- To find equivalent fractions, multiply the numerator and denominator by the same non-zero integer.
- For \(\frac{11}{12} = \frac{\square}{60}\), the missing value is \(\boxed{55}\).
- Other equivalent fractions are generated by scaling: \(\frac{-8}{13} \rightarrow \frac{-16}{26}, \frac{-24}{39}, \frac{-32}{52}\); \(\frac{0}{5} \rightarrow \frac{0}{1}, \frac{0}{10}, \frac{0}{2}\); \(\frac{a}{4} \rightarrow \frac{2a}{8}, \frac{3a}{12}, \frac{4a}{16}\).
- Final answer for \(\frac{11}{12} = \frac{\square}{60}\) is \(\boxed{55}\).
Explanation
- Analysis of Equivalent Fractions
To find equivalent fractions, we use the principle that multiplying the numerator and the denominator of a fraction by the same non-zero constant \(k\) results in an equivalent fraction. That is, for a fraction \(\frac{n}{d}\), \(\frac{n}{d} = \frac{n \times k}{d \times k}\). We will apply this to each case.
- Solving for \(\frac{11}{12}\)
For \(\frac{11}{12}\), we want to find three equivalent fractions, including the one with the denominator 60.
First, for \(\frac{11}{12} = \frac{\square}{60}\):
We observe that \(12 \times 5 = 60\). Thus, we multiply the numerator by 5 as well: \(11 \times 5 = 55\). So, \(\frac{11}{12} = \frac{55}{60}\).
Two other fractions equivalent to \(\frac{11}{12}\) are \(\frac{11 \times 2}{12 \times 2} = \frac{22}{24}\) and \(\frac{11 \times 3}{12 \times 3} = \frac{33}{36}\).
- Solving for \(\frac{-8}{13}\)
For \(\frac{-8}{13}\), we can choose any three constants, such as \(k=2, 3, 4\):
- For \(k=2\): \(\frac{-8 \times 2}{13 \times 2} = \frac{-16}{26}\)
- For \(k=3\): \(\frac{-8 \times 3}{13 \times 3} = \frac{-24}{39}\)
- For \(k=4\): \(\frac{-8 \times 4}{13 \times 4} = \frac{-32}{52}\)
Thus, three fractions equal to \(\frac{-8}{13}\) are \(\frac{-16}{26}, \frac{-24}{39}, \frac{-32}{52}\).
- Solving for \(\frac{0}{5}\)
For \(\frac{0}{5}\), we note that \(\frac{0}{d} = 0\) for any \(d
eq 0\). Therefore, any fraction of the form \(\frac{0}{d}\) (where \(d
eq 0\)) is equivalent to \(\frac{0}{5}\). Three such fractions are \(\frac{0}{1}, \frac{0}{10}, \frac{0}{2}\).
- Solving for \(\frac{a}{4}\)
For \(\frac{a}{4}\), we multiply both the numerator and denominator by non-zero constants \(k=2, 3, 4\):
- For \(k=2\): \(\frac{a \times 2}{4 \times 2} = \frac{2a}{8}\)
- For \(k=3\): \(\frac{a \times 3}{4 \times 3} = \frac{3a}{12}\)
- For \(k=4\): \(\frac{a \times 4}{4 \times 4} = \frac{4a}{16}\)
Thus, three fractions equal to \(\frac{a}{4}\) are \(\frac{2a}{8}, \frac{3a}{12}, \frac{4a}{16}\).
Examples
Equivalent fractions are essential in everyday life when measuring quantities or adjusting recipes. For example, if a recipe calls for \(\frac{1}{2}\) cup of flour, but your measuring cup only measures in fourths, you can use the equivalent fraction \(\frac{2}{4}\) to correctly measure the amount. Understanding these relationships allows us to work flexibly with different units and measurements, ensuring precision in cooking, construction, or budget calculations.