- To divide fractions, multiply the first fraction by the reciprocal of the second: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).
- Part a: \(\frac{2}{5} \times \frac{4}{3} = \boxed{\frac{8}{15}}\)
- Part b: \(\frac{9}{4} \times \frac{4}{-3} = \boxed{-3}\)
- Part c: \(\frac{-5}{7} \times \frac{3}{-1} = \boxed{\frac{15}{7}}\)
- Part d: \(\frac{-5}{3} \times \frac{6}{1} = \boxed{-10}\)
Explanation
- Understanding the Division Rule for Fractions
To divide two fractions, we use the rule that dividing by a fraction is equivalent to multiplying by its reciprocal. For any two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), where \(\frac{c}{d}
eq 0\), the division is defined as \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}.\] We apply this rule to each part of the problem, ensuring that we simplify the resulting fractions and correctly handle the signs of the quotients.
- Solving Part (a)
For part (a), we have \(\frac{2}{5} \div \frac{3}{4}\). The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Thus, the division becomes:
\[\frac{2}{5} \times \frac{4}{3} = \frac{2 \times 4}{5 \times 3} = \frac{8}{15}.\]
This fraction cannot be simplified further.
- Solving Part (b)
For part (b), we have \(\frac{9}{4} \div \left(\frac{-3}{4}\right)\). The reciprocal of \(\frac{-3}{4}\) is \(\frac{4}{-3}\). Thus, the division becomes:
\[\frac{9}{4} \times \left(\frac{4}{-3}\right) = \frac{9 \times 4}{4 \times -3} = \frac{36}{-12}.\]
Dividing 36 by -12 gives \(-3\). So, the quotient is \(-3\).
- Solving Part (c)
For part (c), we have \(\left(\frac{-5}{7}\right) \div \left(\frac{-1}{3}\right)\). The reciprocal of \(\frac{-1}{3}\) is \(\frac{3}{-1}\). Thus, the division becomes:
\[\left(\frac{-5}{7}\right) \times \left(\frac{3}{-1}\right) = \frac{-5 \times 3}{7 \times -1} = \frac{-15}{-7}.\]
Since a negative divided by a negative is positive, the quotient is \(\frac{15}{7}\).
- Solving Part (d)
For part (d), we have \(\left(\frac{-5}{3}\right) \div \frac{1}{6}\). The reciprocal of \(\frac{1}{6}\) is \(\frac{6}{1}\). Thus, the division becomes:
\[\left(\frac{-5}{3}\right) \times \frac{6}{1} = \frac{-5 \times 6}{3 \times 1} = \frac{-30}{3}.\]
Dividing -30 by 3 gives \(-10\). So, the quotient is \(-10\).
Examples
When dividing items or resources, such as splitting a bulk order of materials into smaller packages, understanding how to handle fractions is crucial. For example, if you have \(\frac{10}{3}\) kilograms of sand and need to divide it into bags of \(\frac{2}{3}\) kilograms each, you would calculate \(\frac{10}{3} \div \frac{2}{3} = \frac{10}{3} \times \frac{3}{2} = 5\), meaning you can fill exactly 5 bags. Mastering fraction division ensures precise resource allocation and management in various practical situations.