If the average height of boys in a hall is 180 cm and the average height of girls in the hall is 150 cm, and the combined average height of everyone present is 165 cm, then how many girls are present if there are 78 boys?

Introduction / Context:
This question tests the concept of a weighted average in the context of heights of boys and girls. The overall average is exactly midway between 180 cm and 150 cm, which hints at equal contributions from both groups. You are given the number of boys and asked to find the number of girls that makes the combined average equal to 165 cm.

Given Data / Assumptions:

• Average height of boys = 180 cm.
• Average height of girls = 150 cm.
• Combined average height of all people present = 165 cm.
• Number of boys = 78.
• We must find the number of girls.

Concept / Approach:
If we let the number of girls be G, then total height of boys is 180 * 78 and total height of girls is 150 * G. The overall average is 165, so the combined total height (180 * 78 + 150 * G) divided by (78 + G) must equal 165. Solving this equation gives G. Because the overall average lies exactly halfway between 180 and 150, we can intuitively expect the numbers of boys and girls to be equal, but we confirm this algebraically.

Step-by-Step Solution:
Let the number of girls be G.Total height of boys = 180 * 78.Total height of girls = 150 * G.Combined average height = 165, so (180 * 78 + 150 * G) / (78 + G) = 165.Multiply both sides by (78 + G): 180 * 78 + 150G = 165(78 + G).Compute left side constant part: 180 * 78 = 14040.Right side: 165 * 78 + 165G = 12870 + 165G.So the equation is 14040 + 150G = 12870 + 165G.Rearrange: 14040 - 12870 = 165G - 150G.1170 = 15G, so G = 1170 / 15 = 78.

Verification / Alternative check:
If there are 78 boys and 78 girls, total people = 156. Total boys' height = 180 * 78 = 14040. Total girls' height = 150 * 78 = 11700. Combined total height = 14040 + 11700 = 25740. Average height = 25740 / 156 = 165 cm, which matches the given combined average. This confirms that the number of girls is indeed 78.

Why Other Options Are Wrong:
Any number of girls different from 78 would shift the weighted average away from 165. More girls (who are shorter on average) would pull the average below 165, while fewer girls would push the average closer to 180. Options such as 56, 64 or 87 do not maintain the overall average at 165 when checked through the weighted average formula.

Common Pitfalls:
Some students mis-handle the algebra and get G wrong, or they attempt to guess the number of girls without checking the average. Others mistakenly try to average 180 and 150 without weighting by the numbers of boys and girls. Always write the equation for total height and solve it carefully to avoid these mistakes.

Final Answer:
The number of girls present in the hall is 78.