What is the average height of the class? I. The class average decreases by 1 cm when the tallest person (56 cm) is excluded. II. The class average increases by 1 cm when the shortest person (42 cm) is excluded.

Introduction / Context:
Average problems with removal of extreme values can be solved using algebra linking the original mean, class size, and the extreme values. We need to decide which information is sufficient to determine the class average uniquely.

Given Data / Assumptions:

• Let n be class size and A be the original average.
• Tallest height = 56 cm; shortest height = 42 cm.
• Statement I: removing 56 cm lowers the average by 1 cm.
• Statement II: removing 42 cm raises the average by 1 cm.

Concept / Approach:
Use the identity: when removing an element x from a set of size n with average A, the new average becomes (nA − x)/(n − 1). Set this equal to the described new average and solve.

Step-by-Step Solution:
From I: (nA − 56)/(n − 1) = A − 1 ⇒ nA − 56 = (n − 1)A − (n − 1) ⇒ A − 56 = −(n − 1) ⇒ A = 57 − n.From II: (nA − 42)/(n − 1) = A + 1 ⇒ nA − 42 = (n − 1)A + (n − 1) ⇒ A − 42 = n − 1 ⇒ A = n + 41.Equate A from both: 57 − n = n + 41 ⇒ 2n = 16 ⇒ n = 8 ⇒ A = 49.Each statement alone leaves two variables (n and A); together they yield a unique solution.

Verification / Alternative check:
Check I: removing 56 from eight students lowers mean from 49 to 48 (works). Check II: removing 42 raises mean from 49 to 50 (works).

Why Other Options Are Wrong:
I alone or II alone is insufficient due to two unknowns; together they are necessary and sufficient.

Common Pitfalls:
Mishandling the average formula when an element is removed or forgetting to multiply averages by the number of items.

Final Answer:
Both statements together are necessary; the class average is 49 cm.