When a bus started from the first stop, the number of male passengers to the number of female passengers was 3 : 1. At the stop 16 passengers get down and 6 more female passengers get into. Now the ratio of the male to female passengers becomes 2 : 1. What was the total number of passengers in the bus when it started from the first stop?

Options

A. 64
B. 48
C. 54
D. 72

Answer

To solve the problem step by step, we will denote the number of male passengers as \(3x\) and the number of female passengers as \(x\).

### Step 1: Establish Initial Conditions
The initial ratio of male to female passengers is given as \(3:1\). Therefore, we can express the number of male and female passengers as:
- Male passengers = \(3x\)
- Female passengers = \(x\)

**Total passengers when the bus started:**
\[ \text{Total passengers} = 3x + x = 4x \]

### Step 2: Account for Passengers Getting Down
At the next stop, 16 passengers get down. Since the ratio of male to female passengers is \(3:1\), we can assume that the number of male passengers getting down is \( \frac{3}{4} \times 16 = 12\) and the number of female passengers getting down is \( \frac{1}{4} \times 16 = 4\).

**Remaining passengers after 16 get down:**
- Remaining male passengers = \(3x - 12\)
- Remaining female passengers = \(x - 4\)

### Step 3: Account for New Female Passengers Getting In
After 16 passengers get down, 6 more female passengers get into the bus. Therefore, the new number of female passengers becomes:
\[ \text{New female passengers} = (x - 4) + 6 = x + 2 \]

### Step 4: Set Up the New Ratio
Now, the ratio of male to female passengers is given as \(2:1\). Thus, we can set up the equation:
\[ \frac{3x - 12}{x + 2} = 2 \]

### Step 5: Solve the Equation
Cross-multiplying gives us:
\[ 3x - 12 = 2(x + 2) \]
Expanding the right side:
\[ 3x - 12 = 2x + 4 \]
Now, isolate \(x\):
\[ 3x - 2x = 4 + 12 \]
\[ x = 16 \]

### Step 6: Calculate Total Passengers
Now that we have \(x\), we can find the total number of passengers when the bus started:
\[ \text{Total passengers} = 4x = 4 \times 16 = 64 \]

Thus, the total number of passengers in the bus when it started from the first stop is **64**.