The average age of all students in a class is 15.8 years. The average age of boys alone is 16.4 years and the average age of girls alone is 15.4 years. What is the ratio of the number of boys to the number of girls in the class?

Introduction / Context:
This question involves weighted averages. The overall average age of a group is given, along with the averages of two subgroups (boys and girls). The task is to find the ratio of the number of boys to girls. Such problems are common in data interpretation and arithmetic sections, and they check whether the student can connect the combined average with the weighted contributions from subgroups.

Given Data / Assumptions:

• Average age of the entire class = 15.8 years.
• Average age of boys = 16.4 years.
• Average age of girls = 15.4 years.
• Let the number of boys be B and the number of girls be G.
• We assume all averages are correctly computed from actual ages.

Concept / Approach:
The overall average is a weighted average of subgroup averages. Using total age of boys as 16.4 * B and total age of girls as 15.4 * G, the overall mean is (16.4 * B + 15.4 * G) / (B + G) = 15.8. This gives an equation connecting B and G. Solving for the ratio B : G gives the required answer. This is a standard technique whenever an overall average and subgroup averages are known.

Step-by-Step Solution:
Let B = number of boys and G = number of girls.Total age of boys = 16.4 * B.Total age of girls = 15.4 * G.Total number of students = B + G, and overall average = 15.8.So (16.4 * B + 15.4 * G) / (B + G) = 15.8.Multiply both sides by (B + G): 16.4B + 15.4G = 15.8B + 15.8G.Rearrange terms: 16.4B - 15.8B = 15.8G - 15.4G.0.6B = 0.4G.Divide both sides by 0.2: 3B = 2G.So B / G = 2 / 3.Therefore, the ratio of boys to girls is 2 : 3.

Verification / Alternative check:
Assume B = 2k and G = 3k for some positive k.Total students = 5k.Total age = 16.4 * 2k + 15.4 * 3k = 32.8k + 46.2k = 79k.Overall average = 79k / 5k = 15.8 years, which matches the given overall average.So the ratio 2 : 3 is consistent with all given data.

Why Other Options Are Wrong:
Ratios like 1 : 2, 3 : 4 or 3 : 5 do not satisfy the weighted average equation. If any of these ratios are substituted into the equation, the resulting overall average is different from 15.8. Only B : G = 2 : 3 keeps the overall average equal to 15.8 years.

Common Pitfalls:
Students sometimes try to solve these questions by guessing rather than setting up the weighted average formula. Another mistake is to subtract averages directly without considering how the numbers of students weigh each average. Always represent total ages using average * number of people, add them, and then divide by total people to get the overall average equation.

Final Answer:
The required ratio of boys to girls is 2 : 3.