A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair. While preparing the bill, the bill clerk interchanged the number of black and brown pairs by mistake which increased the bill by 100%. What was the number of pairs of brown socks in the original order?

The correct option is 25.

Let us break down the problem step-by-step to understand how we arrive at this answer.

Step 1: Define the variables for the quantities and prices of the socks.
Let the number of pairs of brown socks ordered be $x$.
The number of pairs of black socks ordered is given as $5$.
Let the price of a single pair of brown socks be $p$.
Since the price of a black pair is thrice that of a brown pair, the price of a single pair of black socks is $3p$.

Step 2: Calculate the original bill amount.
The cost of $5$ pairs of black socks is $5\times 3p=15p$.
The cost of $x$ pairs of brown socks is $x\times p=xp$.
Therefore, the original total bill is:
$B_{\text{original}}=15p+xp=p(15+x)$

Step 3: Calculate the incorrect bill amount.
The clerk interchanged the quantities by mistake.
This means the clerk billed $x$ pairs of black socks and $5$ pairs of brown socks.
The cost of $x$ pairs of black socks is $x\times 3p=3xp$.
The cost of $5$ pairs of brown socks is $5\times p=5p$.
Therefore, the incorrect total bill is:
$B_{\text{incorrect}}=3xp+5p=p(3x+5)$

Step 4: Use the percentage increase relationship to solve for $x$.
We are told that the mistake increased the bill by 100%. An increase of 100% means the new bill is double the original bill:
$B_{\text{incorrect}}=2\times B_{\text{original}}$
Substitute the expressions we found into this relation:
$p(3x+5)=2\times p(15+x)$

Since the price $p$ is a non-zero positive value, we can divide both sides by $p$ to simplify the equation:
$3x+5=2(15+x)$
Expand the right side:
$3x+5=30+2x$
Subtract $2x$ from both sides:
$x+5=30$
Subtract $5$ from both sides:
$x=25$

Thus, the number of pairs of brown socks in the original order was 25.