Let's solve the problem step by step.

Question Overview:

We are tasked with two main parts:

1. Part (a): Find the ratio of the number of students in group $A$ to group $B$ to group $C$.
2. Part (b): Use the information provided to determine the total number of students in the camp if group $C$ has 38 fewer students than group $A$.

Step-by-Step Solution:

Part (a): Finding the ratio of $A : B : C$

1. Ratio from $A$ to $B + C$: The ratio of $A$ to $B + C$ is $4 : 5$. Let the number of students in: $A = 4x, \quad B + C = 5x$

2. Ratio from $C$ to $A + B$: The ratio of $C$ to $A + B$ is $2 : 3$. Let the number of students in: $C = 2y, \quad A + B = 3y$

3. Express $A + B$ from the first relationship: From the first ratio, $A = 4x$ and $B + C = 5x$. So, $A + B = 4x + B = 3y \quad \text{and} \quad C = 2y.$

4. Express $B$ and equate relationships: From $B + C = 5x$ and $C = 2y$: $B = 5x - 2y$

5. Combine to eliminate $y$: From $A + B = 3y$: $4x + (5x - 2y) = 3y$ Simplify: $9x = 5y$ Solve for $y$ in terms of $x$: $y = \frac{9x}{5}$

6. Substitute $y$ back to find $A : B : C$:

   • $A = 4x$
   • $B = 5x - 2y = 5x - 2\left(\frac{9x}{5}\right) = 5x - \frac{18x}{5} = \frac{25x - 18x}{5} = \frac{7x}{5}$
   • $C = 2y = 2\left(\frac{9x}{5}\right) = \frac{18x}{5}$

   The ratio of $A : B : C$ is: $A : B : C = 4x : \frac{7x}{5} : \frac{18x}{5}$ Multiply through by 5 to eliminate fractions: $A : B : C = 20 : 7 : 18$

Final Answer for (a):

$A : B : C = 20 : 7 : 18$

Part (b): Total number of students in the camp

1. Let the number of students in $A$, $B$, and $C$ be $20k$, $7k$, and $18k$ respectively (from the ratio $20 : 7 : 18$).

2. It is given that the number of students in $C$ is 38 less than the number of students in $A$: $18k = 20k - 38$ Solve for $k$: $18k - 20k = -38$ $-2k = -38 \quad \Rightarrow \quad k = 19$

3. Find the total number of students: $A = 20k = 20 \times 19 = 380, \quad B = 7k = 7 \times 19 = 133, \quad C = 18k = 18 \times 19 = 342$ Total students: $380 + 133 + 342 = 855$

Final Answer for (b):

The total number of students in the camp is: $855$

Summary of Answers:

1. Part (a): $A : B : C = 20 : 7 : 18$
2. Part (b): Total students = $855$

Would you like further clarifications or detailed explanations? 😊

Related Questions:

1. What would happen to the ratio $A : B : C$ if the number of students in group $C$ was halved?
2. Can we represent the problem graphically to visualize the ratios more effectively?
3. What mathematical principles are used to solve such ratio problems?
4. How do we verify that the ratios are consistent across all conditions?
5. What if the number of students in $B$ was given directly? How would it affect the calculations?

Tip:

When solving ratio problems, always align your terms by introducing variables systematically to avoid confusion!