A piece of rope is 10 metres long. It is cut into three smaller pieces. What…

Concept: In a Data Sufficiency question, a statement (or combination of statements) is SUFFICIENT only when it pins the quantity asked for down to exactly one value; if the given facts still allow more than one consistent value, the statement(s) are INSUFFICIENT. Each statement is tested alone first, and only if neither works alone do you check whether combining them removes every remaining degree of freedom — a claim in one statement that doesn't specify WHICH item it refers to can leave the combination just as open as before.

1. Let the three pieces have lengths a, b, c metres with a + b + c = 10.

2. Statement I alone: two pieces are equal, so the pieces look like x, x, (10 − 2x). Here x can be any value between 0 and 5 metres (for example x = 1 gives 1, 1, 8; x = 4 gives 4, 4, 2), so the longest piece changes with x — Statement I alone does not fix a unique answer.

3. Statement II alone: one piece is 3 metres, leaving 7 metres to be split between the other two pieces in any way (for example 1 and 6, or 2 and 5, or 3.5 and 3.5) — the longest piece is still not fixed — Statement II alone does not fix a unique answer either.

4. Combine both statements: two of the three pieces are equal, and one piece (unspecified which) is 3 metres. Two different assignments are both consistent with this: (a) the 3 m piece is the one separate from the equal pair, so the equal pair's common length x satisfies x + x + 3 = 10, giving x = 3.5 m, i.e. pieces 3.5 m, 3.5 m, 3 m; (b) the 3 m piece is one of the equal pair itself, so the equal pair is 3 m, 3 m and the remaining piece is 10 − 3 − 3 = 4 m, i.e. pieces 3 m, 3 m, 4 m.

5. Both assignments — {3.5, 3.5, 3} and {3, 3, 4} — satisfy both statements exactly as worded, but they give different longest-piece values (3.5 m versus 4 m). Since the two statements together still admit more than one consistent value, they do not pin down a unique longest piece.

Cross-check: 3.5 + 3.5 + 3 = 10 and 3 + 3 + 4 = 10 — both triples are valid, complete splits of the 10 m rope, and both satisfy 'two pieces equal' and 'one piece is 3 m'. The existence of two genuinely different, equally valid answers is itself the proof that the combined data is insufficient.

Hence, both statements put together are NOT sufficient to determine the length of the longest piece; neither statement alone, nor the two combined, fixes a unique answer.