Adding Mixed Numbers worksheet with fraction addition problems.

To solve the problem of adding mixed numbers and expressing the answers in the lowest terms, we will follow these steps for each problem:

1. Convert mixed numbers to improper fractions.
2. Find a common denominator for the fractions.
3. Add the fractions.
4. Simplify the result (if necessary).
5. Convert back to a mixed number if the result is an improper fraction.

Let's solve each problem step by step.

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

#### Step 1: Convert to improper fractions
- \( 1 \frac{1}{7} = \frac{7 \cdot 1 + 1}{7} = \frac{8}{7} \)
- \( 1 \frac{1}{5} = \frac{5 \cdot 1 + 1}{5} = \frac{6}{5} \)

#### Step 2: Find a common denominator
The denominators are 7 and 5. The least common denominator (LCD) is \( 7 \times 5 = 35 \).

#### Step 3: Rewrite fractions with the common denominator
- \( \frac{8}{7} = \frac{8 \cdot 5}{7 \cdot 5} = \frac{40}{35} \)
- \( \frac{6}{5} = \frac{6 \cdot 7}{5 \cdot 7} = \frac{42}{35} \)

#### Step 4: Add the fractions
\[ \frac{40}{35} + \frac{42}{35} = \frac{40 + 42}{35} = \frac{82}{35} \]

#### Step 5: Convert back to a mixed number
\[ \frac{82}{35} = 2 \frac{12}{35} \]

#### Final Answer:
\[ 1 \frac{1}{7} + 1 \frac{1}{5} = 2 \frac{12}{35} \]

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Problem 2: \( 2 \frac{9}{15} + 2 \frac{3}{5} \)

#### Step 1: Simplify and convert to improper fractions
- \( 2 \frac{9}{15} \): Simplify \( \frac{9}{15} \) to \( \frac{3}{5} \). So, \( 2 \frac{9}{15} = 2 \frac{3}{5} \).
- \( 2 \frac{3}{5} = \frac{5 \cdot 2 + 3}{5} = \frac{13}{5} \)

#### Step 2: Find a common denominator
Both fractions already have the same denominator, 5.

#### Step 3: Add the fractions
\[ \frac{13}{5} + \frac{13}{5} = \frac{13 + 13}{5} = \frac{26}{5} \]

#### Step 4: Convert back to a mixed number
\[ \frac{26}{5} = 5 \frac{1}{5} \]

#### Final Answer:
\[ 2 \frac{9}{15} + 2 \frac{3}{5} = 5 \frac{1}{5} \]

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

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

#### Step 2: Find a common denominator
The denominators are 4 and 5. The LCD is \( 4 \times 5 = 20 \).

#### Step 3: Rewrite fractions with the common denominator
- \( \frac{13}{4} = \frac{13 \cdot 5}{4 \cdot 5} = \frac{65}{20} \)
- \( \frac{13}{5} = \frac{13 \cdot 4}{5 \cdot 4} = \frac{52}{20} \)

#### Step 4: Add the fractions
\[ \frac{65}{20} + \frac{52}{20} = \frac{65 + 52}{20} = \frac{117}{20} \]

#### Step 5: Convert back to a mixed number
\[ \frac{117}{20} = 5 \frac{17}{20} \]

#### Final Answer:
\[ 3 \frac{1}{4} + 2 \frac{3}{5} = 5 \frac{17}{20} \]

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Problem 4: \( 2 \frac{8}{9} + 3 \frac{4}{6} \)

#### Step 1: Simplify and convert to improper fractions
- \( 3 \frac{4}{6} \): Simplify \( \frac{4}{6} \) to \( \frac{2}{3} \). So, \( 3 \frac{4}{6} = 3 \frac{2}{3} \).
- \( 2 \frac{8}{9} = \frac{9 \cdot 2 + 8}{9} = \frac{26}{9} \)
- \( 3 \frac{2}{3} = \frac{3 \cdot 3 + 2}{3} = \frac{11}{3} \)

#### Step 2: Find a common denominator
The denominators are 9 and 3. The LCD is 9.

#### Step 3: Rewrite fractions with the common denominator
- \( \frac{26}{9} \) remains \( \frac{26}{9} \).
- \( \frac{11}{3} = \frac{11 \cdot 3}{3 \cdot 3} = \frac{33}{9} \)

#### Step 4: Add the fractions
\[ \frac{26}{9} + \frac{33}{9} = \frac{26 + 33}{9} = \frac{59}{9} \]

#### Step 5: Convert back to a mixed number
\[ \frac{59}{9} = 6 \frac{5}{9} \]

#### Final Answer:
\[ 2 \frac{8}{9} + 3 \frac{4}{6} = 6 \frac{5}{9} \]

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Final Answers for All Problems

1. \( 1 \frac{1}{7} + 1 \frac{1}{5} = 2 \frac{12}{35} \)
2. \( 2 \frac{9}{15} + 2 \frac{3}{5} = 5 \frac{1}{5} \)
3. \( 3 \frac{1}{4} + 2 \frac{3}{5} = 5 \frac{17}{20} \)
4. \( 2 \frac{8}{9} + 3 \frac{4}{6} = 6 \frac{5}{9} \)
5. \( 2 \frac{9}{12} + 1 \frac{1}{2} = 4 \frac{1}{4} \)
6. \( 4 \frac{3}{7} + 1 \frac{4}{6} = 5 \frac{19}{42} \)
7. \( 2 \frac{2}{6} + 1 \frac{8}{16} = 3 \frac{5}{6} \)
8. \( 2 \frac{6}{20} + 1 \frac{1}{7} = 3 \frac{31}{70} \)
9. \( 3 \frac{1}{14} + 1 \frac{3}{10} = 4 \frac{13}{70} \)
10. \( 1 \frac{7}{12} + 7 \frac{10}{15} = 9 \frac{1}{4} \)
11. \( 3 \frac{3}{8} + 3 \frac{9}{15} = 6 \frac{37}{40} \)
12. \( 3 \frac{6}{12} + 3 \frac{4}{5} = 7 \frac{1}{10} \)
13. \( 2 \frac{1}{3} + 2 \frac{3}{10} = 4 \frac{19}{30} \)
14. \( 3 \frac{16}{25} + 2 \frac{2}{5} = 5 \frac{26}{25} = 6 \frac{1}{25} \)
15. \( 5 \frac{8}{12} + 1 \frac{1}{5} = 6 \frac{23}{30} \)

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Boxed Final Answer

\[
\boxed{
\begin{aligned}
1. & \ 2 \frac{12}{35} \\
2. & \ 5 \frac{1}{5} \\
3. & \ 5 \frac{17}{20} \\
4. & \ 6 \frac{5}{9} \\
5. & \ 4 \frac{1}{4} \\
6. & \ 5 \frac{19}{42} \\
7. & \ 3 \frac{5}{6} \\
8. & \ 3 \frac{31}{70} \\
9. & \ 4 \frac{13}{70} \\
10. & \ 9 \frac{1}{4} \\
11. & \ 6 \frac{37}{40} \\
12. & \ 7 \frac{1}{10} \\
13. & \ 4 \frac{19}{30} \\
14. & \ 6 \frac{1}{25} \\
15. & \ 6 \frac{23}{30}
\end{aligned}
}
\]