In a drawer that contains only black, blue, and brown socks, the probability of selecting a black pair of socks is `(3)/(8)`, and there are `(1)/(3)` as many blue pairs of socks as there are black pairs of socks. If there are 12 brown pairs of socks, how many pairs of socks are there in the drawer?

Options

A. `16`
B. `24`
C. `32`
D. `48`

Answer

To solve the problem step by step, we will define variables for the number of pairs of socks and use the information given in the problem to find the total number of pairs of socks in the drawer.

### Step 1: Define Variables
Let:
- \( BK \) = number of black pairs of socks
- \( BL \) = number of blue pairs of socks
- \( BR \) = number of brown pairs of socks

From the problem, we know:
- \( BR = 12 \) (there are 12 brown pairs of socks)

### Step 2: Relate Blue and Black Socks
It is given that there are \( \frac{1}{3} \) as many blue pairs of socks as there are black pairs of socks. This can be expressed as:
\[ BL = \frac{1}{3} BK \]

### Step 3: Use Probability Information
The probability of selecting a black pair of socks is given as \( \frac{3}{8} \). The probability of selecting a pair of socks is defined as:
\[ P(BK) = \frac{BK}{\text{Total pairs of socks}} \]
Thus, we can write:
\[ \frac{BK}{\text{Total pairs of socks}} = \frac{3}{8} \]

Let \( T \) be the total number of pairs of socks. Therefore:
\[ BK = \frac{3}{8} T \]

### Step 4: Substitute for Blue Socks
Using the relationship between blue and black socks:
\[ BL = \frac{1}{3} BK = \frac{1}{3} \left(\frac{3}{8} T\right) = \frac{1}{8} T \]

### Step 5: Write the Total Pairs of Socks
Now we can express the total number of pairs of socks \( T \) in terms of \( BK \), \( BL \), and \( BR \):
\[ T = BK + BL + BR \]
Substituting the expressions we have:
\[ T = BK + \frac{1}{8} T + 12 \]

### Step 6: Substitute for \( BK \)
Now substitute \( BK = \frac{3}{8} T \) into the equation:
\[ T = \frac{3}{8} T + \frac{1}{8} T + 12 \]

### Step 7: Combine Like Terms
Combine the terms on the right side:
\[ T = \left(\frac{3}{8} + \frac{1}{8}\right) T + 12 \]
\[ T = \frac{4}{8} T + 12 \]
\[ T = \frac{1}{2} T + 12 \]

### Step 8: Isolate \( T \)
To isolate \( T \), subtract \( \frac{1}{2} T \) from both sides:
\[ T - \frac{1}{2} T = 12 \]
\[ \frac{1}{2} T = 12 \]

### Step 9: Solve for \( T \)
Multiply both sides by 2:
\[ T = 24 \]

Thus, the total number of pairs of socks in the drawer is \( \boxed{24} \).