Two fractions can describe the same value when they are equivalent; this means that they represent the same amount despite having different numerators and denominators. Generally, two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if the cross-multiplication of their numerators and denominators is equal, i.e., \(a \cdot d = b \cdot c\).
Here's how you can understand this:
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Common Value: Both fractions provide a way to express the same quantity. For example, \(\frac{1}{2}\) and \(\frac{2}{4}\) both represent one half.
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Cross-Multiplication: To check if two fractions are equal, use cross-multiplication:
- For \(\frac{a}{b} = \frac{c}{d}\), check if \(a \cdot d = b \cdot c\). If they are equal, the fractions are equivalent.
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Simplifying Fractions: Any fraction can be simplified or expanded. For instance, \(\frac{3}{6}\) simplifies to \(\frac{1}{2}\) by dividing both the numerator and the denominator by 3. On the other hand, \(\frac{2}{4}\) also simplifies to \(\frac{1}{2}\).
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Real-World Example: Consider a pizza cut into 8 slices. If you eat 4 slices, you have eaten \(\frac{4}{8}\) of the pizza. If someone else eats 2 out of 4 slices, that's \(\frac{2}{4}\). Both amounts describe the same portion of the pizza, which is one half.
In summary, two fractions with different numerators and denominators can describe the same value when they are equivalent, meaning they simplify to the same fraction or represent the same ratio.