A bag contains 50 P, 25 P and 10 P coins in the ratio 5: 9:4. amounting to Rs. 206. Find the number of coins of each type.

Answer

To solve the problem step by step, we will follow the given information and use the concept of ratios and proportions.

### Step 1: Understand the Ratio
The ratio of the number of coins of 50 P, 25 P, and 10 P is given as 5:9:4. Let's denote:
- Number of 50 P coins = 5x
- Number of 25 P coins = 9x
- Number of 10 P coins = 4x

### Step 2: Calculate the Total Value of Coins
Next, we need to calculate the total value contributed by each type of coin:
- Value of 50 P coins = Number of coins × Value of each coin = \(5x \times 0.50\) (in Rs) = \(2.5x\) Rs
- Value of 25 P coins = \(9x \times 0.25\) (in Rs) = \(2.25x\) Rs
- Value of 10 P coins = \(4x \times 0.10\) (in Rs) = \(0.4x\) Rs

### Step 3: Set Up the Equation for Total Value
According to the problem, the total value of the coins is Rs. 206. Therefore, we can set up the equation:
\[ 2.5x + 2.25x + 0.4x = 206 \]

### Step 4: Combine Like Terms
Now, we combine the terms on the left side:
\[ (2.5 + 2.25 + 0.4)x = 206 \]
\[ 5.15x = 206 \]

### Step 5: Solve for x
To find the value of \(x\), we divide both sides by 5.15:
\[ x = \frac{206}{5.15} \approx 40 \]

### Step 6: Calculate the Number of Coins
Now that we have \(x\), we can find the number of coins of each type:
- Number of 50 P coins = \(5x = 5 \times 40 = 200\)
- Number of 25 P coins = \(9x = 9 \times 40 = 360\)
- Number of 10 P coins = \(4x = 4 \times 40 = 160\)

### Final Answer
The number of coins of each type is:
- 50 P coins: 200
- 25 P coins: 360
- 10 P coins: 160