To add the two mixed numbers \(3 \frac{5}{7}\) and \(7 \frac{1}{2}\), you can follow these steps:
- Convert the mixed numbers into improper fractions.
- Find a common denominator.
- Add the fractions.
- Convert back to a mixed number if necessary.
Step 1: Convert to improper fractions
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For \(3 \frac{5}{7}\): \[ 3 \frac{5}{7} = \frac{3 \times 7 + 5}{7} = \frac{21 + 5}{7} = \frac{26}{7} \]
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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 a common denominator
The denominators are 7 and 2. The least common multiple (LCM) of 7 and 2 is 14.
Convert both fractions to have a denominator of 14:
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For \(\frac{26}{7}\): \[ \frac{26}{7} = \frac{26 \times 2}{7 \times 2} = \frac{52}{14} \]
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For \(\frac{15}{2}\): \[ \frac{15}{2} = \frac{15 \times 7}{2 \times 7} = \frac{105}{14} \]
Step 3: Add the fractions
Now that we have a common denominator: \[ \frac{52}{14} + \frac{105}{14} = \frac{52 + 105}{14} = \frac{157}{14} \]
Step 4: Convert back to a mixed number
To convert \(\frac{157}{14}\) back to a mixed number, divide 157 by 14:
- 14 goes into 157 eleven times (since \(14 \times 11 = 154\)).
- The remainder is \(157 - 154 = 3\).
So, \(\frac{157}{14} = 11 \frac{3}{14}\).
Final Answer
\[ 3 \frac{5}{7} + 7 \frac{1}{2} = 11 \frac{3}{14} \]