Adding and Subtracting Fractions with Three Terms (A) - Free Printable

To solve the problems involving adding and subtracting fractions, we need to follow these steps:

1. Convert mixed numbers to improper fractions if necessary.
2. Find a common denominator for all fractions in the expression.
3. Perform the addition or subtraction.
4. Simplify the result to its lowest terms.

Let's solve each problem step by step.

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Problem 1: \( 2 \frac{5}{6} - \left( 4 \frac{1}{3} - \frac{3}{2} \right) \)

#### Step 1: Convert mixed numbers to improper fractions
- \( 2 \frac{5}{6} = \frac{2 \cdot 6 + 5}{6} = \frac{17}{6} \)
- \( 4 \frac{1}{3} = \frac{4 \cdot 3 + 1}{3} = \frac{13}{3} \)

#### Step 2: Simplify inside the parentheses
\[ 4 \frac{1}{3} - \frac{3}{2} = \frac{13}{3} - \frac{3}{2} \]

- Find the least common denominator (LCD) of 3 and 2, which is 6.
- Convert fractions:
\[ \frac{13}{3} = \frac{13 \cdot 2}{3 \cdot 2} = \frac{26}{6} \]
\[ \frac{3}{2} = \frac{3 \cdot 3}{2 \cdot 3} = \frac{9}{6} \]

- Subtract:
\[ \frac{26}{6} - \frac{9}{6} = \frac{26 - 9}{6} = \frac{17}{6} \]

#### Step 3: Substitute back into the original expression
\[ 2 \frac{5}{6} - \left( 4 \frac{1}{3} - \frac{3}{2} \right) = \frac{17}{6} - \frac{17}{6} \]

- Subtract:
\[ \frac{17}{6} - \frac{17}{6} = 0 \]

#### Final Answer:
\[ \boxed{0} \]

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Problem 2: \( \frac{1}{2} + \frac{13}{8} - \frac{11}{12} \)

#### Step 1: Find the least common denominator (LCD) of 2, 8, and 12
- The LCD is 24.

#### Step 2: Convert all fractions to have the denominator 24
- \( \frac{1}{2} = \frac{1 \cdot 12}{2 \cdot 12} = \frac{12}{24} \)
- \( \frac{13}{8} = \frac{13 \cdot 3}{8 \cdot 3} = \frac{39}{24} \)
- \( \frac{11}{12} = \frac{11 \cdot 2}{12 \cdot 2} = \frac{22}{24} \)

#### Step 3: Perform the addition and subtraction
\[ \frac{1}{2} + \frac{13}{8} - \frac{11}{12} = \frac{12}{24} + \frac{39}{24} - \frac{22}{24} \]
\[ = \frac{12 + 39 - 22}{24} = \frac{29}{24} \]

#### Final Answer:
\[ \boxed{\frac{29}{24}} \]

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Problem 3: \( \frac{3}{10} - \frac{1}{6} + 3 \frac{4}{5} \)

#### Step 1: Convert the mixed number to an improper fraction
- \( 3 \frac{4}{5} = \frac{3 \cdot 5 + 4}{5} = \frac{19}{5} \)

#### Step 2: Find the least common denominator (LCD) of 10, 6, and 5
- The LCD is 30.

#### Step 3: Convert all fractions to have the denominator 30
- \( \frac{3}{10} = \frac{3 \cdot 3}{10 \cdot 3} = \frac{9}{30} \)
- \( \frac{1}{6} = \frac{1 \cdot 5}{6 \cdot 5} = \frac{5}{30} \)
- \( \frac{19}{5} = \frac{19 \cdot 6}{5 \cdot 6} = \frac{114}{30} \)

#### Step 4: Perform the operations
\[ \frac{3}{10} - \frac{1}{6} + 3 \frac{4}{5} = \frac{9}{30} - \frac{5}{30} + \frac{114}{30} \]
\[ = \frac{9 - 5 + 114}{30} = \frac{118}{30} \]

#### Step 5: Simplify the fraction
- Divide numerator and denominator by their greatest common divisor (GCD), which is 2:
\[ \frac{118}{30} = \frac{59}{15} \]

#### Final Answer:
\[ \boxed{\frac{59}{15}} \]

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Problem 4: \( \frac{3}{4} + \frac{2}{7} - \frac{2}{7} \)

#### Step 1: Notice that \( \frac{2}{7} - \frac{2}{7} = 0 \)
\[ \frac{3}{4} + \frac{2}{7} - \frac{2}{7} = \frac{3}{4} + 0 = \frac{3}{4} \]

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

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Problem 5: \( 1 \frac{1}{5} + \frac{17}{2} - \frac{3}{2} \)

#### Step 1: Convert the mixed number to an improper fraction
- \( 1 \frac{1}{5} = \frac{1 \cdot 5 + 1}{5} = \frac{6}{5} \)

#### Step 2: Find the least common denominator (LCD) of 5, 2, and 2
- The LCD is 10.

#### Step 3: Convert all fractions to have the denominator 10
- \( \frac{6}{5} = \frac{6 \cdot 2}{5 \cdot 2} = \frac{12}{10} \)
- \( \frac{17}{2} = \frac{17 \cdot 5}{2 \cdot 5} = \frac{85}{10} \)
- \( \frac{3}{2} = \frac{3 \cdot 5}{2 \cdot 5} = \frac{15}{10} \)

#### Step 4: Perform the operations
\[ 1 \frac{1}{5} + \frac{17}{2} - \frac{3}{2} = \frac{12}{10} + \frac{85}{10} - \frac{15}{10} \]
\[ = \frac{12 + 85 - 15}{10} = \frac{82}{10} \]

#### Step 5: Simplify the fraction
- Divide numerator and denominator by their GCD, which is 2:
\[ \frac{82}{10} = \frac{41}{5} \]

#### Final Answer:
\[ \boxed{\frac{41}{5}} \]

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Problem 6: \( \frac{17}{6} + \frac{5}{3} - 3 \frac{1}{2} \)

#### Step 1: Convert the mixed number to an improper fraction
- \( 3 \frac{1}{2} = \frac{3 \cdot 2 + 1}{2} = \frac{7}{2} \)

#### Step 2: Find the least common denominator (LCD) of 6, 3, and 2
- The LCD is 6.

#### Step 3: Convert all fractions to have the denominator 6
- \( \frac{17}{6} \) remains \( \frac{17}{6} \)
- \( \frac{5}{3} = \frac{5 \cdot 2}{3 \cdot 2} = \frac{10}{6} \)
- \( \frac{7}{2} = \frac{7 \cdot 3}{2 \cdot 3} = \frac{21}{6} \)

#### Step 4: Perform the operations
\[ \frac{17}{6} + \frac{5}{3} - 3 \frac{1}{2} = \frac{17}{6} + \frac{10}{6} - \frac{21}{6} \]
\[ = \frac{17 + 10 - 21}{6} = \frac{6}{6} = 1 \]

#### Final Answer:
\[ \boxed{1} \]

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Problem 7: \( \frac{5}{2} + 1 \frac{7}{9} + \frac{1}{3} \)

#### Step 1: Convert the mixed number to an improper fraction
- \( 1 \frac{7}{9} = \frac{1 \cdot 9 + 7}{9} = \frac{16}{9} \)

#### Step 2: Find the least common denominator (LCD) of 2, 9, and 3
- The LCD is 18.

#### Step 3: Convert all fractions to have the denominator 18
- \( \frac{5}{2} = \frac{5 \cdot 9}{2 \cdot 9} = \frac{45}{18} \)
- \( \frac{16}{9} = \frac{16 \cdot 2}{9 \cdot 2} = \frac{32}{18} \)
- \( \frac{1}{3} = \frac{1 \cdot 6}{3 \cdot 6} = \frac{6}{18} \)

#### Step 4: Perform the operations
\[ \frac{5}{2} + 1 \frac{7}{9} + \frac{1}{3} = \frac{45}{18} + \frac{32}{18} + \frac{6}{18} \]
\[ = \frac{45 + 32 + 6}{18} = \frac{83}{18} \]

#### Final Answer:
\[ \boxed{\frac{83}{18}} \]

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Problem 8: \( 1 \frac{11}{12} - \left( 1 \frac{3}{4} - \frac{1}{8} \right) \)

#### Step 1: Convert mixed numbers to improper fractions
- \( 1 \frac{11}{12} = \frac{1 \cdot 12 + 11}{12} = \frac{23}{12} \)
- \( 1 \frac{3}{4} = \frac{1 \cdot 4 + 3}{4} = \frac{7}{4} \)

#### Step 2: Simplify inside the parentheses
\[ 1 \frac{3}{4} - \frac{1}{8} = \frac{7}{4} - \frac{1}{8} \]

- Find the LCD of 4 and 8, which is 8.
- Convert fractions:
\[ \frac{7}{4} = \frac{7 \cdot 2}{4 \cdot 2} = \frac{14}{8} \]
\[ \frac{1}{8} = \frac{1}{8} \]

- Subtract:
\[ \frac{14}{8} - \frac{1}{8} = \frac{14 - 1}{8} = \frac{13}{8} \]

#### Step 3: Substitute back into the original expression
\[ 1 \frac{11}{12} - \left( 1 \frac{3}{4} - \frac{1}{8} \right) = \frac{23}{12} - \frac{13}{8} \]

- Find the LCD of 12 and 8, which is 24.
- Convert fractions:
\[ \frac{23}{12} = \frac{23 \cdot 2}{12 \cdot 2} = \frac{46}{24} \]
\[ \frac{13}{8} = \frac{13 \cdot 3}{8 \cdot 3} = \frac{39}{24} \]

- Subtract:
\[ \frac{46}{24} - \frac{39}{24} = \frac{46 - 39}{24} = \frac{7}{24} \]

#### Final Answer:
\[ \boxed{\frac{7}{24}} \]

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Problem 9: \( \frac{11}{2} - \left( \frac{2}{7} + \frac{3}{2} \right) \)

#### Step 1: Simplify inside the parentheses
\[ \frac{2}{7} + \frac{3}{2} \]

- Find the LCD of 7 and 2, which is 14.
- Convert fractions:
\[ \frac{2}{7} = \frac{2 \cdot 2}{7 \cdot 2} = \frac{4}{14} \]
\[ \frac{3}{2} = \frac{3 \cdot 7}{2 \cdot 7} = \frac{21}{14} \]

- Add:
\[ \frac{2}{7} + \frac{3}{2} = \frac{4}{14} + \frac{21}{14} = \frac{4 + 21}{14} = \frac{25}{14} \]

#### Step 2: Substitute back into the original expression
\[ \frac{11}{2} - \left( \frac{2}{7} + \frac{3}{2} \right) = \frac{11}{2} - \frac{25}{14} \]

- Find the LCD of 2 and 14, which is 14.
- Convert fractions:
\[ \frac{11}{2} = \frac{11 \cdot 7}{2 \cdot 7} = \frac{77}{14} \]
\[ \frac{25}{14} = \frac{25}{14} \]

- Subtract:
\[ \frac{11}{2} - \frac{25}{14} = \frac{77}{14} - \frac{25}{14} = \frac{77 - 25}{14} = \frac{52}{14} \]

#### Step 3: Simplify the fraction
- Divide numerator and denominator by their GCD, which is 2:
\[ \frac{52}{14} = \frac{26}{7} \]

#### Final Answer:
\[ \boxed{\frac{26}{7}} \]

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Problem 10: \( 3 \frac{1}{3} + 1 \frac{3}{4} - 1 \frac{2}{3} \)

#### Step 1: Convert mixed numbers to improper fractions
- \( 3 \frac{1}{3} = \frac{3 \cdot 3 + 1}{3} = \frac{10}{3} \)
- \( 1 \frac{3}{4} = \frac{1 \cdot 4 + 3}{4} = \frac{7}{4} \)
- \( 1 \frac{2}{3} = \frac{1 \cdot 3 + 2}{3} = \frac{5}{3} \)

#### Step 2: Find the least common denominator (LCD) of 3 and 4
- The LCD is 12.

#### Step 3: Convert all fractions to have the denominator 12
- \( \frac{10}{3} = \frac{10 \cdot 4}{3 \cdot 4} = \frac{40}{12} \)
- \( \frac{7}{4} = \frac{7 \cdot 3}{4 \cdot 3} = \frac{21}{12} \)
- \( \frac{5}{3} = \frac{5 \cdot 4}{3 \cdot 4} = \frac{20}{12} \)

#### Step 4: Perform the operations
\[ 3 \frac{1}{3} + 1 \frac{3}{4} - 1 \frac{2}{3} = \frac{40}{12} + \frac{21}{12} - \frac{20}{12} \]
\[ = \frac{40 + 21 - 20}{12} = \frac{41}{12} \]

#### Final Answer:
\[ \boxed{\frac{41}{12}} \]

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Problem 11: \( \frac{4}{3} - \left( 1 \frac{11}{12} - \frac{5}{4} \right) \)

#### Step 1: Convert the mixed number to an improper fraction
- \( 1 \frac{11}{12} = \frac{1 \cdot 12 + 11}{12} = \frac{23}{12} \)

#### Step 2: Simplify inside the parentheses
\[ 1 \frac{11}{12} - \frac{5}{4} = \frac{23}{12} - \frac{5}{4} \]

- Find the LCD of 12 and 4, which is 12.
- Convert fractions:
\[ \frac{23}{12} = \frac{23}{12} \]
\[ \frac{5}{4} = \frac{5 \cdot 3}{4 \cdot 3} = \frac{15}{12} \]

- Subtract:
\[ \frac{23}{12} - \frac{5}{4} = \frac{23}{12} - \frac{15}{12} = \frac{23 - 15}{12} = \frac{8}{12} \]

- Simplify:
\[ \frac{8}{12} = \frac{2}{3} \]

#### Step 3: Substitute back into the original expression
\[ \frac{4}{3} - \left( 1 \frac{11}{12} - \frac{5}{4} \right) = \frac{4}{3} - \frac{2}{3} \]

- Subtract:
\[ \frac{4}{3} - \frac{2}{3} = \frac{4 - 2}{3} = \frac{2}{3} \]

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

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Problem 12: \( 2 \frac{1}{3} - \frac{2}{3} + 1 \frac{4}{5} \)

#### Step 1: Convert the mixed numbers to improper fractions
- \( 2 \frac{1}{3} = \frac{2 \cdot 3 + 1}{3} = \frac{7}{3} \)
- \( 1 \frac{4}{5} = \frac{1 \cdot 5 + 4}{5} = \frac{9}{5} \)

#### Step 2: Find the least common denominator (LCD) of 3 and 5
- The LCD is 15.

#### Step 3: Convert all fractions to have the denominator 15
- \( \frac{7}{3} = \frac{7 \cdot 5}{3 \cdot 5} = \frac{35}{15} \)
- \( \frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15} \)
- \( \frac{9}{5} = \frac{9 \cdot 3}{5 \cdot 3} = \frac{27}{15} \)

#### Step 4: Perform the operations
\[ 2 \frac{1}{3} - \frac{2}{3} + 1 \frac{4}{5} = \frac{35}{15} - \frac{10}{15} + \frac{27}{15} \]
\[ = \frac{35 - 10 + 27}{15} = \frac{52}{15} \]

#### Final Answer:
\[ \boxed{\frac{52}{15}} \]

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Final Answers:

1. \( \boxed{0} \)
2. \( \boxed{\frac{29}{24}} \)
3. \( \boxed{\frac{59}{15}} \)
4. \( \boxed{\frac{3}{4}} \)
5. \( \boxed{\frac{41}{5}} \)
6. \( \boxed{1} \)
7. \( \boxed{\frac{83}{18}} \)
8. \( \boxed{\frac{7}{24}} \)
9. \( \boxed{\frac{26}{7}} \)
10. \( \boxed{\frac{41}{12}} \)
11. \( \boxed{\frac{2}{3}} \)
12. \( \boxed{\frac{52}{15}} \)