Question

What is the average height of the class?
(I) Average height of the class decreases by 1 cm if we exclude the tallest person of the class whose height is 56 cm.
(II) Average height of the class increases by 1 cm if we exclude the shortest person of the class whose height is 42 cm.

• A. Answer (1) if data in Statement I alone is sufficient to answer the question but the data in Statement II alone is not sufficient to answer the question.
• B. Answer (2) if data in Statement II alone is sufficient to answer the question but the data in Statement I alone is not sufficient to answer the question.
• C. Answer (3) if data in Statement I and II together are necessary to answer the question.
• D. Answer (4) if data in Statement I and II together are not sufficient to answer the question.

Hint:

Set up S = nA and use both exclusion conditions as two equations; check if either alone can solve for the two unknowns n and A.

Answer 1

The Correct Option is C

Work with the total height sum $S$ and the class size $n$ directly, instead of jumping straight to the average.

The average height is $A = S/n$. Statement I says removing the tallest student (height $56$) drops the average by $1$:

\[\frac{S-56}{n-1} = \frac{S}{n} - 1\]

Cross-multiplying and simplifying (the $S/n$ terms cancel out the same way whichever order you substitute) reduces this to a single relation between $n$ and $A = S/n$, namely $n+A=57$. This is one equation with two unknowns, so $n$ and $A$ individually stay undetermined from Statement I alone.

Statement II says removing the shortest student (height $42$) raises the average by $1$:

\[\frac{S-42}{n-1} = \frac{S}{n} + 1\]

which reduces in the same way to $n-A=41$, again one equation with the same two unknowns, so Statement II alone is equally insufficient on its own.

But the two conditions together give two independent equations in the two unknowns $n$ and $A$: $n+A=57$ and $n-A=41$. Two independent linear equations in two unknowns have exactly one solution, found by adding them: $2n=98$, so $n=49$ students, and then $A = 57-49 = 8$ cm.

Since each statement alone leaves an underdetermined system, but together they form a solvable 2-by-2 linear system with a unique answer, the two statements together are necessary and sufficient, while neither is enough alone. This matches option (3): the average height, \(\boxed{8 \text{ cm}}\), needs both statements combined.