Ram bought 6 black and X white balls. The price of a black ball is `5//2` of a white balls's price . On the time of billing, clerk by mistake inter-changes the number of both black and white balls. Then bill increases by `45%` . Find the value of X ?

Options

A. `15`
B. `20`
C. `25`
D. `30`

Answer

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

### Step 1: Define the Variables
Let the price of a white ball be \( P \). Then, the price of a black ball is given as \( \frac{5}{2}P \).

### Step 2: Calculate the Original Bill
Ram bought 6 black balls and \( X \) white balls. The original bill can be calculated as:
\[ \text{Original Bill} = \text{(Number of Black Balls)} \times \text{(Price of Black Ball)} + \text{(Number of White Balls)} \times \text{(Price of White Ball)} \]
Substituting the values:
\[ \text{Original Bill} = 6 \times \left(\frac{5}{2}P\right) + X \times P = 15P + XP = (15 + X)P \]

### Step 3: Calculate the Wrong Bill
Due to the clerk's mistake, the number of black balls is now \( X \) and the number of white balls is 6. The wrong bill is:
\[ \text{Wrong Bill} = X \times \left(\frac{5}{2}P\right) + 6 \times P = \frac{5}{2}XP + 6P = \left(\frac{5}{2}X + 6\right)P \]

### Step 4: Set Up the Equation for the Increase in Bill
According to the problem, the wrong bill is 45% more than the original bill. Therefore, we can set up the equation:
\[ \text{Wrong Bill} = \text{Original Bill} + 0.45 \times \text{Original Bill} \]
This can be rewritten as:
\[ \text{Wrong Bill} = 1.45 \times \text{Original Bill} \]
Substituting our expressions for the bills:
\[ \left(\frac{5}{2}X + 6\right)P = 1.45 \times (15 + X)P \]

### Step 5: 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 6: Rearranging the Equation
Now, let's rearrange the equation to isolate \( X \):
\[ \frac{5}{2}X - 1.45X = 21.75 - 6 \]
Calculating the right side:
\[ \frac{5}{2}X - 1.45X = 15.75 \]
Converting \( \frac{5}{2} \) to decimal gives \( 2.5 \):
\[ 2.5X - 1.45X = 15.75 \]
This simplifies to:
\[ 1.05X = 15.75 \]

### Step 7: Solve for \( X \)
Now, divide both sides by \( 1.05 \):
\[ X = \frac{15.75}{1.05} = 15 \]

### Final Answer
Thus, the value of \( X \) is \( 15 \).

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