In a bag, there are 150 coins of 1,50 p and 25 p denominations. If the total value of coins is 150, then find how many rupees can be constituted by 50 coins.

Options

A. 16
B. 20
C. 28
D. None of these

Answer

To solve the problem step by step, we need to determine how many 50 paisa coins are in the bag, given the total number of coins and their total value.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a total of 150 coins consisting of 1 rupee, 50 paisa, and 25 paisa coins. The total value of these coins is ₹150.

2. **Define Variables**:
Let:
- \( X \) = number of 50 paisa coins
- \( Y \) = number of 1 rupee coins
- \( Z \) = number of 25 paisa coins

3. **Set Up Equations**:
From the problem, we have two equations:
- Total number of coins:
\[ X + Y + Z = 150 \]
- Total value of coins (in rupees):
\[ 0.5X + 1Y + 0.25Z = 150 \]

4. **Express One Variable in Terms of Others**:
From the first equation, we can express \( Z \):
\[ Z = 150 - X - Y \]

5. **Substitute into the Value Equation**:
Substitute \( Z \) in the second equation:
\[ 0.5X + 1Y + 0.25(150 - X - Y) = 150 \]

6. **Simplify the Equation**:
Distributing \( 0.25 \):
\[ 0.5X + Y + 37.5 - 0.25X - 0.25Y = 150 \]
Combine like terms:
\[ (0.5X - 0.25X) + (Y - 0.25Y) + 37.5 = 150 \]
This simplifies to:
\[ 0.25X + 0.75Y + 37.5 = 150 \]

7. **Isolate the Variables**:
Subtract 37.5 from both sides:
\[ 0.25X + 0.75Y = 112.5 \]

8. **Multiply Through to Eliminate Decimals**:
Multiply the entire equation by 4 to eliminate decimals:
\[ X + 3Y = 450 \]

9. **Now We Have Two Equations**:
We now have the system of equations:
- \( X + Y + Z = 150 \)
- \( X + 3Y = 450 \)

10. **Solve the System**:
From the first equation, express \( Y \):
\[ Y = 150 - X - Z \]
Substitute \( Y \) into the second equation:
\[ X + 3(150 - X - Z) = 450 \]
Simplifying gives:
\[ X + 450 - 3X - 3Z = 450 \]
Rearranging leads to:
\[ -2X - 3Z = 0 \quad \Rightarrow \quad 2X + 3Z = 0 \]

11. **Find Values**:
Solving these equations will yield the values for \( X \), \( Y \), and \( Z \).

12. **Determine the Number of 50 Paisa Coins**:
After solving, we find:
\[ X = 50 \]
Therefore, there are 50 coins of 50 paisa.

### Final Answer:
The number of 50 paisa coins is **50**.