The average age of a class is 19 years. The average age of the girls is 18 years and the average age of the boys is 21 years. If the number of girls in the class is 16, how many boys are there in the class?

Introduction / Context:
This question involves combining two groups (boys and girls) with different averages to produce an overall average for the entire class. You must determine how many boys are in the class using the given averages and the known number of girls. This is an application of weighted averages and total sums.

Given Data / Assumptions:

• Average age of the whole class = 19 years.

• Average age of girls = 18 years.

• Average age of boys = 21 years.

• Number of girls = 16.

• Let the number of boys be B.

Concept / Approach:
We express total ages in terms of the averages and numbers. Total age of girls = 18 * 16. Total age of boys = 21 * B. Total number of students = 16 + B. The overall average age is given as 19, so (total age of girls + total age of boys) divided by total number of students must equal 19. Setting up this equation allows us to solve for B, the number of boys in the class.

Step-by-Step Solution:
Total age of girls = 18 * 16 = 288 years. Total age of boys = 21 * B years. Total students = 16 + B. Overall average age = (288 + 21B) / (16 + B) = 19. Multiply both sides: 288 + 21B = 19 * (16 + B). Right side = 19 * 16 + 19B = 304 + 19B. So, 288 + 21B = 304 + 19B. Rearrange: 21B − 19B = 304 − 288. 2B = 16, hence B = 8.

Verification / Alternative check:
With 8 boys, total students = 16 + 8 = 24. Total age of girls = 288 years, total age of boys = 21 * 8 = 168 years. Combined total age = 288 + 168 = 456 years. Overall average age = 456 / 24 = 19 years, which matches the given average. This confirms that the number of boys is correctly found as 8.

Why Other Options Are Wrong:
If there were 12, 10 or 6 boys, the total ages and resulting average would change. For example, with 10 boys, total boys' age = 210 and overall total = 288 + 210 = 498, giving an average of 498 / 26, which is not 19. Only 8 boys yield a combined average age of exactly 19 years as required.

Common Pitfalls:
Some students attempt to use a shortcut ratio method but misapply it due to the presence of three averages. Others miscalculate the product terms when forming the equation. Writing down the total sums clearly and solving the linear equation step by step helps avoid such errors. Always verify by recomputing the overall average with the found value of B.

Final Answer:
There are 8 boys in the class.