In a school examination, the average score of the boys is 81 and the average score of the girls is 83. The combined average score of all students in the school is 81.8. What is the ratio of the number of girls to the number of boys who took the examination?

Introduction / Context:
This question explores the relationship between group averages and the overall average of a combined group. By forming an equation using weighted averages, we can determine the relative number of boys and girls in the examination. This concept is commonly tested in aptitude exams under the topic of alligation and averages.

Given Data / Assumptions:

• Average score of boys = 81.

• Average score of girls = 83.

• Combined average score of all students = 81.8.

• Let the number of boys be B and the number of girls be G.

• All scores are assumed to be on the same scale and measured in marks.

Concept / Approach:
The overall average is a weighted average of the averages of the boys and the girls. We can express the total marks of boys as 81 * B and the total marks of girls as 83 * G. The combined total marks are then divided by the total number of students B + G to give the overall average 81.8. Setting up this equation and simplifying will allow us to find the ratio G : B directly without needing the exact numbers of students.

Step-by-Step Solution:
Total marks of boys = 81 * B. Total marks of girls = 83 * G. Combined average = (81B + 83G) / (B + G) = 81.8. Multiply both sides by (B + G): 81B + 83G = 81.8B + 81.8G. Rearrange terms: 81B − 81.8B + 83G − 81.8G = 0. This simplifies to −0.8B + 1.2G = 0. So, 1.2G = 0.8B. Divide both sides by 0.4: 3G = 2B. Therefore, G : B = 2 : 3.

Verification / Alternative check:
To verify the ratio, assume a convenient multiple, for example B = 3k and G = 2k. Then total marks = 81 * 3k + 83 * 2k = 243k + 166k = 409k. Total students = 5k. Combined average = 409k / 5k = 81.8, which matches the given overall average. Thus the assumed ratio 2 : 3 for girls to boys is correct.

Why Other Options Are Wrong:
Ratios such as 3 : 2, 4 : 3 or 3 : 4 do not satisfy the weighted average equation. Substituting those ratios into the same calculation would yield average scores that are either higher or lower than 81.8. Only the ratio 2 : 3 gives exactly the required overall average.

Common Pitfalls:
Sometimes students confuse the ratio of averages with the ratio of the numbers of students, which are different concepts. Another common mistake is to try to solve using guesswork without setting up the weighted average equation. Writing out the total marks expression and carefully simplifying the equation is the safest and most systematic method.

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