To add \( 4 \frac{1}{2} \) and \( 1 \frac{3}{5} \), we first convert the mixed numbers to improper fractions.
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Convert \( 4 \frac{1}{2} \): \[ 4 \frac{1}{2} = 4 \times 2 + 1 = 8 + 1 = 9 \quad \Rightarrow \quad \frac{9}{2} \]
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Convert \( 1 \frac{3}{5} \): \[ 1 \frac{3}{5} = 1 \times 5 + 3 = 5 + 3 = 8 \quad \Rightarrow \quad \frac{8}{5} \]
Now we add the two improper fractions \( \frac{9}{2} \) and \( \frac{8}{5} \). To do this, we need a common denominator. The least common multiple of 2 and 5 is 10.
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Convert \( \frac{9}{2} \) to a fraction with a denominator of 10: \[ \frac{9}{2} = \frac{9 \times 5}{2 \times 5} = \frac{45}{10} \]
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Convert \( \frac{8}{5} \) to a fraction with a denominator of 10: \[ \frac{8}{5} = \frac{8 \times 2}{5 \times 2} = \frac{16}{10} \]
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Now, add the two fractions: \[ \frac{45}{10} + \frac{16}{10} = \frac{45 + 16}{10} = \frac{61}{10} \]
Now we convert \( \frac{61}{10} \) back to a mixed number:
- Divide 61 by 10:
- \( 61 \div 10 = 6 \) (whole number part)
- Remainder: \( 61 - (10 \times 6) = 1 \)
Thus, \( \frac{61}{10} = 6 \frac{1}{10} \).
The sum \( 4 \frac{1}{2} + 1 \frac{3}{5} = 6 \frac{1}{10} \) in simplest form is: \[ \boxed{6 \frac{1}{10}} \]