To add the mixed numbers \(3 \frac{5}{7}\) and \(7 \frac{1}{2}\), we can follow these steps:
- Convert each mixed number into an improper fraction.
- Find a common denominator.
- Add the fractions.
- Simplify if necessary.
Step 1: Convert to improper fractions
For \(3 \frac{5}{7}\): \[ 3 \frac{5}{7} = \frac{3 \times 7 + 5}{7} = \frac{21 + 5}{7} = \frac{26}{7} \]
For \(7 \frac{1}{2}\): \[ 7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2} \]
Step 2: Find the common denominator
The denominators are 7 and 2. The least common multiple (LCM) of 7 and 2 is 14.
Step 3: Convert each fraction to have the common denominator
For \(\frac{26}{7}\): \[ \frac{26}{7} \times \frac{2}{2} = \frac{52}{14} \]
For \(\frac{15}{2}\): \[ \frac{15}{2} \times \frac{7}{7} = \frac{105}{14} \]
Step 4: Add the fractions
Now we can add: \[ \frac{52}{14} + \frac{105}{14} = \frac{52 + 105}{14} = \frac{157}{14} \]
Step 5: Convert back to a mixed number
To convert \(\frac{157}{14}\) back to a mixed number:
- Divide 157 by 14, which is 11 with a remainder. \[ 157 \div 14 = 11 \quad \text{(because } 11 \times 14 = 154\text{)} \]
- The remainder is \(157 - 154 = 3\).
Thus, we can write: \[ \frac{157}{14} = 11 \frac{3}{14} \]
Final answer
The sum of \(3 \frac{5}{7} + 7 \frac{1}{2}\) is: \[ \boxed{11 \frac{3}{14}} \]