In a school examination, the average score of the boys is 81 and the average score of the girls is 83. If the overall average score of all the students in the school is 81.8, what is the ratio of the number of girls to the number of boys who appeared in the examination?

Introduction:
This problem uses the concept of weighted averages to relate the average scores of boys and girls to the overall average score. From these averages, we are asked to determine the ratio of the number of girls to the number of boys.

Given Data / Assumptions:
Average score of boys = 81. Average score of girls = 83. Overall average score = 81.8. Let the number of boys be B and the number of girls be G. We need the ratio G : B.

Concept / Approach:
For two groups with different averages, the combined average is: Combined average = (Average of boys * B + Average of girls * G) / (B + G). We substitute the given averages and the overall average to form an equation in B and G, and then simplify to find the ratio G : B.

Step-by-Step Solution:
Step 1: Use the weighted average formula. (81 * B + 83 * G) / (B + G) = 81.8. Step 2: Cross multiply: 81B + 83G = 81.8(B + G). 81B + 83G = 81.8B + 81.8G. Step 3: Rearrange terms. 81B - 81.8B + 83G - 81.8G = 0. -0.8B + 1.2G = 0. Step 4: Simplify the equation. 1.2G = 0.8B. Divide both sides by 0.4: 3G = 2B. So, G / B = 2 / 3. Therefore, the ratio of the number of girls to the number of boys is 2 : 3.

Verification / Alternative Check:
Assume B = 3 and G = 2 (keeping the ratio 2 : 3). Total score of boys = 3 * 81 = 243. Total score of girls = 2 * 83 = 166. Combined total = 243 + 166 = 409. Total students = 5. Combined average = 409 / 5 = 81.8, which matches the given overall average.

Why Other Options Are Wrong:
3 : 2 and 4 : 3: These imply more girls than boys relative to the given averages and do not yield an overall average of 81.8. 3 : 4 and 5 : 4: Substituting these ratios in the weighted average formula does not give 81.8 as the overall average.

Common Pitfalls:
Many students try to use the alligation rule without careful calculation or mix up the direction of the ratio. Others attempt to average the two averages directly, which is incorrect because the counts of boys and girls are different. Always start from the definition of weighted average and form a correct linear equation.

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