In a school, the ratio of the number of boys to the number of girls is 3 : 2.\nIf 20% of the boys and 25% of the girls are scholarship holders, what percentage of all students are not scholarship holders?

Introduction / Context:
This problem combines ratio and percentage concepts with a real life context about scholarship holders in a school. It tests your ability to move from ratios to actual fractions of a total, apply different percentages to subgroups, and then compute the overall percentage of students who do not receive scholarships. These skills are very important in data interpretation and quantitative reasoning.

Given Data / Assumptions:

• Ratio of boys to girls = 3 : 2.
• 20% of the boys are scholarship holders.
• 25% of the girls are scholarship holders.
• We assume all students are either boys or girls and there are no other categories.
• We must find the percentage of total students who are not scholarship holders.

Concept / Approach:
First convert the boy girl ratio into fractional parts of the total students. Then apply the given scholarship percentages separately to boys and girls to find the fraction of scholarship holders in each gender group. Add these fractions to find the total proportion of scholarship holders in the school. The complement of this proportion (that is, one minus the scholarship fraction) gives the proportion of non scholarship students, which we then express as a percentage.

Step-by-Step Solution:
Step 1: Let the total number of students be 5k, where 3k are boys and 2k are girls, because the ratio is 3 : 2. Step 2: Number of scholarship boys = 20% of 3k = (20 / 100) * 3k = 0.6k. Step 3: Number of scholarship girls = 25% of 2k = (25 / 100) * 2k = 0.5k. Step 4: Total scholarship students = 0.6k + 0.5k = 1.1k. Step 5: Total students = 5k, so the fraction with scholarships = 1.1k / 5k = 1.1 / 5 = 0.22. Step 6: Therefore, the fraction without scholarships = 1 - 0.22 = 0.78, which is 78%.

Verification / Alternative check:
Assume k = 100 for easy numbers. Then total students = 500. Boys = 300, girls = 200. Scholarship boys = 20% of 300 = 60. Scholarship girls = 25% of 200 = 50. Total scholarship students = 110. Non scholarship students = 500 - 110 = 390. Percentage of non scholarship students = (390 / 500) * 100 = 78%. This numerical approach confirms the algebraic result.

Why Other Options Are Wrong:
56% and 70%: These values underestimate the non scholarship percentage and come from miscalculations or ignoring the ratio base.
80%: Slightly higher than the correct value; this can occur if one mistakenly averages the complementary percentages incorrectly.
62%: Has no support in the correct ratio and percentage setup and is a clear miscalculation.

Common Pitfalls:
Many learners mistakenly take a simple average of 20% and 25% to get total scholarship percentage, ignoring the fact that the groups (boys and girls) are not equal in size. Others may incorrectly assume the total students are 100 and break the ratio incorrectly. Always convert ratios into actual parts of a common multiple and then apply the given percentages logically.

Final Answer:
78% of the students in the school are not scholarship holders.