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 solve this problem, we need to determine the minimum amount of money you should bet across the 5 days to ensure a profit of at least 120 rupees, given that the reward-to-money ratios for each day are known.

### 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:

1. **Identify the minimum bet required to achieve a profit of at least 120 rupees on each day:**

   Let \( x_i \) be the amount of money bet on day \( i \).
   - **Day 1:** Profit \( = 4x_1 \geq 120 \) \( \Rightarrow x_1 = \frac{120}{4} = 30 \) rupees
   - **Day 2:** Profit \( = 3x_2 \geq 120 \) \( \Rightarrow x_2 = \frac{120}{3} = 40 \) rupees
   - **Day 3:** Profit \( = 2x_3 \geq 120 \) \( \Rightarrow x_3 = \frac{120}{2} = 60 \) rupees
   - **Day 4:** Profit \( = 6x_4 \geq 120 \) \( \Rightarrow x_4 = \frac{120}{6} = 20 \) rupees
   - **Day 5:** Profit \( = 2x_5 \geq 120 \) \( \Rightarrow x_5 = \frac{120}{2} = 60 \) rupees

2. **Determine the minimum bet required:**

   To minimize the initial amount of money needed, we should aim to win on the day that requires the smallest bet.

   - **Day 4** requires the smallest bet: \( 20 \) rupees.

### Conclusion:
The minimum amount of money you should initially have to ensure a net profit of at least 120 rupees is **20 rupees**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What if the reward ratios change each day?
2. How would the strategy change if multiple days could be played?
3. What happens if the profit requirement increases to 200 rupees?
4. Can you determine the expected value of betting on a specific day?
5. How would the solution differ if there was a penalty for losing?

**Tip:** Always start by analyzing the smallest bet required to meet your goal, as it minimizes risk and cost.

Calculate the minimum initial bet required across 5 days at a casino event to ensure a net profit of at least 120 rupees. Learn the strategy and calculations needed to meet the profit goal based on daily reward-to-money ratios.

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