To solve the expression \((3 \frac{2}{3} + 17 \frac{1}{5}) - (11 \frac{3}{5} + 2 \frac{1}{3})\), we will first convert the mixed numbers to improper fractions, perform the necessary addition and subtraction, and then convert back to a mixed number if necessary.
Step 1: Convert mixed numbers to improper fractions.
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Convert \(3 \frac{2}{3}\): \[ 3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \]
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Convert \(17 \frac{1}{5}\): \[ 17 \frac{1}{5} = \frac{17 \times 5 + 1}{5} = \frac{85 + 1}{5} = \frac{86}{5} \]
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Convert \(11 \frac{3}{5}\): \[ 11 \frac{3}{5} = \frac{11 \times 5 + 3}{5} = \frac{55 + 3}{5} = \frac{58}{5} \]
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Convert \(2 \frac{1}{3}\): \[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]
Step 2: Combine the fractions for the additions.
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Calculate \(3 \frac{2}{3} + 17 \frac{1}{5}\): \[ \frac{11}{3} + \frac{86}{5} \] Find a common denominator, which is 15: \[ \frac{11}{3} = \frac{11 \times 5}{3 \times 5} = \frac{55}{15} \] \[ \frac{86}{5} = \frac{86 \times 3}{5 \times 3} = \frac{258}{15} \] Now add the two: \[ \frac{55}{15} + \frac{258}{15} = \frac{55 + 258}{15} = \frac{313}{15} \]
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Calculate \(11 \frac{3}{5} + 2 \frac{1}{3}\): \[ \frac{58}{5} + \frac{7}{3} \] Again, use a common denominator of 15: \[ \frac{58}{5} = \frac{58 \times 3}{5 \times 3} = \frac{174}{15} \] \[ \frac{7}{3} = \frac{7 \times 5}{3 \times 5} = \frac{35}{15} \] Now add these: \[ \frac{174}{15} + \frac{35}{15} = \frac{174 + 35}{15} = \frac{209}{15} \]
Step 3: Subtract the two results.
Now we subtract the two sums: \[ \frac{313}{15} - \frac{209}{15} = \frac{313 - 209}{15} = \frac{104}{15} \]
Step 4: Convert back to a mixed number.
To convert \(\frac{104}{15}\) to a mixed number:
- Divide \(104\) by \(15\):
- \(104 \div 15 = 6\) remainder \(14\). So, \(\frac{104}{15} = 6 \frac{14}{15}\).
Final Answer
\[ (3 \frac{2}{3} + 17 \frac{1}{5}) - (11 \frac{3}{5} + 2 \frac{1}{3}) = 6 \frac{14}{15} \]