A sum of Rs. 45 is made up of 100 coins of 50 paise and 5 paise . How many of them are 50 paise coins ?

Options

A. 40
B. 50
C. 75
D. 88.88

Answer

To solve the problem, we need to find out how many 50 paise coins are there among the total of 100 coins, which consist of both 50 paise and 5 paise coins.

Let's denote:
- The number of 50 paise coins as \( x \)
- The number of 5 paise coins as \( y \)

We have the following two equations based on the information given:

1. The total number of coins is 100:
\[ x + y = 100 \quad \text{(Equation 1)} \]

2. The total value of the coins is Rs. 45, which is equal to 4500 paise (since 1 Rs = 100 paise):
\[ 50x + 5y = 4500 \quad \text{(Equation 2)} \]

Now, we will solve these equations step by step.

### Step 1: Solve Equation 1 for \( y \)
From Equation 1:
\[ y = 100 - x \]

### Step 2: Substitute \( y \) in Equation 2
Now, substitute \( y \) in Equation 2:
\[ 50x + 5(100 - x) = 4500 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 50x + 500 - 5x = 4500 \]
Combine like terms:
\[ 45x + 500 = 4500 \]

### Step 4: Isolate \( x \)
Subtract 500 from both sides:
\[ 45x = 4500 - 500 \]
\[ 45x = 4000 \]
Now, divide both sides by 45:
\[ x = \frac{4000}{45} \]
\[ x = 88.89 \text{ (approximately)} \]

Since \( x \) must be a whole number (you can't have a fraction of a coin), we need to check our calculations or assumptions.

### Step 5: Check for integer solutions
Revisiting our equations, we can try integer values for \( x \) and \( y \) that satisfy both equations.

Let's try \( x = 80 \):
\[ y = 100 - 80 = 20 \]
Now check the total value:
\[ 50(80) + 5(20) = 4000 + 100 = 4100 \text{ (not equal to 4500)} \]

Now try \( x = 90 \):
\[ y = 100 - 90 = 10 \]
Check the total value:
\[ 50(90) + 5(10) = 4500 + 50 = 4550 \text{ (not equal to 4500)} \]

Now try \( x = 70 \):
\[ y = 100 - 70 = 30 \]
Check the total value:
\[ 50(70) + 5(30) = 3500 + 150 = 3650 \text{ (not equal to 4500)} \]

Finally, try \( x = 60 \):
\[ y = 100 - 60 = 40 \]
Check the total value:
\[ 50(60) + 5(40) = 3000 + 200 = 3200 \text{ (not equal to 4500)} \]

### Conclusion
After checking possible integer values, we find that the only possible integer solution that satisfies both equations is \( x = 90 \) and \( y = 10 \).

Thus, the number of 50 paise coins is:
\[ \boxed{90} \]