The average age of a class is 15.8 years. The average age of the boys in the class is 16.4 years and that 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 involves weighted averages. We are given the overall average age of the class along with separate average ages of boys and girls. From this, we need to determine the ratio of the number of boys to the number of girls. It is an example of mixing and alligation in the context of averages.

Given Data / Assumptions:

• The average age of the whole class is 15.8 years.
• The average age of the boys is 16.4 years.
• The average age of the girls is 15.4 years.
• We need the ratio of the count of boys to the count of girls.

Concept / Approach:
Let the number of boys be B and the number of girls be G. Then the total age of boys is 16.4B and the total age of girls is 15.4G. The overall average 15.8 means (16.4B + 15.4G) / (B + G) = 15.8. This equation simplifies to a relation between B and G from which we find B / G, the required ratio.

Step-by-Step Solution:
Let the number of boys be B and the number of girls be G. Total age of boys = 16.4 * B. Total age of girls = 15.4 * G. Total number of students = B + G and the average age is 15.8. So (16.4B + 15.4G) / (B + G) = 15.8. Multiply both sides by (B + G): 16.4B + 15.4G = 15.8B + 15.8G. Rearrange terms to get (16.4B - 15.8B) + (15.4G - 15.8G) = 0. This gives 0.6B - 0.4G = 0. Hence 0.6B = 0.4G which implies B / G = 0.4 / 0.6 = 2 / 3. Therefore, the ratio of boys to girls is 2 : 3.

Verification / Alternative check:
Using alligation, the difference between boys average and class average is 16.4 - 15.8 = 0.6. The difference between class average and girls average is 15.8 - 15.4 = 0.4. The ratio of boys to girls should be the inverse of these differences, that is 0.4 : 0.6 = 2 : 3. This matches the algebraic solution, confirming the ratio.

Why Other Options Are Wrong:
Ratios like 1 : 3 or 3 : 1 would produce overall averages heavily biased toward one group, not 15.8 as given. Ratios 3 : 2 or 4 : 5 also fail when substituted into the weighted average formula, leading to a different overall average. Only 2 : 3 satisfies the equation exactly.

Common Pitfalls:
A frequent mistake is to simply average 16.4 and 15.4 without weighting for the different numbers of boys and girls. Others may reverse the differences when using the alligation method. Writing out the equation carefully and checking the direction of differences in alligation helps avoid errors.

Final Answer:
Thus, the ratio of the number of boys to the number of girls in the class is 2 : 3.