The ratio of the number of ladles to that of gents at a party was `3 : 2`. When 20 more gents joined the party, the ratio was reversed. The number of ladles present at the party was:

Options

A. 36
B. 32
C. 24
D. 16

Answer

To solve the problem step by step, we will denote the number of ladies as \( L \) and the number of gents as \( G \).

### Step 1: Set up the initial ratio
The problem states that the ratio of the number of ladies to gents is \( 3:2 \). This can be expressed as:
\[ \frac{L}{G} = \frac{3}{2} \]
From this, we can express \( L \) in terms of \( G \):
\[ L = \frac{3}{2}G \]

### Step 2: Account for the change in the number of gents
When 20 more gents join the party, the number of gents becomes \( G + 20 \). The new ratio of ladies to gents is reversed, which means it becomes \( 2:3 \). We can express this new ratio as:
\[ \frac{L}{G + 20} = \frac{2}{3} \]
From this, we can express \( L \) in terms of \( G + 20 \):
\[ L = \frac{2}{3}(G + 20) \]

### Step 3: Set the two expressions for \( L \) equal to each other
Now we have two expressions for \( L \):
1. \( L = \frac{3}{2}G \)
2. \( L = \frac{2}{3}(G + 20) \)

We can set these equal to each other:
\[ \frac{3}{2}G = \frac{2}{3}(G + 20) \]

### Step 4: Solve for \( G \)
To eliminate the fractions, we can multiply both sides by 6 (the least common multiple of 2 and 3):
\[ 6 \cdot \frac{3}{2}G = 6 \cdot \frac{2}{3}(G + 20) \]
This simplifies to:
\[ 9G = 4(G + 20) \]
Expanding the right side:
\[ 9G = 4G + 80 \]
Now, isolate \( G \):
\[ 9G - 4G = 80 \]
\[ 5G = 80 \]
\[ G = 16 \]

### Step 5: Find the number of ladies \( L \)
Now that we have \( G \), we can find \( L \) using the first equation:
\[ L = \frac{3}{2}G = \frac{3}{2} \times 16 = 24 \]

### Final Answer
The number of ladies present at the party was \( 24 \).

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