Operations with Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\) is not equal to zero. In Grade 8, you will learn how to perform four basic operations with rational numbers: addition, subtraction, multiplication, and division.

1. Addition of Rational Numbers

To add two rational numbers, follow these steps:

Find a common denominator for the fractions.
Add the numerators while keeping the common denominator.
Simplify the resulting fraction if possible.

Example: Add \(\frac{3}{4}\) and \(\frac{1}{2}\).

Step 1: The common denominator for 4 and 2 is 4.

Step 2: Convert \(\frac{1}{2}\) to \(\frac{2}{4}\).

Step 3: Add the numerators: \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\).

Step 4: The fraction \(\frac{5}{4}\) is already in its simplest form.

2. Subtraction of Rational Numbers

To subtract two rational numbers, follow these steps:

Find a common denominator for the fractions.
Subtract the numerators while keeping the common denominator.
Simplify the resulting fraction if possible.

Example: Subtract \(\frac{5}{6}\) from \(\frac{7}{8}\).

Step 1: The common denominator for 6 and 8 is 24.

Step 2: Convert \(\frac{5}{6}\) to \(\frac{20}{24}\) and \(\frac{7}{8}\) to \(\frac{21}{24}\).

Step 3: Subtract the numerators: \(\frac{21}{24} - \frac{20}{24} = \frac{1}{24}\).

Step 4: The fraction \(\frac{1}{24}\) is already in its simplest form.

3. Multiplication of Rational Numbers

To multiply two rational numbers, follow these steps:

Multiply the numerators together.
Multiply the denominators together.
Simplify the resulting fraction if possible.

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

Step 1: Multiply the numerators: \(2 \times 3 = 6\).

Step 2: Multiply the denominators: \(3 \times 5 = 15\).

Step 3: The resulting fraction is \(\frac{6}{15}\).

Step 4: Simplify \(\frac{6}{15}\) to \(\frac{2}{5}\).

4. Division of Rational Numbers

To divide two rational numbers, follow these steps:

Take the reciprocal of the second fraction.
Multiply the first fraction by this reciprocal.
Simplify the resulting fraction if possible.

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

Step 1: The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).

Step 2: Multiply \(\frac{4}{5}\) by \(\frac{3}{2}\): \(\frac{4}{5} \times \frac{3}{2} = \frac{12}{10}\).

Step 3: Simplify \(\frac{12}{10}\) to \(\frac{6}{5}\).