A box contains 420 coins in 1 rupee, 50 paisa and 20 paisa coins, the ratio of their rupee values being 13: 11 : 7. The number of 50 paisa coins is

Options

A. a)42
B. b)78
C. c)66
D. d)132

Answer

To solve the problem, we need to find the number of 50 paisa coins in a box containing a total of 420 coins of 1 rupee, 50 paisa, and 20 paisa coins, with their rupee values in the ratio of 13:11:7.

### Step-by-Step Solution:

1. **Understanding the Value Ratios**:
The ratio of the rupee values of the coins is given as 13:11:7. This means:
- Let the number of 1 rupee coins be \( x \).
- Let the number of 50 paisa coins be \( y \).
- Let the number of 20 paisa coins be \( z \).

The values of these coins in rupees can be expressed as:
- Value of 1 rupee coins = \( 1 \cdot x = x \) rupees
- Value of 50 paisa coins = \( 0.5 \cdot y = 0.5y \) rupees
- Value of 20 paisa coins = \( 0.2 \cdot z = 0.2z \) rupees

According to the ratio provided:
\[ x : 0.5y : 0.2z = 13 : 11 : 7 \]

2. **Setting Up Equations**:
From the ratio, we can set up the following equations based on a common multiplier \( k \):
- \( x = 13k \)
- \( 0.5y = 11k \) → \( y = \frac{11k}{0.5} = 22k \)
- \( 0.2z = 7k \) → \( z = \frac{7k}{0.2} = 35k \)

3. **Total Number of Coins**:
The total number of coins is given as 420:
\[ x + y + z = 420 \]
Substituting the expressions for \( x \), \( y \), and \( z \):
\[ 13k + 22k + 35k = 420 \]
\[ 70k = 420 \]

4. **Solving for \( k \)**:
Dividing both sides by 70:
\[ k = \frac{420}{70} = 6 \]

5. **Finding the Number of Coins**:
Now we can find the number of each type of coin:
- Number of 1 rupee coins \( x = 13k = 13 \cdot 6 = 78 \)
- Number of 50 paisa coins \( y = 22k = 22 \cdot 6 = 132 \)
- Number of 20 paisa coins \( z = 35k = 35 \cdot 6 = 210 \)

6. **Conclusion**:
The number of 50 paisa coins is \( y = 132 \).

### Final Answer:
The number of 50 paisa coins is **132**.