2}{15}$
5. $\frac{2}{25}$
6. $\frac{13}{30}$
7. $\frac{74}{45}$ or $1\frac{29}{45}$
8. $\frac{23}{36}$
9. $\frac{51}{10}$ or $5\frac{1}{10}$

Let me know if you'd like these explained in a visual or diagrammatic way!