In a conference hall there are people in blue and yellow dresses. The ratio of the number of women in blue to the number of men in yellow is 3 : 2 and the ratio of the number of men in blue to the number of women in yellow is 3 : 5. If the ratio of the number of people in blue to the number of people in yellow is 21 : 23, then what is the ratio of the number of men to the number of women in the conference hall?

Options

A. `19 : 21`
B. `21 : 29`
C. `17 : 27`
D. Cannot be determined

Answer

To solve the problem step by step, we need to define the variables based on the given ratios and then set up equations to find the required ratio of men to women in the conference hall.

### Step 1: Define the Variables
Let:
- The number of women in blue dresses = \(3x\)
- The number of men in yellow dresses = \(2x\)
- The number of men in blue dresses = \(3y\)
- The number of women in yellow dresses = \(5y\)

### Step 2: Set Up the Equations
From the problem, we know:
1. The ratio of the number of people in blue to the number of people in yellow is \(21:23\).
- Therefore, the total number of people in blue = \(3x + 3y\)
- The total number of people in yellow = \(2x + 5y\)
- This gives us the equation:
\[ \frac{3x + 3y}{2x + 5y} = \frac{21}{23} \]

### Step 3: Cross Multiply to Eliminate the Fraction
Cross-multiplying gives:
\[ 23(3x + 3y) = 21(2x + 5y) \]
Expanding both sides:
\[ 69x + 69y = 42x + 105y \]

### Step 4: Rearrange the Equation
Rearranging the equation:
\[ 69x - 42x = 105y - 69y \]
This simplifies to:
\[ 27x = 36y \]
Dividing both sides by 9:
\[ 3x = 4y \quad \text{(Equation 1)} \]

### Step 5: Set Up the Second Equation
From the ratio of the total number of people in blue to yellow, we have:
\[ 2x + 5y = 23 \]
Using \(x + y = 7\) (which we derived from the first equation), we can express \(x\) in terms of \(y\):
\[ x = 7 - y \]

### Step 6: Substitute and Solve
Substituting \(x = 7 - y\) into \(2x + 5y = 23\):
\[ 2(7 - y) + 5y = 23 \]
Expanding gives:
\[ 14 - 2y + 5y = 23 \]
Combining like terms:
\[ 14 + 3y = 23 \]
Subtracting 14 from both sides:
\[ 3y = 9 \]
Dividing by 3:
\[ y = 3 \]

### Step 7: Find \(x\)
Substituting \(y = 3\) back into \(x + y = 7\):
\[ x + 3 = 7 \implies x = 4 \]

### Step 8: Calculate the Total Number of Men and Women
Now we can find the total number of men and women:
- Total men = \(2x + 3y = 2(4) + 3(3) = 8 + 9 = 17\)
- Total women = \(3x + 5y = 3(4) + 5(3) = 12 + 15 = 27\)

### Step 9: Find the Ratio of Men to Women
The ratio of the number of men to the number of women is:
\[ \text{Ratio} = \frac{\text{Total Men}}{\text{Total Women}} = \frac{17}{27} \]

### Final Answer
The ratio of the number of men to the number of women in the conference hall is \(17:27\).

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