Sure! Here’s a brief overview of multiplying and dividing rational numbers, as typically covered in a unit on this topic.
Rational Numbers
A rational number is any number that can be expressed as the quotient or fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b \neq 0\).
Multiplying Rational Numbers
To multiply two rational numbers, you simply multiply the numerators together and the denominators together.
Formula: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]
Example: Multiply \(\frac{2}{3}\) and \(\frac{4}{5}\): \[ \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \]
Dividing Rational Numbers
To divide one rational number by another, you multiply by the reciprocal of the second number.
Formula: \[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \]
Example: Divide \(\frac{3}{4}\) by \(\frac{2}{5}\): \[ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} \]
Important Tips
- Always simplify your answer if possible, which means reducing the fraction to its lowest terms.
- Be mindful of negative signs. The product and quotient of two numbers will follow these rules regarding signs:
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
Example with Negative Numbers
Multiplying: \(-\frac{3}{4} \times \frac{2}{3}\): \[ -\frac{3}{4} \times \frac{2}{3} = -\frac{3 \times 2}{4 \times 3} = -\frac{6}{12} = -\frac{1}{2} \]
Dividing: \(-\frac{5}{6} \div \frac{1}{2}\): \[ -\frac{5}{6} \div \frac{1}{2} = -\frac{5}{6} \times 2 = -\frac{5 \times 2}{6 \times 1} = -\frac{10}{6} = -\frac{5}{3} \]
This should give you a solid understanding of multiplying and dividing rational numbers! If there are specific exercises or problems you'd like help with, feel free to ask!
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