The ratio of number of male and female in a bus is `3 : 1. 16 `passengers got down and 6 females got into the bus, then ratio becomes `2 : 1.` Find the total number of passengers in the bus initially.

Options

A. 64
B. 59
C. 62
D. 63

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define Variables
Let the total number of passengers initially in the bus be \( x \).

### Step 2: Establish Initial Ratios
According to the problem, the ratio of males to females is \( 3:1 \). This means if we let the number of females be \( k \), then the number of males will be \( 3k \). Therefore, we can express the total number of passengers as:
\[ x = 3k + k = 4k \]

### Step 3: Account for Passengers Getting Down
When 16 passengers got down, the total number of passengers in the bus becomes:
\[ x - 16 \]
The number of females remains \( k \) since we do not know how many of the 16 passengers that got down were females.

### Step 4: Account for Females Getting In
After 16 passengers got down, 6 females got into the bus. Therefore, the new number of females in the bus becomes:
\[ k + 6 \]

### Step 5: Establish New Ratio
After these changes, the new ratio of males to females is \( 2:1 \). The number of males remains \( 3k \). Thus, we can set up the following equation based on the new ratio:
\[ \frac{3k}{k + 6} = 2 \]

### Step 6: Cross-Multiply to Solve for \( k \)
Cross-multiplying gives us:
\[ 3k = 2(k + 6) \]
Expanding this, we get:
\[ 3k = 2k + 12 \]
Subtracting \( 2k \) from both sides, we find:
\[ k = 12 \]

### Step 7: Calculate Total Passengers
Now that we have \( k \), we can find the total number of passengers initially:
\[ x = 4k = 4 \times 12 = 48 \]

### Step 8: Account for Passengers Getting Down
Since we need the total number of passengers before 16 got down, we add those back:
\[ x + 16 = 48 + 16 = 64 \]

### Final Answer
Thus, the total number of passengers in the bus initially is:
\[ \boxed{64} \]

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