In a bag, there are `50` paise coins, Rs `1` coins, and Rs `2` coins. The total value of these coins is Rs `30`. The number of Rs `2` coins is half the number of Rs `1` coins, which is half the number of `50` paise coins. Find the number of Rs `1`coins.

Options

A. `20`
B. `16`
C. `15`
D. `10`

Explanation

To solve the problem, we will define variables for the number of coins and set up equations based on the information given.

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( a \) be the number of 50 paise coins.
- Let \( b \) be the number of Rs 1 coins.
- Let \( c \) be the number of Rs 2 coins.

2. **Set Up the Equations:**
- The total value of the coins is Rs 30. The value can be expressed in terms of the number of coins:
\[ \frac{50}{100}a + 1b + 2c = 30 \]
- Simplifying the equation:
\[ 0.5a + b + 2c = 30 \]

3. **Relate the Number of Coins:**
- According to the problem, the number of Rs 2 coins is half the number of Rs 1 coins:
\[ c = \frac{b}{2} \]
- Also, the number of Rs 1 coins is half the number of 50 paise coins:
\[ b = \frac{a}{2} \]
- From this, we can express \( a \) in terms of \( b \):
\[ a = 2b \]

4. **Substitute the Values:**
- Substitute \( a \) and \( c \) in the total value equation:
\[ 0.5(2b) + b + 2\left(\frac{b}{2}\right) = 30 \]
- Simplifying this gives:
\[ b + b + b = 30 \]
- Therefore:
\[ 3b = 30 \]

5. **Solve for \( b \):**
- Divide both sides by 3:
\[ b = 10 \]

6. **Conclusion:**
- The number of Rs 1 coins is \( b = 10 \).