In a class, the average age of the boys is 16 years and the average age of the girls is 15 years. What is the average age of the whole class?

Introduction / Context:
This question is designed to test your understanding of how averages work when combining two subgroups into one larger group. We know the average ages of boys and girls separately, and we are asked to find the average age of the whole class. However, the problem contains a subtle but important missing piece of information, which makes it a test of logical reasoning as well as basic arithmetic.

Given Data / Assumptions:

• The average age of the boys in the class is 16 years.
• The average age of the girls in the class is 15 years.
• The total number of students in the class is not given.
• The numbers of boys and girls are also not given.
• We are asked to find the average age of all students taken together.

Concept / Approach:
The overall average of a combined group depends on both the subgroup averages and the sizes of those subgroups. In mathematical terms, if there are b boys with average 16 and g girls with average 15, then the overall average A is given by:
A = (16b + 15g) / (b + g). Without knowing either b or g, we cannot simplify this expression to a single unique number. Different values of b and g will generally produce different overall averages, even though the subgroup averages remain the same.

Step-by-Step Solution:
Step 1: Assume the number of boys is b and the number of girls is g. Step 2: Total age of all boys = 16b years. Step 3: Total age of all girls = 15g years. Step 4: Total age of the whole class = 16b + 15g. Step 5: Total number of students = b + g. Step 6: Overall average age A = (16b + 15g) / (b + g). Step 7: Without a specific relation between b and g, A cannot be simplified to a single fixed number. Step 8: Choose two different combinations, for example b = 1, g = 1 and b = 10, g = 1, to see that the resulting overall averages are different.

Verification / Alternative check:
Take b = 1, g = 1. The overall average = (16 * 1 + 15 * 1) / (1 + 1) = 31 / 2 = 15.5 years. Now take b = 10, g = 1. The overall average = (16 * 10 + 15 * 1) / 11 = (160 + 15) / 11 = 175 / 11 ≈ 15.91 years. Since both situations respect the given subgroup averages but produce different class averages, there is no unique answer for the average age of the whole class.

Why Other Options Are Wrong:
The value 15 years would be the overall average only if there were no boys, which contradicts the given condition that there are boys with average 16 years. The values 20 and 21 years are clearly impossible because they exceed the maximum subgroup average of 16 years. The value 16 would be possible only if there were no girls, which again contradicts the given information. Therefore none of these numerical options can be guaranteed as the class average.

Common Pitfalls:
A very common mistake is to simply average the two subgroup averages, for example taking (16 + 15) / 2 = 15.5, without considering the relative sizes of the subgroups. This is incorrect unless the number of boys and girls is the same, which is not stated in the problem. Another mistake is to assume that the overall average must be somewhere between 15 and 16 and then to guess a value, which is not logically justified. Always remember that to compute a combined average you must know subgroup sizes or have enough information to deduce them.

Final Answer:
Since the numbers of boys and girls are not known, the overall average age of the whole class cannot be uniquely determined. The correct response is data inadequate.