Let's analyze each scenario one by one.
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Scenario 1: Complete \( \frac{3}{12} \) lessons today and \( \frac{4}{12} \) lessons per day for 7 days:
- Total completed today: \( \frac{3}{12} = 0.25 \)
- Total lessons over 7 days: \( 7 \times \frac{4}{12} = \frac{28}{12} \approx 2.33 \)
- Total so far: \( 0.25 + 2.33 = 2.58 \) (not enough)
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Scenario 2: Complete \( 4.5 \) lessons today and \( 4.5 \) lessons per day for 7 days:
- Total completed today: \( 4.5 \)
- Total lessons over 7 days: \( 7 \times 4.5 = 31.5 \)
- Total so far: \( 4.5 + 31.5 = 36 \) (too many)
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Scenario 3: Complete \( 4.5 \) lessons today and \( 3.5 \) lessons per day for 4 days:
- Total completed today: \( 4.5 \)
- Total lessons over 4 days: \( 4 \times 3.5 = 14 \)
- Total so far: \( 4.5 + 14 = 18.5 \) (too many)
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Scenario 4: Complete \( 4.5 \) lessons today and \( 3.5 \) lessons per day for 4 days:
- This is a repeat of Scenario 3.
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Scenario 5: Complete \( \frac{3}{12} \) lessons today and \( \frac{4}{12} \) lessons per day for 3 days:
- Total completed today: \( \frac{3}{12} = 0.25 \)
- Total lessons over 3 days: \( 3 \times \frac{4}{12} = 1 \)
- Total so far: \( 0.25 + 1 = 1.25 \) (not enough)
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Scenario 6: Complete \( 3.5 \) lessons today and \( 4.5 \) lessons per day for 3 days:
- Total completed today: \( 3.5 \)
- Total lessons over 3 days: \( 3 \times 4.5 = 13.5 \)
- Total so far: \( 3.5 + 13.5 = 17 \) (too many)
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Scenario 7: Complete \( 4.5 \) lessons today and \( 3.5 \) lessons per day for 3 days:
- Total completed today: \( 4.5 \)
- Total lessons over 3 days: \( 3 \times 3.5 = 10.5 \)
- Total so far: \( 4.5 + 10.5 = 15 \) (just right!)
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Scenario 8: Complete \( 4.5 \) lessons today and \( 3.5 \) lessons per day for 3 days:
- This is a repeat of Scenario 7.
The only valid scenario is Scenario 7 which states, "You need to complete 15 math lessons. If you complete \( 4.5 \) lessons today and \( 3.5 \) lessons per day after that, you can complete the lessons in 3 more days."