The ratio of number of ladies to gents at a party was 1 : 2, but when 2 ladies and 2 gents left, the ratio became 1 : 3. How many people were originally present at the party?

Options

A. 6
B. 9
C. 12
D. 10

Answer

To solve the problem step by step, we will denote the number of ladies and gents at the party using a variable. Let's go through the solution:

### Step 1: Define the Variables
Let the number of ladies be represented as \( k \). Since the ratio of ladies to gents is 1:2, the number of gents will be \( 2k \).

### Step 2: Set Up the Initial Condition
From the problem, we know:
- Number of ladies = \( k \)
- Number of gents = \( 2k \)

### Step 3: Account for the Changes
When 2 ladies and 2 gents leave the party, the new number of ladies becomes \( k - 2 \) and the new number of gents becomes \( 2k - 2 \).

### Step 4: Set Up the New Ratio
According to the problem, after the departure of 2 ladies and 2 gents, the new ratio of ladies to gents becomes 1:3. This can be expressed as:
\[ \frac{k - 2}{2k - 2} = \frac{1}{3} \]

### Step 5: Cross-Multiply to Solve for \( k \)
Cross-multiplying gives us:
\[ 3(k - 2) = 1(2k - 2) \]
Expanding both sides:
\[ 3k - 6 = 2k - 2 \]

### Step 6: Rearrange the Equation
Now, we will rearrange the equation to isolate \( k \):
\[ 3k - 2k = 6 - 2 \]
This simplifies to:
\[ k = 4 \]

### Step 7: Calculate the Total Number of People
Now that we have \( k \), we can find the total number of people at the party:
- Total number of ladies = \( k = 4 \)
- Total number of gents = \( 2k = 2 \times 4 = 8 \)

Thus, the total number of people originally present at the party is:
\[ k + 2k = 4 + 8 = 12 \]

### Final Answer
Therefore, the total number of people originally present at the party is **12**.

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