In a hall, the average height of the boys is 180 cm and the average height of the girls is 150 cm. If the combined average height of the entire gathering is 165 cm and there are 78 boys present, how many girls are present in the hall?

Introduction:
This question deals with weighted averages of two groups, boys and girls, whose individual averages are known. You are asked to find the missing number of girls when the overall average height is given.

Given Data / Assumptions:
Average height of boys = 180 cm. Average height of girls = 150 cm. Combined average height of the gathering = 165 cm. Number of boys = 78. Number of girls = unknown (let this be g).

Concept / Approach:
We use the idea of weighted average. If one group has B members with average X and another group has G members with average Y, then the combined average is: Combined average = (B * X + G * Y) / (B + G). Here, B = 78, X = 180, Y = 150, and the combined average is 165. We form an equation in terms of G and solve it.

Step-by-Step Solution:
Step 1: Let the number of girls be g. Step 2: Total height of boys = 78 * 180. 78 * 180 = 14040 cm. Step 3: Total height of girls = g * 150. Step 4: Combined average is given as 165 cm, so (14040 + 150g) / (78 + g) = 165. Step 5: Cross multiply: 14040 + 150g = 165 * (78 + g). 14040 + 150g = 165 * 78 + 165g. 14040 + 150g = 12870 + 165g. 14040 - 12870 = 165g - 150g. 1170 = 15g. g = 1170 / 15 = 78.

Verification / Alternative Check:
If there are 78 boys and 78 girls, total persons = 156. Total height of boys = 14040 cm. Total height of girls = 78 * 150 = 11700 cm. Combined total = 14040 + 11700 = 25740 cm. Combined average = 25740 / 156 = 165 cm, which matches the given value, so the calculation is correct.

Why Other Options Are Wrong:
56, 64, 87, 92: Substituting any of these values for g in the weighted average formula does not give a combined average of 165 cm. Either the average comes out greater than 165 or less than 165.

Common Pitfalls:
A frequent mistake is to assume the number of girls is simply proportional to the averages without forming the correct equation. Another error is to attempt to average the averages directly. Always use the formula for combined average with total sums: total of boys plus total of girls divided by the total number of people.

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