Solving Ratio Using Average Heights

This problem involves finding the ratio between the number of girls and boys based on their average heights and the overall class average height. We can solve this efficiently using the concept of alligation, which is derived from the weighted average formula.

Applying Alligation Method

1. Identify Given Averages
   • Average height of boys ($H_B$): 165 cm
   • Average height of girls ($H_G$): 150 cm
   • Average height of the entire class ($H_C$): 160 cm
2. Calculate Differences from Class Average

   Find the absolute difference between the average height of each group (boys and girls) and the average height of the entire class:

   • Difference for Boys = $|H_B - H_C| = |165 - 160| = 5$
   • Difference for Girls = $|H_G - H_C| = |150 - 160| = 10$
3. Determine Ratio of Numbers

   The ratio of the number of boys ($N_B$) to the number of girls ($N_G$) is inversely proportional to these differences:

   $ \frac{N_B}{N_G} = \frac{\text{Difference for Girls}}{\text{Difference for Boys}} $

   Substituting the values:

   $ \frac{N_B}{N_G} = \frac{10}{5} = \frac{2}{1} $

   This means the ratio of boys to girls ($N_B : N_G$) is 2 : 1.

4. Find Girls to Boys Ratio

   The question asks for the ratio of the number of girls to the number of boys ($N_G : N_B$). Since $N_B : N_G = 2 : 1$, we invert this ratio:

   $ N_G : N_B = 1 : 2 $

Therefore, the ratio of the number of girls to the number of boys in the class is 1 : 2.