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Simplifying fractions makes working with them easier and often prettier. In this concept, you aim to turn a complicated fraction into its simplest form. By doing this, you reduce the fraction to the smallest possible numerator and denominator that still represent the same value.
To simplify a fraction, you need to find a common number that both the numerator and the denominator can be divided by. This is called the Greatest Common Divisor (GCD). Once you find the GCD, you divide both the numerator and the denominator by this number.
For example, consider the fraction \(-\frac{12}{18}\). The GCD of 12 and 18 is 6. You divide both the numerator and the denominator by 6:\
\[ \frac{-12 \div 6}{18 \div 6} = -\frac{2}{3} \] By simplifying \(-\frac{12}{18}\), you get \(-\frac{2}{3}\) which is an easier fraction to work with.

The Greatest Common Divisor (GCD) is an essential concept in simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD helps you simplify fractions effectively.
Here's how you can find the GCD of two numbers, for example, 12 and 18:

• List the factors of each number.
• Factors of 12: 1, 2, 3, 4, 6, 12.
• Factors of 18: 1, 2, 3, 6, 9, 18.

The common factors are 1, 2, 3, and 6. The greatest of these is 6, so the GCD of 12 and 18 is 6.
Using the GCD to simplify fractions is straightforward. Divide both the numerator and the denominator by their GCD. For \(-\frac{12}{18}\), the GCD is 6, and so when you divide both parts by 6, you get \(-\frac{2}{3}\).

Multiplying fractions is straightforward once you know the trick: multiply the numerators together and the denominators together. This concept is crucial for generating equivalent fractions, among other operations.
To create equivalent fractions, you multiply both the numerator and the denominator of the simplified fraction by the same integer. This keeps the value of the fraction the same.
For example, take \(-\frac{2}{3}\). To find equivalent fractions, you can multiply by 2, 3, or any other integer:

• Multiply by 2: \[ \frac{-2 \times 2}{3 \times 2} = -\frac{4}{6} \]
• Multiply by 3: \[ \frac{-2 \times 3}{3 \times 3} = -\frac{6}{9} \]
• Multiply by 4: \[ \frac{-2 \times 4}{3 \times 4} = -\frac{8}{12} \]

By multiplying both parts of the fraction by the same number, you keep the fraction equal in value to the original but in a different form.