The average age of all the 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 is a classic mixture and average problem. Instead of liquids or salaries, we are dealing with ages of boys and girls. The overall class average lies between the average of boys and the average of girls. Using the concept of weighted averages and a simple equation, we can find the ratio of the number of boys to the number of girls.

Given Data / Assumptions:

• The overall average age of all students = 15.8 years.
• The average age of boys = 16.4 years.
• The average age of girls = 15.4 years.
• We are asked to find the ratio of the number of boys to the number 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 gives a relationship: (16.4B + 15.4G) / (B + G) = 15.8. This equation can be solved for the ratio B : G. Conceptually, the overall average is closer to the girls' average because there are slightly more girls than boys, which we will see in the final 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.Overall 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: 16.4B - 15.8B = 15.8G - 15.4G.This gives 0.6B = 0.4G.So B / G = 0.4 / 0.6 = 2 / 3.Thus, the ratio of boys to girls is 2 : 3.

Verification / Alternative check:
We can test with small numbers in this ratio. Suppose B = 2 and G = 3. Then total boys' age = 2 * 16.4 = 32.8. Total girls' age = 3 * 15.4 = 46.2. Combined total age = 32.8 + 46.2 = 79. Average age = 79 / 5 = 15.8, which perfectly matches the given overall average. That confirms the correctness of the ratio 2 : 3.

Why Other Options Are Wrong:
Ratios such as 1 : 2 or 3 : 4 do not satisfy the equation derived from the averages. If we plug them in, the recalculated overall average will not be 15.8. The ratio must balance the contributions of the 16.4 and 15.4 ages to reach 15.8. Only 2 : 3 achieves that exact balance.

Common Pitfalls:
Students sometimes mis-handle decimals or try to use an alligation diagram but mix up which difference corresponds to boys or girls. Another error is to cross-multiply incorrectly or forget to divide both sides properly when solving 0.6B = 0.4G. Writing the equation clearly and simplifying step by step prevents these mistakes.

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