- The fraction \(\frac{0}{6}\) has a value of 0.
- For any non-zero denominator \(D\), the equation \(\frac{0}{6} = \frac{n}{D}\) requires \(n = 0\).
- Applying this to the given denominators 12, 30, and 48, we find the corresponding numerators are all 0.
- The final answer is \(\boxed{0, 0, 0}\).
Explanation
- Analyzing the Problem
We are given the fraction \(\frac{0}{6}\). To find an equivalent fraction \(\frac{n}{D}\) where \(D\) is a new denominator, we rely on the property that two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if and only if \(a \cdot d = b \cdot c\). For the fraction \(\frac{0}{6}\), the value is \(0\) because \(0 \div 6 = 0\). Therefore, any fraction \(\frac{n}{D}\) equivalent to \(\frac{0}{6}\) must satisfy \(\frac{n}{D} = 0\), which implies that the numerator \(n\) must be \(0\) (as long as the denominator \(D
eq 0\)).
- Finding the Equivalent Fractions
To find the equivalent fraction for the denominator 12, we set \(\frac{0}{6} = \frac{n_1}{12}\). Multiplying both sides by 12, we get \(n_1 = 12 \times \frac{0}{6} = 12 \times 0 = 0\). Thus, \(\frac{0}{6} = \frac{0}{12}\). Similarly, for the denominator 30, we set \(\frac{0}{6} = \frac{n_2}{30}\), which gives \(n_2 = 30 \times 0 = 0\), so \(\frac{0}{6} = \frac{0}{30}\). Finally, for the denominator 48, we set \(\frac{0}{6} = \frac{n_3}{48}\), yielding \(n_3 = 48 \times 0 = 0\), so \(\frac{0}{6} = \frac{0}{48}\).
- Conclusion
We have determined that for any non-zero denominator \(D\), the fraction equivalent to \(\frac{0}{6}\) has a numerator of \(0\). Thus, the missing numerators for the denominators 12, 30, and 48 are all 0. The results are:
\(\frac{0}{6} = \frac{0}{12}\), \(\frac{0}{6} = \frac{0}{30}\), and \(\frac{0}{6} = \frac{0}{48}\). The requested values are \(\boxed{0, 0, 0}\).
Examples
In real life, if you have zero apples to divide among 6 people, each person gets 0 apples. If you have the same zero apples to divide among 12, 30, or 48 people, each person still gets 0 apples. This demonstrates that zero divided by any non-zero quantity is always zero, regardless of how many groups you are dividing it into.