Final answer:
Working backward from the final one coin Patrick has, after giving away coins to three students, it's determined that Patrick originally started with 15 euros.
Step-by-step explanation:
To solve this problem, we will work backward from what we know about the final distribution of coins. Patrick ends up with one coin after giving away parts of his original amount to three students. We'll use variables to represent the amount of coins after each distribution to the students. Let's call the final amount Patrick has C.
Following the problem's description:
- After giving coins to the youngest student, Patrick has 1 coin left.
- Before this, he gave half of his coins plus one to the youngest student. This means that twice the final amount (2C) plus one was the amount he had before giving to the youngest, so before giving to the youngest he had 2 × 1 + 1 = 3 coins.
- Applying the same logic to the second student, twice the amount before the youngest (2 × 3 + 1) was what he had before giving to the second student, which is 7 coins.
- Finally, he did the same with the eldest student, so before giving to the eldest, he had 2 × 7 + 1 = 15 coins.
Therefore, Patrick started with 15 euros.