If we add three times the number of boys in a class of a school to half the number of girls, we get the same result as if we add the number of boys to twice the number of girls in the same class. Find the number of boys and girls if you also know that the total number of children in the class is from 25 to 30.

Let's denote the number of boys in the class by \( b \) and the number of girls by \( g \).

The problem gives us two key pieces of information:

1. Three times the number of boys plus half the number of girls is equal to the number of boys plus twice the number of girls:
   \[
   3b + \frac{1}{2}g = b + 2g
   \]

2. The total number of children in the class (boys plus girls) is between 25 and 30:
   \[
   b + g = n \quad \text{where } 25 \leq n \leq 30
   \]

### Solving the Equations

**Step 1:** Simplify the first equation.
\[
3b + \frac{1}{2}g = b + 2g
\]
Subtract \( b \) from both sides:
\[
2b + \frac{1}{2}g = 2g
\]
Multiply through by 2 to eliminate the fraction:
\[
4b + g = 4g
\]
Subtract \( g \) from both sides:
\[
4b = 3g
\]
Divide both sides by 3:
\[
b = \frac{3}{4}g
\]

**Step 2:** Substitute \( b = \frac{3}{4}g \) into the second equation \( b + g = n \).
\[
\frac{3}{4}g + g = n
\]
Combine like terms:
\[
\frac{7}{4}g = n
\]
Multiply both sides by 4:
\[
7g = 4n
\]
Divide both sides by 7:
\[
g = \frac{4n}{7}
\]

Since \( g \) and \( n \) must be integers, \( 4n \) must be divisible by 7.

Let's check for \( n \) values between 25 and 30:

- **\( n = 28 \)** is the only value that makes \( g \) an integer:
  \[
  g = \frac{4 \times 28}{7} = 16
  \]

Given \( g = 16 \), we find \( b \):
\[
b = \frac{3}{4} \times 16 = 12
\]

So, the class has **12 boys** and **16 girls**.

### Verification
- Total students: \( b + g = 12 + 16 = 28 \) (within the 25 to 30 range).
- Check the initial condition:
  \[
  3b + \frac{1}{2}g = 3(12) + \frac{1}{2}(16) = 36 + 8 = 44
  \]
  \[
  b + 2g = 12 + 2(16) = 12 + 32 = 44
  \]
Both conditions match, confirming our solution is correct.

Would you like to see any additional details, or have any questions?

---

1. What other values of \( n \) could be checked to verify the solution?
2. How does the requirement \( 4n \) be divisible by 7 ensure \( g \) is an integer?
3. Can the same approach be used for other values of total children?
4. What would happen if the total number of children was less than 25?
5. Is there another way to set up the equations to solve this problem?
6. What is the significance of \( b = \frac{3}{4}g \) in the solution process?
7. Could this problem be solved graphically?
8. How can we interpret the relationship between boys and girls in terms of ratios?

**Tip:** Always check for divisibility when dealing with fractions in equations, especially when integers are required as solutions.

Explore how to solve a problem involving the number of boys and girls in a school class, where specific equations are manipulated to find the correct solution. This problem involves key algebraic concepts such as solving linear equations and manipulating fractions. Suitable for students in Grades 7-9.

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