Ram ordered for 6 black toys and some additional brown toys. The price of a black toy is 2`1/2` times that of a brown toy. While preparing the bill, the clerk interchanged the number of black toys with the number of brown toys which increased the bill by 45%. The number of brown toys is:

Options

A. 8
B. 6
C. 15
D. 12

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Define Variables
Let the price of one brown toy be \( P \). According to the problem, the price of one black toy is \( 2 \frac{1}{2} \) times the price of a brown toy. This can be expressed as:
\[ \text{Price of one black toy} = \frac{5}{2} P \]

### Step 2: Calculate the Total Price for Black and Brown Toys
Ram ordered 6 black toys and \( x \) brown toys. Therefore, the total price for the black toys is:
\[ \text{Total price for black toys} = 6 \times \frac{5}{2} P = 15P \]
The total price for the brown toys is:
\[ \text{Total price for brown toys} = x \times P = xP \]

### Step 3: Calculate the Original Bill
The original bill (before the interchange) is:
\[ \text{Original Bill} = 15P + xP = (15 + x)P \]

### Step 4: Calculate the New Bill After Interchanging
After the clerk interchanged the number of toys, the new bill becomes:
\[ \text{New Bill} = x \times \frac{5}{2} P + 6 \times P = \frac{5}{2} x P + 6P \]

### Step 5: Set Up the Equation for the Increase in Bill
According to the problem, the new bill is 45% more than the original bill. Therefore, we can set up the equation:
\[ \text{New Bill} = \text{Original Bill} + 0.45 \times \text{Original Bill} \]
This can be simplified to:
\[ \text{New Bill} = 1.45 \times \text{Original Bill} \]
Substituting the expressions for the bills:
\[ \frac{5}{2} x P + 6P = 1.45 \times (15 + x)P \]

### Step 6: Simplify the Equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{5}{2} x + 6 = 1.45(15 + x) \]
Expanding the right side:
\[ \frac{5}{2} x + 6 = 21.75 + 1.45x \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ \frac{5}{2} x - 1.45x = 21.75 - 6 \]
Converting \( \frac{5}{2} \) to decimal gives \( 2.5 \):
\[ 2.5x - 1.45x = 15.75 \]
This simplifies to:
\[ 1.05x = 15.75 \]

### Step 8: Solve for \( x \)
Dividing both sides by \( 1.05 \):
\[ x = \frac{15.75}{1.05} = 15 \]

### Conclusion
The number of brown toys is \( \boxed{15} \).

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