Calculations: Adding and subtracting common fractions with different denominators When the denominators are you will need to find the equivalent fraction to get the denominators the same to add or subtract the fractions. Worked examples Calculate: 1. $$ \frac{2}{5}+\frac{1}{3} $$ 2. $$ 2 \frac{1}{2}+1 \frac{1}{5} $$ 3. $$ 3 \frac{2}{3}-1 \frac{3}{7} $$ Answers 1. $$ \frac{2}{5}+\frac{1}{3}=\frac{6}{15}+\frac{5}{15} $$ 2. $$ 2 \frac{1}{2}+1 \frac{1}{5}=\frac{5}{2}+\frac{6}{5} $$ 3. $$ 3 \frac{2}{3}-1 \frac{3}{7}=\frac{11}{3}-\frac{10}{7} $$ $$ =\frac{11}{15} $$ $$ =\frac{25}{10}+\frac{12}{10} $$ $$ =\frac{77}{21}-\frac{30}{21} $$ $$ =\frac{37}{10}=3 \frac{7}{10} $$ $$ =\frac{47}{21}=2 \frac{5}{21} $$

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Let's solve each of the problems step by step, ensuring we find equivalent fractions with common denominators for addition and subtraction.

### Problem 1: $$ \frac{2}{5} + \frac{1}{3} $$

1. **Find a common denominator**: The denominators are 5 and 3. The least common multiple (LCM) of 5 and 3 is 15.
2. **Convert each fraction**:
   - For $$ \frac{2}{5} $$: 
     $$
     \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}
     $$
   - For $$ \frac{1}{3} $$: 
     $$
     \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
     $$
3. **Add the fractions**:
   $$
   \frac{6}{15} + \frac{5}{15} = \frac{11}{15}
   $$

### Problem 2: $$ 2 \frac{1}{2} + 1 \frac{1}{5} $$

1. **Convert mixed numbers to improper fractions**:
   - For $$ 2 \frac{1}{2} $$:
     $$
     2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}
     $$
   - For $$ 1 \frac{1}{5} $$:
     $$
     1 \frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}
     $$
2. **Find a common denominator**: The denominators are 2 and 5. The LCM of 2 and 5 is 10.
3. **Convert each fraction**:
   - For $$ \frac{5}{2} $$:
     $$
     \frac{5}{2} = \frac{5 \times 5}{2 \times 5} = \frac{25}{10}
     $$
   - For $$ \frac{6}{5} $$:
     $$
     \frac{6}{5} = \frac{6 \times 2}{5 \times 2} = \frac{12}{10}
     $$
4. **Add the fractions**:
   $$
   \frac{25}{10} + \frac{12}{10} = \frac{37}{10} = 3 \frac{7}{10}
   $$

### Problem 3: $$ 3 \frac{2}{3} - 1 \frac{3}{7} $$

1. **Convert mixed numbers to improper fractions**:
   - For $$ 3 \frac{2}{3} $$:
     $$
     3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{11}{3}
     $$
   - For $$ 1 \frac{3}{7} $$:
     $$
     1 \frac{3}{7} = \frac{1 \times 7 + 3}{7} = \frac{10}{7}
     $$
2. **Find a common denominator**: The denominators are 3 and 7. The LCM of 3 and 7 is 21.
3. **Convert each fraction**:
   - For $$ \frac{11}{3} $$:
     $$
     \frac{11}{3} = \frac{11 \times 7}{3 \times 7} = \frac{77}{21}
     $$
   - For $$ \frac{10}{7} $$:
     $$
     \frac{10}{7} = \frac{10 \times 3}{7 \times 3} = \frac{30}{21}
     $$
4. **Subtract the fractions**:
   $$
   \frac{77}{21} - \frac{30}{21} = \frac{47}{21} = 2 \frac{5}{21}
   $$

### Final Answers:
1. $$ \frac{2}{5} + \frac{1}{3} = \frac{11}{15} $$
2. $$ 2 \frac{1}{2} + 1 \frac{1}{5} = 3 \frac{7}{10} $$
3. $$ 3 \frac{2}{3} - 1 \frac{3}{7} = 2 \frac{5}{21} $$
1. $$ \frac{2}{5} + \frac{1}{3} = \frac{11}{15} $$ 2. $$ 2 \frac{1}{2} + 1 \frac{1}{5} = 3 \frac{7}{10} $$ 3. $$ 3 \frac{2}{3} - 1 \frac{3}{7} = 2 \frac{5}{21} $$

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