Recovering absent students’ heights from class averages: The average height of 40 students is 163 cm. On a particular day, three students A, B, and C were absent, and the average of the remaining 37 students was 162 cm. If A and B have equal height and C is exactly 2 cm shorter than A, what is the height of student A (in cm)?

Introduction / Context:
This problem uses the definition of the arithmetic mean to reconstruct the combined height of absent students from two given averages. Once we know the total of the absent trio, the stated relations between A, B, and C allow us to solve for A’s exact height.

Given Data / Assumptions:

• Total students = 40; class average = 163 cm
• Present students = 37; present average = 162 cm
• A and B have equal height; let A = B = h
• C is 2 cm less than A → C = h - 2

Concept / Approach:
Average = total / count. Hence, total height of all 40 and total of the 37 present can be found. Their difference is the combined height of A, B, and C. Then solve the linear equation from h + h + (h - 2).

Step-by-Step Solution:
Total (40) = 40 * 163 = 6520 cmTotal (37) = 37 * 162 = 5994 cmSum(A,B,C) = 6520 - 5994 = 526 cmLet A = B = h, C = h - 2 → 3h - 2 = 526 → 3h = 528 → h = 176 cm

Verification / Alternative check:
Heights would be 176, 176, and 174. Their sum is 526, which matches the computed absent total. Substituting back confirms the two given averages.

Why Other Options Are Wrong:
166, 180, 186, and 172 cm do not satisfy the linear relation 3h - 2 = 526, hence they cannot produce the correct totals.

Common Pitfalls:
Forgetting that C is exactly 2 cm less than A (not B), or using 40 instead of 37 when recomputing the present group’s total. Keep totals and counts consistent.

Final Answer:
176 cm