In a school, 391 boys and 323 girls are divided into the largest possible equal classes so that each class contains the same number of boys and the same number of girls. How many such classes are formed?

Introduction / Context:
Although this question appears under averages, it actually tests your understanding of highest common factor and equal grouping. The school wants to divide 391 boys and 323 girls into as many equal classes as possible such that each class has the same number of boys and the same number of girls. This is a classic application of greatest common divisor.

Given Data / Assumptions:

• Number of boys = 391.
• Number of girls = 323.
• All boys are to be divided into equal classes, each class having the same number of boys.
• All girls are to be divided into equal classes, each class having the same number of girls.
• The number of classes must be the same for boys and girls, and it must be as large as possible.

Concept / Approach:
If there are N classes, then the number of boys per class is 391 / N and the number of girls per class is 323 / N. For both of these to be whole numbers, N must be a common divisor of both 391 and 323. To make the classes as large in number as possible, N must be the greatest common divisor of 391 and 323.

Step-by-Step Solution:
Step 1: List factors or use the Euclidean algorithm to find the greatest common divisor of 391 and 323. Step 2: Compute 391 - 323 = 68. Step 3: Now find the greatest common divisor of 323 and 68. Next 323 - 4 * 68 = 323 - 272 = 51. Step 4: Find the greatest common divisor of 68 and 51: 68 - 51 = 17. Step 5: Now find the greatest common divisor of 51 and 17. Since 51 = 3 * 17, the remainder is 0 and the greatest common divisor is 17. Step 6: Therefore the largest possible number of equal classes is 17.

Verification / Alternative check:
If there are 17 classes, each class has 391 / 17 = 23 boys and 323 / 17 = 19 girls. Both are whole numbers, and it is not possible to have more classes than 17 and still keep integer numbers of boys and girls in every class. Trying a higher number like 19 or 23 will not divide both 391 and 323 exactly.

Why Other Options Are Wrong:
Option 23 divides 391 but not 323, so you cannot create 23 equal classes for both groups. Option 19 divides 323 but not 391. Option 44 is larger than both square roots and clearly cannot be a common divisor. Only 17 divides both numbers exactly and is the greatest such divisor.

Common Pitfalls:
Students sometimes confuse the meaning of equal classes and look for the greatest common divisor of the total students instead of treating boys and girls separately. Others pick a smaller common factor without checking if there is a larger one. Always use a systematic approach like the Euclidean algorithm to find the greatest common divisor.

Final Answer:
The school can form 17 equal classes.