To solve the division of the mixed numbers [tex]\(3 \frac{1}{2} \div 1 \frac{2}{4}\)[/tex], we will follow these steps:
1. Convert the mixed numbers to improper fractions:
- For [tex]\(3 \frac{1}{2}\)[/tex]:
- The whole number part is 3.
- The fractional part is [tex]\(\frac{1}{2}\)[/tex].
- To convert this mixed number to an improper fraction: Multiply the whole number by the denominator of the fraction and add the numerator.
[tex]\[
3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}
\][/tex]
- For [tex]\(1 \frac{2}{4}\)[/tex]:
- The whole number part is 1.
- The fractional part is [tex]\(\frac{2}{4}\)[/tex].
- Simplify [tex]\(\frac{2}{4}\)[/tex] to [tex]\(\frac{1}{2}\)[/tex].
- Now convert this mixed number to an improper fraction:
[tex]\[
1 \frac{2}{4} = 1 + \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}
\][/tex]
2. Perform the division of the improper fractions:
Division of fractions is equivalent to multiplying by the reciprocal of the divisor. Therefore, we change the problem from division to multiplication:
[tex]\[
\frac{7}{2} \div \frac{3}{2} = \frac{7}{2} \times \frac{2}{3}
\][/tex]
3. Multiply the fractions:
- Multiply the numerators together: [tex]\(7 \times 2 = 14\)[/tex]
- Multiply the denominators together: [tex]\(2 \times 3 = 6\)[/tex]
[tex]\[
\frac{7}{2} \times \frac{2}{3} = \frac{14}{6}
\][/tex]
4. Simplify the resulting fraction:
Simplify [tex]\(\frac{14}{6}\)[/tex] by dividing the numerator and denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{14}{6} = \frac{14 \div 2}{6 \div 2} = \frac{7}{3}
\][/tex]
5. Convert the improper fraction back to a mixed number (optional, but for a more comprehensive answer):
[tex]\[
\frac{7}{3} = 2 \frac{1}{3}
\][/tex]
Therefore, the final result of [tex]\(3 \frac{1}{2} \div 1 \frac{2}{4}\)[/tex] is [tex]\(4 \frac{2}{3}\)[/tex] or approximately [tex]\(4.67\)[/tex].
