A thief steals four gallons of liquid soap kept in a train compartment's bathroom from a container that is full of liquid soap. He then fills it with water to avoid detection. Unable to resist the temptation he steals 4 gallons of the mixture again, and fills it with water. When the liquid soap is checked at a station it is found that the ratio of the liquid soap now left in the container to the water in it is 36 : 13. What was the initial amount of the liquid soap in the container if it is known that the liquid soap is neither used nor augmented by anybody else during the entire period?

Options

A. 7 gallons
B. 14 gallons
C. 21 gallons
D. 28 gallons

Answer

To solve the problem step by step, we will denote the initial amount of liquid soap in the container as \( x \) gallons.

### Step 1: Understanding the situation
Initially, the container has \( x \) gallons of liquid soap. The thief steals 4 gallons of liquid soap, leaving \( x - 4 \) gallons of soap in the container. He then fills the container with 4 gallons of water.

**Hint:** Identify the initial amount and the amount stolen in the first instance.

### Step 2: After the first theft
After the first theft, the amount of liquid soap left is:
\[ \text{Liquid Soap after first theft} = x - 4 \text{ gallons} \]
The total volume in the container remains \( x \) gallons (since he fills it with water).

### Step 3: The second theft
The thief steals 4 gallons of the mixture (which now consists of both soap and water). The ratio of liquid soap to the total mixture before the second theft can be calculated. The total mixture is still \( x \) gallons, and the amount of liquid soap is \( x - 4 \) gallons.

**Hint:** Calculate the ratio of liquid soap to the total mixture before the second theft.

### Step 4: Calculate the ratio before the second theft
The ratio of liquid soap to the total mixture is:
\[ \text{Ratio of soap} = \frac{x - 4}{x} \]
The thief steals 4 gallons of this mixture. The amount of liquid soap in the 4 gallons stolen can be calculated as follows:
\[ \text{Liquid Soap in 4 gallons} = 4 \times \frac{x - 4}{x} \]
Thus, the amount of liquid soap remaining after the second theft is:
\[ \text{Liquid Soap after second theft} = (x - 4) - 4 \times \frac{x - 4}{x} \]

### Step 5: Simplifying the expression
Now, simplifying the expression for the remaining liquid soap:
\[ \text{Liquid Soap after second theft} = (x - 4) - \frac{4(x - 4)}{x} \]
\[ = (x - 4) - \left(4 - \frac{16}{x}\right) \]
\[ = x - 4 - 4 + \frac{16}{x} \]
\[ = x - 8 + \frac{16}{x} \]

### Step 6: Setting up the ratio after the second theft
After the second theft, the ratio of liquid soap to water is given as \( 36:13 \). This means:
\[ \frac{\text{Liquid Soap}}{\text{Water}} = \frac{36}{13} \]
Let’s denote the amount of water in the container after the second theft as \( W \):
\[ W = x - \text{Liquid Soap after second theft} \]
Using the total volume:
\[ W = x - \left(x - 8 + \frac{16}{x}\right) = 8 - \frac{16}{x} \]

### Step 7: Setting up the equation
Now we can set up the equation based on the ratio:
\[ \frac{x - 8 + \frac{16}{x}}{8 - \frac{16}{x}} = \frac{36}{13} \]

### Step 8: Cross-multiplying
Cross-multiplying gives:
\[ 13\left(x - 8 + \frac{16}{x}\right) = 36\left(8 - \frac{16}{x}\right) \]

### Step 9: Expanding and simplifying
Expanding both sides:
\[ 13x - 104 + \frac{208}{x} = 288 - \frac{576}{x} \]
Multiplying through by \( x \) to eliminate the fraction:
\[ 13x^2 - 104x + 208 = 288x - 576 \]

### Step 10: Rearranging the equation
Rearranging gives:
\[ 13x^2 - 392x + 784 = 0 \]

### Step 11: Solving the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{392 \pm \sqrt{(-392)^2 - 4 \cdot 13 \cdot 784}}{2 \cdot 13} \]
Calculating the discriminant and solving will yield the initial amount of liquid soap.

### Final Answer
After solving, we find that the initial amount of liquid soap in the container is:
\[ x = 28 \text{ gallons} \]