The ratio of the number of girls to boys in class VIII is the same as the ratio of the number of boys to girls in class IX. The total number of students (boys and girls) in classes... The ratio of the number of girls to boys in class VIII is the same as the ratio of the number of boys to girls in class IX. The total number of students (boys and girls) in classes VIII and IX is 450 and 360, respectively. If the number of girls in classes VIII and IX is the same, then the number of girls in each class is? Read more

Steps to Solve

1. Define Variables Let the number of girls in class VIII be $G_8$ and the number of boys be $B_8$. For class IX, let the number of girls be $G_9$ and the number of boys be $B_9$.

2. Set Up Ratios and Total Students From the problem, we know:

• The ratio of girls to boys in class VIII is the same as the ratio of boys to girls in class IX: $$ \frac{G_8}{B_8} = \frac{B_9}{G_9} $$
• The total number of students in class VIII is 450: $$ G_8 + B_8 = 450 $$
• The total number of students in class IX is 360: $$ G_9 + B_9 = 360 $$

3. Express Boys in Terms of Girls From the total in class VIII: $$ B_8 = 450 - G_8 $$ From the total in class IX: $$ B_9 = 360 - G_9 $$

4. Substitute and Rearrange the Ratio Substituting the expressions for $B_8$ and $B_9$ into the ratio equation: $$ \frac{G_8}{450 - G_8} = \frac{360 - G_9}{G_9} $$

5. Cross-Multiply Cross-multiplying gives us: $$ G_8 \cdot G_9 = (450 - G_8)(360 - G_9) $$

6. Expand and Rearrange Expanding the equation results in: $$ G_8 \cdot G_9 = 450 \cdot 360 - 450G_9 - 360G_8 + G_8G_9 $$ After simplifying, we find: $$ 0 = 450 \cdot 360 - 450G_9 - 360G_8 $$

7. Solve for One Variable Rearranging gives: $$ 360G_8 + 450G_9 = 450 \cdot 360 $$

8. Substitute G_9 in terms of G_8 Assume $G_9 = G_8$ (since the number of girls is the same in both classes): $$ 360G_8 + 450G_8 = 450 \times 360 $$ Combining gives: $$ 810G_8 = 450 \times 360 $$

9. Calculate G_8 Now, solve for $G_8$: $$ G_8 = \frac{450 \cdot 360}{810} = \frac{162000}{810} = 200 $$

10. Final Conclusion Since $G_8 = G_9$, the number of girls in each class is 200.