To solve this problem, we need to determine the distribution of students across the different houses and destinations (Museum and Red Fort), taking into account the given ratios of males to females.

1. Distribute Total Students Among Houses

We know:

• Total students across all houses: $800$
• Students in the Red House: $350$

Let the number of students in the Blue House be $x$, and in the Green House be $y$. Thus, we have: $350 + x + y = 800$ $x + y = 450 \quad \text{(Equation 1)}$

2. Distribute Students Traveling to Museum

We know:

• Blue House has $150$ students traveling to the Museum.
• The total number of males and females from Red and Green houses together is equal.
• The Red House has an equal number of males and females.

Let:

• Number of students traveling to the Museum from Red House be $r_m$.
• Number of students traveling to the Museum from Green House be $g_m$.

Thus:

• $r_m$ males and $r_m$ females travel from the Red House (since males = females in Red House).
• Let $m_g$ be the number of males traveling from the Green House and $f_g$ be the number of females.

The equation for gender equality in Red and Green Houses traveling to the Museum is: $2r_m + m_g + f_g \quad \text{(total males and females from Red and Green Houses to Museum)}$

Since the total number of males and females are equal: $2r_m + m_g + f_g = 2r_m + 2g_m$

3. Distribute Students Traveling to Red Fort

• The Green House has $120$ students traveling to the Red Fort.
• The number of students from the Blue House traveling to the Red Fort is $120/2 = 60$.
• Let $r_f$ be the number of students traveling to the Red Fort from the Red House.

From the problem statement: $r_f = 0.6 \times \text{Total number of students traveling to the Red Fort}$

Let the total number of students traveling to the Red Fort be $T$: $T = 60 + 120 + r_f$ $r_f = 0.6 \times T$

Substituting $r_f = 0.6T$: $T = 60 + 120 + 0.6T$ $T - 0.6T = 180$ $0.4T = 180 \quad \Rightarrow \quad T = 450$

Thus, $r_f = 0.6 \times 450 = 270$.

4. Ratio of Males to Females

Given:

• Red House: $\frac{3}{2}$
• Blue House: $\frac{1}{1}$
• Green House: $\frac{5}{1}$

Calculate the number of males and females using the ratios and distribution calculated earlier.

Since the number of males and females in the Red House is equal, $r_m$ can be computed by: $\frac{3}{2} \times \text{Number of Females} = \text{Number of Males}$

And similar calculations apply to the other houses.

Summary

With the provided details:

• The number of students per house traveling to Museum/Red Fort.
• Gender ratio.
• Total students per house.

These details can now be computed accurately and broken down according to the given ratios and constraints.

Would you like the detailed calculations for each step or any further clarifications?

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Related Questions

1. What would happen if the total number of students in the school changed?
2. How does changing the ratio of males to females in one house affect the others?
3. What if the number of students traveling to the Museum changes?
4. How would the distribution change if another destination were added?
5. What if the ratio in the Green House changed to $4:1$?
6. How do you calculate the exact number of students per house for both destinations?
7. What if the students from the Blue House had a different travel destination?
8. How does changing the total number of students in the Blue House affect the distribution?

Tip: When solving problems with multiple constraints, breaking down the problem into smaller parts and solving step by step ensures accuracy.