- Simplify the fraction \(\frac{4}{6}\) by dividing both parts by 2 to get \(\frac{2}{3}\).
- Find a common denominator of 9 to compare \(\frac{7}{9}\) and \(\frac{2}{3}\), which results in \(\frac{7}{9}\) and \(\frac{6}{9}\).
- Compare the numerators: since 7 is greater than 6, \(\frac{7}{9} > \frac{6}{9}\).
- Conclude that Ben's piece is larger: \(\boxed{\text{Ben}}\).
Explanation
- Problem Analysis
We are given two identical sandwiches, one shared with Ben and one shared with Corey. Ben receives \(\frac{7}{9}\) of his sandwich, and Corey receives \(\frac{4}{6}\) of his. We need to determine which friend received a larger piece by comparing the two fractions \(\frac{7}{9}\) and \(\frac{4}{6}\).
- Simplifying the Fractions
To compare \(\frac{4}{6}\) with \(\frac{7}{9}\), we first simplify \(\frac{4}{6}\). Dividing both the numerator and the denominator by their greatest common divisor, which is 2, we get:
\[\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}.\]
Now we compare \(\frac{7}{9}\) and \(\frac{2}{3}\).
- Comparing the Fractions
To compare \(\frac{7}{9}\) and \(\frac{2}{3}\), we find a common denominator. The least common multiple of 9 and 3 is 9. We can convert \(\frac{2}{3}\) to have a denominator of 9 by multiplying both the numerator and denominator by 3:
\[\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}.\]
Now the comparison is between \(\frac{7}{9}\) and \(\frac{6}{9}\). Since \(7 > 6\), it follows that \(\frac{7}{9} > \frac{6}{9}\).
- Conclusion
Since \(\frac{7}{9} > \frac{6}{9}\) and \(\frac{6}{9}\) is equivalent to Corey's share of \(\frac{4}{6}\), we conclude that Ben's piece is larger than Corey's piece.
Thus, Ben received a bigger piece. \(\boxed{\text{Ben}}\)
Examples
Comparing fractions is a fundamental skill in everyday life, especially when dealing with food, measurements, or time. For example, if you are baking and a recipe calls for \(\frac{3}{4}\) cup of flour, but you only have a \(\frac{2}{3}\) cup measuring tool, knowing which amount is larger helps you adjust your ingredient quantities accurately to ensure your baking turns out perfectly. Mastering this skill helps you make better decisions in cooking, construction, and budgeting.