Find the number oif different sums that can be formed with one rupee, one half rupee and one quarter rupee coins.

Answer

To solve the problem of finding the number of different sums that can be formed with one rupee, one half rupee, and one quarter rupee coins, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coins**:
We have three types of coins:
- One rupee (1)
- Half rupee (0.5)
- Quarter rupee (0.25)

2. **Understand the Combinations**:
We can use any combination of these coins to form different sums. We can choose to use none, one, two, or all three coins.

3. **Use the Formula for Different Sums**:
The number of different sums that can be formed using \( n \) coins is given by the formula:
\[ 2^n - 1 \]
where \( n \) is the number of different types of coins. Here, \( n = 3 \).

4. **Calculate Using the Formula**:
Plugging in the value of \( n \):
\[ 2^3 - 1 = 8 - 1 = 7 \]
This means there are 7 different sums that can be formed.

5. **Verify by Listing Combinations**:
Let's list the possible combinations to ensure we have accounted for all sums:
- Using no coins: 0 (not counted)
- Using one coin:
- 1 (one rupee)
- 0.5 (half rupee)
- 0.25 (quarter rupee)
- Using two coins:
- 1 + 0.5 = 1.5
- 1 + 0.25 = 1.25
- 0.5 + 0.25 = 0.75
- Using all three coins:
- 1 + 0.5 + 0.25 = 1.75

The different sums we can form are:
- 0.25
- 0.5
- 0.75
- 1
- 1.25
- 1.5
- 1.75

This confirms that there are indeed 7 different sums.

### Conclusion:
The total number of different sums that can be formed with one rupee, one half rupee, and one quarter rupee coins is **7**.