Problem Setup: Ratio of Girls to Boys

Let G8 and B8 be the number of girls and boys in Class VIII.

Let G9 and B9 be the number of girls and boys in Class IX.

We are given:

• Total students in Class VIII: $G8 + B8 = 450$
• Total students in Class IX: $B9 + G9 = 360$
• The ratio of girls to boys in Class VIII is the same as the ratio of boys to girls in Class IX: $\frac{G8}{B8} = \frac{B9}{G9}$
• The number of girls in both classes is the same: $G8 = G9$

We need to find the value of $G8$ (or $G9$). Let $G = G8 = G9$.

Calculating Student Numbers

From the total students, we can express the number of boys in terms of $G$:

• For Class VIII: $B8 = 450 - G8 = 450 - G$
• For Class IX: $B9 = 360 - G9 = 360 - G$

Solving the Ratio Equation

Substitute these expressions into the ratio equation $\frac{G8}{B8} = \frac{B9}{G9}$:

$ \frac{G}{450 - G} = \frac{360 - G}{G} $

Cross-multiply:

$ G \times G = (450 - G) \times (360 - G) $

$ G^2 = 450 \times 360 - 450G - 360G + G^2 $

$ G^2 = 162000 - 810G + G^2 $

Subtract $G^2$ from both sides:

$ 0 = 162000 - 810G $

Rearrange to solve for $G$:

$ 810G = 162000 $

$ G = \frac{162000}{810} $

$ G = \frac{16200}{81} $

$ G = 200 $

Conclusion

The number of girls in each class ($G8$ and $G9$) is 200.