- Fractions are equivalent if one is a scaled version of the other by a common non-zero integer factor.
- For \(\frac{11}{12}\), three equivalent fractions are \(\frac{22}{24}, \frac{33}{36}, \frac{55}{60}\); the missing numerator is \(\boxed{33}\).
- For \(\frac{-8}{13}\), three equivalent fractions are \(\frac{-16}{26}, \frac{-24}{39}, \frac{-32}{52}\).
- For \(\frac{0}{5}\), three equivalent fractions are \(\frac{0}{10}, \frac{0}{15}, \frac{0}{20}\).
- For \(\frac{a}{4}\), three equivalent fractions are \(\frac{2a}{8}, \frac{3a}{12}, \frac{4a}{16}\).
Explanation
- Understanding the Fundamental Property of Fractions
To find equivalent fractions for any given fraction \(\frac{n}{d}\), we use the fundamental property that multiplying both the numerator and the denominator by the same non-zero integer \(k\) produces an equivalent fraction. That is, \(\frac{n}{d} = \frac{n \times k}{d \times k}\). We will apply this principle to find three equivalent fractions for each of the given expressions.
- Equivalent Fractions for \(\frac{11}{12}\)
For the fraction \(\frac{11}{12}\), we want to find three equivalent fractions.
- By multiplying by \(k=2\): \(\frac{11 \times 2}{12 \times 2} = \frac{22}{24}\).
- By multiplying by \(k=3\): \(\frac{11 \times 3}{12 \times 3} = \frac{33}{36}\).
- By multiplying by \(k=5\): \(\frac{11 \times 5}{12 \times 5} = \frac{55}{60}\).
The requested specific fraction \(\frac{11}{12} = \frac{?}{36}\) corresponds to our calculation with \(k=3\), so the numerator is 33. Thus, \(\frac{11}{12} = \frac{33}{36}\).
- Equivalent Fractions for \(\frac{-8}{13}\)
For the fraction \(\frac{-8}{13}\), we choose three multipliers:
- By multiplying by \(k=2\): \(\frac{-8 \times 2}{13 \times 2} = \frac{-16}{26}\).
- By multiplying by \(k=3\): \(\frac{-8 \times 3}{13 \times 3} = \frac{-24}{39}\).
- By multiplying by \(k=4\): \(\frac{-8 \times 4}{13 \times 4} = \frac{-32}{52}\).
These are three fractions equal to \(\frac{-8}{13}\).
- Equivalent Fractions for \(\frac{0}{5}\)
For the fraction \(\frac{0}{5}\), any multiplication by a non-zero integer \(k\) results in:
- \(\frac{0 \times 2}{5 \times 2} = \frac{0}{10}\).
- \(\frac{0 \times 3}{5 \times 3} = \frac{0}{15}\).
- \(\frac{0 \times 4}{5 \times 4} = \frac{0}{20}\).
All these fractions are equal to \(\frac{0}{5} = 0\).
- Equivalent Fractions for \(\frac{a}{4}\) and Final Conclusion
For the expression \(\frac{a}{4}\), where \(a\) is a non-zero constant, we multiply by integers \(k=2, 3, 4\):
- By multiplying by \(k=2\): \(\frac{a \times 2}{4 \times 2} = \frac{2a}{8}\).
- By multiplying by \(k=3\): \(\frac{a \times 3}{4 \times 3} = \frac{3a}{12}\).
- By multiplying by \(k=4\): \(\frac{a \times 4}{4 \times 4} = \frac{4a}{16}\).
These are three fractions equivalent to \(\frac{a}{4}\). The numerator for \(\frac{11}{12} = \frac{\boxed{33}}{36}\) is 33.
Examples
Equivalent fractions are useful in many everyday tasks, such as cooking and construction. For example, if a recipe calls for \(\frac{3}{4}\) of a cup of sugar, but you only have a \(\frac{1}{8}\)-cup measuring scoop, you can use the property of equivalent fractions to find that \(\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}\). By using your \(\frac{1}{8}\) scoop six times, you measure the correct amount for your recipe. This demonstrates how simple mathematical principles help us adjust proportions and solve practical problems with the tools we have on hand.