A man ordered 4 pairs of black socks and some pairs of brown socks. The price of a black pair is double that of a brown pair. While preparing the bill, the clerk did a mistake and interchanged the number of black and brown pairs. This increased the bill by 50%. The ratio of the number of black and brown pairs of socks in the original order was

Options

A. 4:1
B. 2:1
C. 1:4
D. 1:2

Answer

To solve the problem step by step, we will follow the logic laid out in the video transcript.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the number of pairs of brown socks ordered be \( x \).
- Let the price of one pair of brown socks be \( y \).
- Therefore, the price of one pair of black socks will be \( 2y \) (since the price of a black pair is double that of a brown pair).

2. **Calculate the Original Bill**:
- The original order consists of 4 pairs of black socks and \( x \) pairs of brown socks.
- The total cost for the black socks is \( 4 \times 2y = 8y \).
- The total cost for the brown socks is \( x \times y = xy \).
- Thus, the original bill (OB) can be expressed as:
\[ OB = 8y + xy \]

3. **Calculate the Interchanged Bill**:
- Due to the clerk's mistake, the number of black and brown pairs was interchanged.
- Now, the order consists of \( x \) pairs of black socks and 4 pairs of brown socks.
- The total cost for the new order of black socks is \( x \times 2y = 2xy \).
- The total cost for the new order of brown socks is \( 4 \times y = 4y \).
- Thus, the interchanged bill (IB) can be expressed as:
\[ IB = 2xy + 4y \]

4. **Set Up the Equation**:
- According to the problem, the interchanged bill is 50% more than the original bill. Therefore:
\[ IB = 1.5 \times OB \]
- Substituting the expressions for OB and IB:
\[ 2xy + 4y = 1.5(8y + xy) \]

5. **Simplify the Equation**:
- Expanding the right side:
\[ 2xy + 4y = 12y + 1.5xy \]
- Rearranging gives:
\[ 2xy - 1.5xy + 4y - 12y = 0 \]
- This simplifies to:
\[ 0.5xy - 8y = 0 \]

6. **Factor Out Common Terms**:
- Factoring out \( y \):
\[ y(0.5x - 8) = 0 \]
- Since \( y \neq 0 \) (as there is a price), we have:
\[ 0.5x - 8 = 0 \]
- Solving for \( x \):
\[ 0.5x = 8 \implies x = 16 \]

7. **Find the Ratio of Black to Brown Socks**:
- The original order had 4 pairs of black socks and \( x = 16 \) pairs of brown socks.
- Therefore, the ratio of the number of black to brown pairs is:
\[ \text{Ratio} = \frac{4}{16} = \frac{1}{4} \]

### Final Answer:
The ratio of the number of black and brown pairs of socks in the original order is \( \frac{1}{4} \).