A box has 210 coins of denominations one-rupee and fifty paise only. The ratio of their respective values is 13:11. The number of one-rupee coins is

Options

A. 65
B. 66
C. 77
D. 78

Answer

To solve the problem, we need to find the number of one-rupee coins in a box containing 210 coins of denominations one-rupee and fifty paise, given that the ratio of their respective values is 13:11.

### Step-by-Step Solution:

1. **Define Variables**:
Let the number of one-rupee coins be \( x \) and the number of fifty paise coins be \( y \).

2. **Set Up the Equations**:
From the problem, we know:
\[ x + y = 210 \quad \text{(1)} \]
The value of one-rupee coins is \( 1 \times x = x \) rupees.
The value of fifty paise coins is \( 0.5 \times y = \frac{y}{2} \) rupees.
The ratio of their values is given as:
\[ \frac{x}{\frac{y}{2}} = \frac{13}{11} \quad \text{(2)} \]

3. **Simplify the Value Ratio**:
From equation (2), we can cross-multiply:
\[ 11x = 13 \times \frac{y}{2} \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 22x = 13y \quad \text{(3)} \]

4. **Substitute Equation (1) into Equation (3)**:
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 210 - x \]
Substitute this into equation (3):
\[ 22x = 13(210 - x) \]

5. **Solve for \( x \)**:
Distributing on the right side:
\[ 22x = 2730 - 13x \]
Adding \( 13x \) to both sides:
\[ 22x + 13x = 2730 \]
\[ 35x = 2730 \]
Dividing both sides by 35:
\[ x = \frac{2730}{35} = 78 \]

6. **Conclusion**:
The number of one-rupee coins is \( x = 78 \).

### Final Answer:
The number of one-rupee coins is **78**.