Let's break down this problem about the ratios of people in blue and red dresses into smaller, manageable parts to find the ratio of men to women in the party hall.

We are given the following ratios:

• Ratio of men in blue dresses (M\_blue) to women in red dresses (W\_red) is 3 ∶ 7.
• Ratio of men in red dresses (M\_red) to women in blue dresses (W\_blue) is 2 ∶ 1.
• Ratio of total people in blue dresses (M\_blue + W\_blue) to total people in red dresses (M\_red + W\_red) is 35 ∶ 76.

We need to find the ratio of the total number of men (M\_blue + M\_red) to the total number of women (W\_blue + W\_red).

Setting Up Ratios with Variables

Let's represent the quantities using variables based on the given ratios:

• From the ratio M\_blue : W\_red = 3 : 7, we can write M\_blue = 3a and W\_red = 7a for some constant 'a'.
• From the ratio M\_red : W\_blue = 2 : 1, we can write M\_red = 2b and W\_blue = 1b (or just b) for some constant 'b'.

Now we have the number of men and women in each color dress in terms of 'a' and 'b':

• Men in blue (M\_blue) = $3a$
• Women in red (W\_red) = $7a$
• Men in red (M\_red) = $2b$
• Women in blue (W\_blue) = $b$

Using the Total People Ratio

The third ratio given is the ratio of people in blue to people in red, which is 35 ∶ 76.

• Total people in blue = M\_blue + W\_blue = $3a + b$
• Total people in red = M\_red + W\_red = $2b + 7a$

So, the ratio is:

$\frac{\text{Total people in blue}}{\text{Total people in red}} = \frac{3a + b}{2b + 7a} = \frac{35}{76}$

Solving for the Relationship between 'a' and 'b'

Now we can cross-multiply and solve the equation to find the relationship between the constants 'a' and 'b':

$76 \times (3a + b) = 35 \times (2b + 7a)$

Distribute the numbers on both sides:

$228a + 76b = 70b + 245a$

Gather the 'a' terms on one side and the 'b' terms on the other side:

$76b - 70b = 245a - 228a$

$6b = 17a$

This equation $6b = 17a$ tells us the relationship between 'a' and 'b'. We can write this as a ratio: $\frac{a}{b} = \frac{6}{17}$.

This means we can set $a = 6k$ and $b = 17k$ for some common constant $k$. Since we are dealing with ratios of numbers of people, $k$ must be a positive value.

Calculating the Number of People in Terms of k

Now substitute $a = 6k$ and $b = 17k$ back into the expressions for the number of people in each category:

• Men in blue (M\_blue) = $3a = 3(6k) = 18k$
• Women in red (W\_red) = $7a = 7(6k) = 42k$
• Men in red (M\_red) = $2b = 2(17k) = 34k$
• Women in blue (W\_blue) = $b = 1(17k) = 17k$

Let's verify the third ratio with these values:

• Total people in blue = $18k + 17k = 35k$
• Total people in red = $34k + 42k = 76k$

The ratio is $35k : 76k$, which simplifies to $35 : 76$. This matches the given information, confirming our calculations are correct so far.

Finding the Ratio of Men to Women

The final step is to find the ratio of the total number of men to the total number of women in the party hall.

• Total Men = M\_blue + M\_red = $18k + 34k = 52k$
• Total Women = W\_blue + W\_red = $17k + 42k = 59k$

The ratio of the number of men to the number of women is:

$\text{Total Men} : \text{Total Women} = 52k : 59k$

Since $k$ is a common positive constant, we can cancel it out:

Ratio of Men to Women = $52 : 59$

Summary of Quantities

The ratio of the number of men to the number of women is 52 : 59.

Revision Table: Understanding Ratios

Additional Information: Solving Ratio Problems

Ratio problems like this one often involve multiple ratios relating different groups. A common strategy is to introduce variables to represent the quantities based on the given ratios. If multiple variables (like 'a' and 'b' in our case) are introduced from different ratios, you usually need to use another piece of information (like the total ratio of blue to red dresses) to find a relationship between these variables. Once that relationship is found, all quantities can be expressed in terms of a single variable (like 'k'), allowing you to calculate the final required ratio.

• Always define your variables clearly (e.g., M\_blue for men in blue).
• Set up equations directly from the given ratios.
• Use additional information to find relationships between different variables used.
• Express all required quantities in terms of a single variable before calculating the final ratio.