Question

Bags I, II, and III contain at least one ball each and together have 10 balls. How many balls are in each bag? Decide whether the statements are sufficient.
(1) Bag I contains five balls more than Bag III.
(2) Bag II contains half as many balls as Bag I.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.

Hint:

Check each statement on its own for more than one valid (I, II, III) triple before trying them combined.

Answer 1

The Correct Option is C

A data sufficiency question like this is not asking us to actually find the split of balls right away, it is asking whether the given clues pin down one and only one possible split. The trick is to test each statement on its own, by hunting for more than one set of values that fits, before checking both together.

1. Statement (1) alone (Bag I has 5 more balls than Bag III): writing $I=III+5$ and plugging into $I+II+III=10$ gives $II+2\,III=5$. Both $(III,II)=(1,3)$ and $(III,II)=(2,1)$ work, giving two different splits, (6,3,1) and (7,1,2). Since more than one answer fits, statement (1) alone is not sufficient.
2. Statement (2) alone (Bag II is half of Bag I): writing $II=I/2$ forces $I$ to be even. Checking $I=2,4,6$ all give valid positive values for Bag III (7, 4, 1 respectively). Multiple splits work here too, so statement (2) alone is not sufficient.
3. Statements (1) and (2) together: now both $I=III+5$ and $II=I/2$ must hold at once. Combining them with the total of 10 leaves only one value of $I$ that works, $I=6$, which fixes $II=3$ and $III=1$. No other combination satisfies both conditions simultaneously, so together the statements give one unique split.

Since each statement by itself allows more than one possible distribution of balls, but the two together allow exactly one, the two statements are needed jointly and neither works alone.

Let's summarize:

• Statement (1) alone: two valid splits exist, not sufficient.
• Statement (2) alone: three valid splits exist, not sufficient.
• Both together: exactly one valid split, (6, 3, 1), sufficient.

So both statements together are required, matching option (C).