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equivalent fractions

Equivalent fractions are fractions that have different numerators and denominators but represent the same value. For instance, \frac{1}{2}, \frac{2}{4}, and \frac{4}{8} are all equivalent because they each represent the same proportion, one-half.

Creating Equivalent Fractions

You can generate equivalent fractions by either multiplying or dividing the numerator and denominator of a fraction by the same non-zero number. This process doesn’t change the overall value of the fraction because you are essentially multiplying or dividing by 1 (e.g., \frac{2}{2} = 1).

Multiplication

To find an equivalent fraction, you can multiply the numerator and the denominator by the same number.

Example:
To find fractions equivalent to \frac{3}{4}:

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

Thus, \frac{3}{4}, \frac{6}{8}, and \frac{9}{12} are equivalent fractions.

Division

To simplify a fraction to its simplest equivalent form, you divide both the numerator and the denominator by their greatest common factor (GCF).

Example:
Let’s simplify the fraction \frac{18}{24}.

The greatest common factor of 18 and 24 is 6.

  • Divide the numerator and denominator by 6:
    \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

Therefore, \frac{18}{24} is equivalent to \frac{3}{4}.

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