Subtract mixed numbers with like denominators and regrouping.

To solve the given problems involving subtraction of mixed numbers with like denominators, we will follow these steps:

1. Separate the whole numbers and fractions.
2. Subtract the whole numbers.
3. Subtract the fractions.
4. If necessary, regroup (borrow) from the whole number part to make the subtraction of fractions possible.
5. Combine the results.

Let's solve each problem step by step.

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Problem 1: \( 5 \frac{3}{10} - 3 \frac{9}{10} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 5 - 3 \)
- Fractions: \( \frac{3}{10} - \frac{9}{10} \)

#### Step 2: Subtract the whole numbers
\[ 5 - 3 = 2 \]

#### Step 3: Subtract the fractions
\[ \frac{3}{10} - \frac{9}{10} \]
Since \( \frac{3}{10} \) is less than \( \frac{9}{10} \), we need to regroup:
- Borrow 1 from the whole number 2, which becomes 1.
- Add \( 1 \) (as \( \frac{10}{10} \)) to \( \frac{3}{10} \):
\[ 1 + \frac{3}{10} = 1 \frac{3}{10} = \frac{10}{10} + \frac{3}{10} = \frac{13}{10} \]
- Now subtract:
\[ \frac{13}{10} - \frac{9}{10} = \frac{4}{10} = \frac{2}{5} \]

#### Step 4: Combine the results
\[ 1 + \frac{2}{5} = 1 \frac{2}{5} \]

#### Final Answer:
\[ \boxed{1 \frac{2}{5}} \]

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Problem 2: \( 5 \frac{1}{2} - 1 \frac{1}{2} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 5 - 1 \)
- Fractions: \( \frac{1}{2} - \frac{1}{2} \)

#### Step 2: Subtract the whole numbers
\[ 5 - 1 = 4 \]

#### Step 3: Subtract the fractions
\[ \frac{1}{2} - \frac{1}{2} = 0 \]

#### Step 4: Combine the results
\[ 4 + 0 = 4 \]

#### Final Answer:
\[ \boxed{4} \]

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Problem 3: \( 6 \frac{2}{9} - 4 \frac{6}{9} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 6 - 4 \)
- Fractions: \( \frac{2}{9} - \frac{6}{9} \)

#### Step 2: Subtract the whole numbers
\[ 6 - 4 = 2 \]

#### Step 3: Subtract the fractions
\[ \frac{2}{9} - \frac{6}{9} \]
Since \( \frac{2}{9} \) is less than \( \frac{6}{9} \), we need to regroup:
- Borrow 1 from the whole number 2, which becomes 1.
- Add \( 1 \) (as \( \frac{9}{9} \)) to \( \frac{2}{9} \):
\[ 1 + \frac{2}{9} = 1 \frac{2}{9} = \frac{9}{9} + \frac{2}{9} = \frac{11}{9} \]
- Now subtract:
\[ \frac{11}{9} - \frac{6}{9} = \frac{5}{9} \]

#### Step 4: Combine the results
\[ 1 + \frac{5}{9} = 1 \frac{5}{9} \]

#### Final Answer:
\[ \boxed{1 \frac{5}{9}} \]

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Problem 4: \( 8 \frac{1}{3} - 2 \frac{2}{3} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 8 - 2 \)
- Fractions: \( \frac{1}{3} - \frac{2}{3} \)

#### Step 2: Subtract the whole numbers
\[ 8 - 2 = 6 \]

#### Step 3: Subtract the fractions
\[ \frac{1}{3} - \frac{2}{3} \]
Since \( \frac{1}{3} \) is less than \( \frac{2}{3} \), we need to regroup:
- Borrow 1 from the whole number 6, which becomes 5.
- Add \( 1 \) (as \( \frac{3}{3} \)) to \( \frac{1}{3} \):
\[ 1 + \frac{1}{3} = 1 \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]
- Now subtract:
\[ \frac{4}{3} - \frac{2}{3} = \frac{2}{3} \]

#### Step 4: Combine the results
\[ 5 + \frac{2}{3} = 5 \frac{2}{3} \]

#### Final Answer:
\[ \boxed{5 \frac{2}{3}} \]

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Problem 5: \( 8 \frac{2}{7} - 2 \frac{6}{7} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 8 - 2 \)
- Fractions: \( \frac{2}{7} - \frac{6}{7} \)

#### Step 2: Subtract the whole numbers
\[ 8 - 2 = 6 \]

#### Step 3: Subtract the fractions
\[ \frac{2}{7} - \frac{6}{7} \]
Since \( \frac{2}{7} \) is less than \( \frac{6}{7} \), we need to regroup:
- Borrow 1 from the whole number 6, which becomes 5.
- Add \( 1 \) (as \( \frac{7}{7} \)) to \( \frac{2}{7} \):
\[ 1 + \frac{2}{7} = 1 \frac{2}{7} = \frac{7}{7} + \frac{2}{7} = \frac{9}{7} \]
- Now subtract:
\[ \frac{9}{7} - \frac{6}{7} = \frac{3}{7} \]

#### Step 4: Combine the results
\[ 5 + \frac{3}{7} = 5 \frac{3}{7} \]

#### Final Answer:
\[ \boxed{5 \frac{3}{7}} \]

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Problem 6: \( 9 \frac{5}{12} - 4 \frac{8}{12} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 9 - 4 \)
- Fractions: \( \frac{5}{12} - \frac{8}{12} \)

#### Step 2: Subtract the whole numbers
\[ 9 - 4 = 5 \]

#### Step 3: Subtract the fractions
\[ \frac{5}{12} - \frac{8}{12} \]
Since \( \frac{5}{12} \) is less than \( \frac{8}{12} \), we need to regroup:
- Borrow 1 from the whole number 5, which becomes 4.
- Add \( 1 \) (as \( \frac{12}{12} \)) to \( \frac{5}{12} \):
\[ 1 + \frac{5}{12} = 1 \frac{5}{12} = \frac{12}{12} + \frac{5}{12} = \frac{17}{12} \]
- Now subtract:
\[ \frac{17}{12} - \frac{8}{12} = \frac{9}{12} = \frac{3}{4} \]

#### Step 4: Combine the results
\[ 4 + \frac{3}{4} = 4 \frac{3}{4} \]

#### Final Answer:
\[ \boxed{4 \frac{3}{4}} \]

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Problem 7: \( 7 \frac{6}{12} - 1 \frac{11}{12} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 7 - 1 \)
- Fractions: \( \frac{6}{12} - \frac{11}{12} \)

#### Step 2: Subtract the whole numbers
\[ 7 - 1 = 6 \]

#### Step 3: Subtract the fractions
\[ \frac{6}{12} - \frac{11}{12} \]
Since \( \frac{6}{12} \) is less than \( \frac{11}{12} \), we need to regroup:
- Borrow 1 from the whole number 6, which becomes 5.
- Add \( 1 \) (as \( \frac{12}{12} \)) to \( \frac{6}{12} \):
\[ 1 + \frac{6}{12} = 1 \frac{6}{12} = \frac{12}{12} + \frac{6}{12} = \frac{18}{12} \]
- Now subtract:
\[ \frac{18}{12} - \frac{11}{12} = \frac{7}{12} \]

#### Step 4: Combine the results
\[ 5 + \frac{7}{12} = 5 \frac{7}{12} \]

#### Final Answer:
\[ \boxed{5 \frac{7}{12}} \]

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Problem 8: \( 9 \frac{1}{9} - 1 \frac{2}{9} \)

#### Step 1: Separate the whole numbers and fractions
- Whole numbers: \( 9 - 1 \)
- Fractions: \( \frac{1}{9} - \frac{2}{9} \)

#### Step 2: Subtract the whole numbers
\[ 9 - 1 = 8 \]

#### Step 3: Subtract the fractions
\[ \frac{1}{9} - \frac{2}{9} \]
Since \( \frac{1}{9} \) is less than \( \frac{2}{9} \), we need to regroup:
- Borrow 1 from the whole number 8, which becomes 7.
- Add \( 1 \) (as \( \frac{9}{9} \)) to \( \frac{1}{9} \):
\[ 1 + \frac{1}{9} = 1 \frac{1}{9} = \frac{9}{9} + \frac{1}{9} = \frac{10}{9} \]
- Now subtract:
\[ \frac{10}{9} - \frac{2}{9} = \frac{8}{9} \]

#### Step 4: Combine the results
\[ 7 + \frac{8}{9} = 7 \frac{8}{9} \]

#### Final Answer:
\[ \boxed{7 \frac{8}{9}} \]

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Final Answers:

1. \( \boxed{1 \frac{2}{5}} \)
2. \( \boxed{4} \)
3. \( \boxed{1 \frac{5}{9}} \)
4. \( \boxed{5 \frac{2}{3}} \)
5. \( \boxed{5 \frac{3}{7}} \)
6. \( \boxed{4 \frac{3}{4}} \)
7. \( \boxed{5 \frac{7}{12}} \)
8. \( \boxed{7 \frac{8}{9}} \)