A casino has launched an event for 5 days in which each day you can bet some money. The special thing about this offer is that it guarantees that at the end of the event you have won at least one day.

However the offer comes with some terms and conditions.

1) In order to participate you have to start from the 1st day itself.

2) Once you have won a reward you cannot participate for the next upcoming days The reward to money ratio for each day is given as

Day 1 2 3 4 5

Ratio 4 3 2 6 2

Your aim is to obtain a net profit of at least 120 rupees. Find the minimum number of rupees you should initially have to ensure that your

aim is achieved.

To find the minimum amount of money you should initially have to achieve a net profit of at least 120 rupees, we need to consider the given constraints and reward-to-money ratios for each day.

### Given Information:

- **Day 1**: Ratio = 4
- **Day 2**: Ratio = 3
- **Day 3**: Ratio = 2
- **Day 4**: Ratio = 6
- **Day 5**: Ratio = 2
- **Target Profit**: 120 rupees

### Approach:

Since you must start on Day 1 and can only continue betting on subsequent days if you lose on the previous days, we need to consider the sequential nature of the bets. The goal is to minimize the total initial money required while ensuring that a profit of 120 rupees is achieved if you win on any of the days.

#### Calculate the required bet for each day:
- **Day 1:** Profit = 4 × Bet\(_1\) ≥ 120  
  Minimum Bet\(_1\) = \( \frac{120}{4} = 30 \) rupees

- **Day 2:** Profit = 3 × Bet\(_2\) ≥ 120  
  Minimum Bet\(_2\) = \( \frac{120}{3} = 40 \) rupees

- **Day 3:** Profit = 2 × Bet\(_3\) ≥ 120  
  Minimum Bet\(_3\) = \( \frac{120}{2} = 60 \) rupees

- **Day 4:** Profit = 6 × Bet\(_4\) ≥ 120  
  Minimum Bet\(_4\) = \( \frac{120}{6} = 20 \) rupees

- **Day 5:** Profit = 2 × Bet\(_5\) ≥ 120  
  Minimum Bet\(_5\) = \( \frac{120}{2} = 60 \) rupees

### Strategy:
If you bet and lose on a day, you must have enough money left to bet the minimum amount on the next day. Hence, the strategy must account for the cumulative cost of bets if you do not win until the last possible day.

1. **Day 1:** Bet 30 rupees. If you win, you achieve the target, and you stop. If you lose, you continue.
2. **Day 2:** Bet 40 rupees. If you win, you achieve the target and stop. If you lose, you continue.
3. **Day 3:** Bet 60 rupees. If you win, you achieve the target and stop. If you lose, you continue.
4. **Day 4:** Bet 20 rupees. If you win, you achieve the target and stop. If you lose, you continue.
5. **Day 5:** Bet 60 rupees. If you win, you achieve the target and stop.

### Calculation of Total Initial Money:

- **If you lose on Day 1:** You must have 30 rupees for Day 1, 40 rupees for Day 2, 60 rupees for Day 3, 20 rupees for Day 4, and 60 rupees for Day 5.

  \( \text{Total} = 30 + 40 + 60 + 20 + 60 = 210 \) rupees.

### Conclusion:
The minimum initial money required to guarantee a net profit of at least 120 rupees is **210 rupees**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the strategy change if you are allowed to bet on any day regardless of previous outcomes?
2. What if the profit requirement was lower or higher?
3. How would the strategy be different if you knew the outcome of each day in advance?
4. Can you formulate a general rule for any similar situation with different ratios?
5. How does the risk change if you alter your betting order?

**Tip:** When betting sequentially, always ensure you have enough capital to sustain potential losses on each day until a guaranteed win.

Discover the minimum initial money required to ensure a net profit of at least 120 rupees in a casino event spanning 5 days with varying reward-to-money ratios each day. Learn the optimal betting strategy and calculations involved. Suitable for advanced high school students and beyond.

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