- Use common denominators to compare fractions: 1. \(\frac{2}{3} > \frac{1}{4}\), 2. \(\frac{6}{8} > \frac{1}{3}\), 3. \(\frac{4}{6} > \frac{2}{4}\), 4. \(\frac{3}{6} < \frac{5}{9}\), 5. \(\frac{1}{5} < \frac{3}{10}\), 6. \(\frac{6}{8} > \frac{2}{3}\), 7. \(\frac{5}{8} < \frac{3}{4}\), 8. \(\frac{4}{10} = \frac{2}{5}\), 9. \(\frac{2}{5} > \frac{5}{15}\), 10. \(\frac{5}{6} > \frac{2}{3}\), 11. \(\frac{1}{4} = \frac{2}{8}\), 12. \(\frac{1}{3} < \frac{2}{5}\), 13. \(\frac{2}{7} < \frac{3}{9}\), 14. \(\frac{3}{5} > \frac{1}{4}\), 15. \(\frac{6}{8} = \frac{3}{4}\), 16. \(\frac{4}{5} > \frac{6}{8}\), 17. \(\frac{3}{4} > \frac{5}{7}\), 18. \(\frac{5}{8} < \frac{7}{9}\).
- Dominic's logic is not necessarily flawed because fractions represent portions of different "wholes". If the cake is sufficiently larger than the cupcake, \(\frac{3}{8}\) of the cake can be more than \(\frac{2}{3}\) of the cupcake.
- Final answers: 1. \(>\), 2. \(>\), 3. \(>\), 4. \(<\), 5. \(<\), 6. \(>\), 7. \(<\), 8. \(=\), 9. \(>\), 10. \(>\), 11. \(=\), 12. \(<\), 13. \(<\), 14. \(>\), 15. \(=\), 16. \(>\), 17. \(>\), 18. \(<\). Dominic's logic is not necessarily flawed.
Explanation
- Comparing Fractions Strategy
To compare two fractions, we can find a common denominator or use a common numerator. A common denominator is the least common multiple of the two denominators. For example, to compare \(\frac{2}{3}\) and \(\frac{1}{4}\), the least common multiple of 3 and 4 is 12. We convert \(\frac{2}{3} = \frac{8}{12}\) and \(\frac{1}{4} = \frac{3}{12}\). Since \(8 > 3\), we conclude \(\frac{2}{3} > \frac{1}{4}\). We will apply this method to all pairs.
- Step-by-Step Comparisons
Let's compare the provided fractions using common denominators:
- \(\frac{2}{3} = \frac{8}{12}\), \(\frac{1}{4} = \frac{3}{12} \implies \frac{2}{3} > \frac{1}{4}\)
- \(\frac{6}{8} = \frac{18}{24}\), \(\frac{1}{3} = \frac{8}{24} \implies \frac{6}{8} > \frac{1}{3}\)
- \(\frac{4}{6} = \frac{8}{12}\), \(\frac{2}{4} = \frac{6}{12} \implies \frac{4}{6} > \frac{2}{4}\)
- \(\frac{3}{6} = \frac{9}{18}\), \(\frac{5}{9} = \frac{10}{18} \implies \frac{3}{6} < \frac{5}{9}\)
- \(\frac{1}{5} = \frac{2}{10}\), \(\frac{3}{10} = \frac{3}{10} \implies \frac{1}{5} < \frac{3}{10}\)
- \(\frac{6}{8} = \frac{18}{24}\), \(\frac{2}{3} = \frac{16}{24} \implies \frac{6}{8} > \frac{2}{3}\)
- \(\frac{5}{8} = \frac{5}{8}\), \(\frac{3}{4} = \frac{6}{8} \implies \frac{5}{8} < \frac{3}{4}\)
- \(\frac{4}{10} = \frac{2}{5}\), \(\frac{2}{5} = \frac{2}{5} \implies \frac{4}{10} = \frac{2}{5}\)
- \(\frac{2}{5} = \frac{6}{15}\), \(\frac{5}{15} = \frac{5}{15} \implies \frac{2}{5} > \frac{5}{15}\)
- \(\frac{5}{6} = \frac{5}{6}\), \(\frac{2}{3} = \frac{4}{6} \implies \frac{5}{6} > \frac{2}{3}\)
- \(\frac{1}{4} = \frac{2}{8}\), \(\frac{2}{8} = \frac{2}{8} \implies \frac{1}{4} = \frac{2}{8}\)
- \(\frac{1}{3} = \frac{5}{15}\), \(\frac{2}{5} = \frac{6}{15} \implies \frac{1}{3} < \frac{2}{5}\)
- \(\frac{2}{7} = \frac{6}{21}\), \(\frac{3}{9} = \frac{7}{21} \implies \frac{2}{7} < \frac{3}{9}\)
- \(\frac{3}{5} = \frac{12}{20}\), \(\frac{1}{4} = \frac{5}{20} \implies \frac{3}{5} > \frac{1}{4}\)
- \(\frac{6}{8} = \frac{3}{4}\), \(\frac{3}{4} = \frac{3}{4} \implies \frac{6}{8} = \frac{3}{4}\)
- \(\frac{4}{5} = \frac{16}{20}\), \(\frac{6}{8} = \frac{15}{20} \implies \frac{4}{5} > \frac{6}{8}\)
- \(\frac{3}{4} = \frac{21}{28}\), \(\frac{5}{7} = \frac{20}{28} \implies \frac{3}{4} > \frac{5}{7}\)
- \(\frac{5}{8} = \frac{45}{72}\), \(\frac{7}{9} = \frac{56}{72} \implies \frac{5}{8} < \frac{7}{9}\)
- Evaluating Dominic's Statement
Dominic's logic involves comparing \(\frac{2}{3}\) of a cupcake with \(\frac{3}{8}\) of a cake. A fraction is always a portion of a specified "whole". If the whole of the cake is significantly larger than the whole of the cupcake, then \(\frac{3}{8}\) of the cake could easily represent a larger amount than \(\frac{2}{3}\) of the cupcake. Therefore, Dominic's logic is not necessarily flawed; it depends entirely on the size of the objects. If he prefers the portion of the cake because he knows the cake's size is large enough to make that portion greater, his reasoning is sound.
Examples
Imagine you are a chef deciding how to serve desserts. You have different sized containers for cake and muffins. To ensure customers get consistent portions, you must understand that \(\frac{1}{2}\) of a small muffin is much less than \(\frac{1}{4}\) of a large whole cake. When comparing fractions, always remember to verify that they refer to the same whole, or adjust your expectations based on the actual size of the objects. This principle is vital in cooking, construction, and resource management.