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

Introduction / Context:
This problem uses weighted averages to connect the overall average age of a class with the separate averages for boys and girls. You are given all three averages and asked to find the ratio of the number of boys to the number of girls. Such questions help learners understand how subgroup averages combine to produce a single overall average.

Given Data / Assumptions:
- Let the number of boys be B and the number of girls be G.
- Overall average age of all students = 15.8 years.
- Average age of boys = 16.4 years.
- Average age of girls = 15.4 years.
- We need to find the ratio B : G.

Concept / Approach:
The total age of the boys is 16.4 * B and the total age of the girls is 15.4 * G. The overall average times the total number of students gives another expression for the total age, namely 15.8 * (B + G). Equating these two expressions and simplifying gives a linear relation between B and G that can be converted into a ratio.

Step-by-Step Solution:
Step 1: Total age of boys = 16.4 * B.Step 2: Total age of girls = 15.4 * G.Step 3: Total age of all students = 16.4B + 15.4G.Step 4: Overall average age = 15.8, so total age can also be written as 15.8 * (B + G).Step 5: Equate the two expressions: 16.4B + 15.4G = 15.8(B + G).Step 6: Expand the right side: 16.4B + 15.4G = 15.8B + 15.8G.Step 7: Rearrange terms: 16.4B - 15.8B = 15.8G - 15.4G.Step 8: This gives 0.6B = 0.4G.Step 9: Divide both sides by 0.2 to simplify: 3B = 2G.Step 10: Therefore, B / G = 2 / 3 and the ratio B : G = 2 : 3.

Verification / Alternative Check:
Assume there are 2 boys and 3 girls to match the ratio 2 : 3. Total age of boys is 2 * 16.4 = 32.8, and total age of girls is 3 * 15.4 = 46.2. Combined total age = 79.0. Total students = 5. Average age = 79.0 / 5 = 15.8, which matches the given overall average. This numerical check confirms that the ratio 2 : 3 is correct.

Why Other Options Are Wrong:
Ratios such as 1 : 2, 3 : 4 or 3 : 5 do not satisfy the equation 16.4B + 15.4G = 15.8(B + G). Substituting these ratios will give an overall average different from 15.8, so they are inconsistent with the data in the problem.

Common Pitfalls:
Some students treat this as a simple average of 16.4 and 15.4 without weighting, which ignores the different numbers of boys and girls. Others might set up the equation incorrectly by mixing up the coefficients. Carefully forming the total age expressions and simplifying step by step avoids these errors.

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