multiplying and dividing rational numbers

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Multiplying and dividing rational numbers involves working with fractions. A rational number is any number that can be expressed in the form of \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \) is not zero. Here's how you can multiply and divide rational numbers:

Multiplying Rational Numbers

To multiply two rational numbers, you simply multiply their numerators together and their denominators together.

For example, let’s multiply \( \frac{a}{b} \) and \( \frac{c}{d} \):

\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]

Example:

Multiply \( \frac{3}{4} \) by \( \frac{2}{5} \).

\[ \frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} \]

Now, simplify \( \frac{6}{20} \):

\[ \frac{6 \div 2}{20 \div 2} = \frac{3}{10} \]

So, \( \frac{3}{4} \times \frac{2}{5} = \frac{3}{10} \).

Dividing Rational Numbers

To divide two rational numbers, you multiply the first rational number by the reciprocal (or the inverse) of the second rational number.

For example, to divide \( \frac{a}{b} \) by \( \frac{c}{d} \):

\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \]

Example:

Divide \( \frac{3}{4} \) by \( \frac{2}{5} \).

\[ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} \]

In this case, \( \frac{15}{8} \) is already in its simplest form, so:

\[ \frac{3}{4} \div \frac{2}{5} = \frac{15}{8} \]

Summary

  1. Multiplication: Multiply the numerators and multiply the denominators.
  2. Division: Multiply by the reciprocal of the second fraction.

Always remember to simplify the result when possible!