In a class, the ratio of the number of boys to the number of girls is 2 : 3. The average weight of the boys is 18 kg and the average weight of the girls is 21 kg. What is the average weight (in kilograms) of all the students in the class together?

Introduction / Context:
This problem tests the concept of weighted averages. The class is composed of boys and girls in a fixed ratio, and each group has its own average weight. The aim is to find the combined average weight of all students. Instead of assuming a specific number of students, we can use the ratio directly and treat it as a scaled model of the class.

Given Data / Assumptions:

• Ratio of boys to girls = 2 : 3.
• Average weight of boys = 18 kg.
• Average weight of girls = 21 kg.
• All students are either boys or girls, no other categories.
• We must find the overall average weight of all students.

Concept / Approach:
When groups have different averages, the combined average is a weighted average. The weights are the sizes of the groups. Because the ratio of boys to girls is 2 : 3, we can assume there are 2k boys and 3k girls for some positive integer k. Then we compute total weight of boys and girls using their averages and divide the combined total by the total number of students 5k. The value of k cancels, leaving a simple expression that depends only on the given averages and ratio.

Step-by-Step Solution:
Step 1: Assume numbers based on the ratio. Let number of boys = 2k and number of girls = 3k. Total number of students = 2k + 3k = 5k. Step 2: Compute total weight of boys. Average weight of boys = 18 kg, so total weight of boys = 2k * 18 = 36k. Step 3: Compute total weight of girls. Average weight of girls = 21 kg, so total weight of girls = 3k * 21 = 63k. Step 4: Combine totals. Total weight of all students = 36k + 63k = 99k. Step 5: Find overall average. Overall average weight = total weight / total students = 99k / 5k. The k cancels, leaving 99 / 5 kg. Thus, average weight = 99/5 kg.
Verification / Alternative check:
We can choose a convenient value for k, for example k = 1. Then we have 2 boys and 3 girls, 5 students total. Total boy weight = 2 * 18 = 36. Total girl weight = 3 * 21 = 63. Combined total = 36 + 63 = 99. Average = 99 / 5 = 19.8, which matches 99/5.
Why Other Options Are Wrong:
101/5, 109/6 and 96/5 correspond to different combinations of group sizes or different averages and do not match the computed weighted average. The value 20 is close but rounded and does not equal the exact value 99/5.
Common Pitfalls:
A common error is to take the simple average of 18 and 21 and compute (18 + 21) / 2 = 19.5, which ignores the 2 : 3 ratio. Another mistake is to pick arbitrary numbers of boys and girls that do not respect the given ratio, leading to inconsistent totals.
Final Answer:
The average weight of all the boys and girls together is 99/5 kilograms.