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It might initially seem confusing that Derek’s [tex]\(\frac{1}{2}\)[/tex] piece of pie is smaller than Eduardo’s [tex]\(\frac{1}{6}\)[/tex] piece of pie, but the key lies in the size of the whole pies they come from. Let’s explore this scenario by considering the sizes of the whole pies.
### Step-by-Step Explanation:
1. Understand the Fractions:
- Eduardo has [tex]\(\frac{1}{6}\)[/tex] of a pie.
- Derek has [tex]\(\frac{1}{2}\)[/tex] of a pie.
2. Consider the Sizes of the Whole Pies:
- For Eduardo’s piece to be larger than Derek’s even though [tex]\(\frac{1}{6}\)[/tex] is numerically less than [tex]\(\frac{1}{2}\)[/tex], the whole pie from which Eduardo has a piece must be significantly larger than the whole pie from which Derek has a piece.
3. Compare the Sizes:
- Let’s say Eduardo’s whole pie is large. For instance, think of Eduardo’s pie as being two times larger than Derek’s whole pie. So, if Eduardo’s whole pie is of size 6 units, Derek’s whole pie could be of size 2 units.
4. Calculate the Actual Sizes of Their Pieces:
- Eduardo’s piece is [tex]\(\frac{1}{6}\)[/tex] of his pie. If his whole pie is 6 units, his piece would be [tex]\(6 \times \frac{1}{6} = 1\)[/tex] unit.
- Derek’s piece is [tex]\(\frac{1}{2}\)[/tex] of his pie. If his whole pie is 2 units, his piece would be [tex]\(2 \times \frac{1}{2} = 1\)[/tex] unit.
With these unit sizes, Eduardo and Derek's pieces are both 1 unit (no contradiction nor reason for different sizes yet by comparison). Let's assume a different proportion.
Reintroduction for larger than:[tex]\[ Assume Eduardo’s pie is 12 units, Derek’s pie is 3 units (thus properly reflecting the fraction in Eduardo's whole thus): - Eduardo’s piece is now \(= \frac{12}{6} = 2\) units. - Derek’s pie \(= \frac{3}{2} = 1.5\) units. 5. Conclusion: - By this comparison now reflecting the context correctly, Eduardo’s \(\frac{1}{6}\) units, indeed becomes \(2\) units in practice representation. - Derek’s \(\frac{1}{2}\) renders as \(1.5\) less overall, confirming the possible scenario from larger whole basis contextual. Drawing this scenario: - Eduardo’s Pie: \[ \begin{array}{l} \text{1 whole pie (12 units)} \\ [-] ______|\;\_\;\_\;_\;_6_separet space \_\;\_\;_\;\_\; | \end{array} \][/tex]
Eduardo’s [tex]\(\frac{12}{6}= 2\)[/tex]: Circled larger piece
- Derek’s Pie:
[tex]\[ \begin{array}{l} \text{1 whole pie (3 units)} \\ \_\;_ |-------_1.5 delim_ \end{array} \][/tex]
This demonstrates showing Derek’s pie being less within such provided slices context.
### Step-by-Step Explanation:
1. Understand the Fractions:
- Eduardo has [tex]\(\frac{1}{6}\)[/tex] of a pie.
- Derek has [tex]\(\frac{1}{2}\)[/tex] of a pie.
2. Consider the Sizes of the Whole Pies:
- For Eduardo’s piece to be larger than Derek’s even though [tex]\(\frac{1}{6}\)[/tex] is numerically less than [tex]\(\frac{1}{2}\)[/tex], the whole pie from which Eduardo has a piece must be significantly larger than the whole pie from which Derek has a piece.
3. Compare the Sizes:
- Let’s say Eduardo’s whole pie is large. For instance, think of Eduardo’s pie as being two times larger than Derek’s whole pie. So, if Eduardo’s whole pie is of size 6 units, Derek’s whole pie could be of size 2 units.
4. Calculate the Actual Sizes of Their Pieces:
- Eduardo’s piece is [tex]\(\frac{1}{6}\)[/tex] of his pie. If his whole pie is 6 units, his piece would be [tex]\(6 \times \frac{1}{6} = 1\)[/tex] unit.
- Derek’s piece is [tex]\(\frac{1}{2}\)[/tex] of his pie. If his whole pie is 2 units, his piece would be [tex]\(2 \times \frac{1}{2} = 1\)[/tex] unit.
With these unit sizes, Eduardo and Derek's pieces are both 1 unit (no contradiction nor reason for different sizes yet by comparison). Let's assume a different proportion.
Reintroduction for larger than:[tex]\[ Assume Eduardo’s pie is 12 units, Derek’s pie is 3 units (thus properly reflecting the fraction in Eduardo's whole thus): - Eduardo’s piece is now \(= \frac{12}{6} = 2\) units. - Derek’s pie \(= \frac{3}{2} = 1.5\) units. 5. Conclusion: - By this comparison now reflecting the context correctly, Eduardo’s \(\frac{1}{6}\) units, indeed becomes \(2\) units in practice representation. - Derek’s \(\frac{1}{2}\) renders as \(1.5\) less overall, confirming the possible scenario from larger whole basis contextual. Drawing this scenario: - Eduardo’s Pie: \[ \begin{array}{l} \text{1 whole pie (12 units)} \\ [-] ______|\;\_\;\_\;_\;_6_separet space \_\;\_\;_\;\_\; | \end{array} \][/tex]
Eduardo’s [tex]\(\frac{12}{6}= 2\)[/tex]: Circled larger piece
- Derek’s Pie:
[tex]\[ \begin{array}{l} \text{1 whole pie (3 units)} \\ \_\;_ |-------_1.5 delim_ \end{array} \][/tex]
This demonstrates showing Derek’s pie being less within such provided slices context.
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