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

Step 1: Set up the variables.
Let the number of balls in Bags I, II, and III be \(I\), \(II\), and \(III\) respectively, each at least 1, with
\[ I + II + III = 10 \]

Step 2: Test statement (1) alone.
Statement (1) says \(I = III + 5\). Substituting into the total:
\[ (III+5) + II + III = 10 \implies II + 2\,III = 5 \]
With \(II \ge 1\) and \(III \ge 1\), try \(III = 1\): \(II = 3\), giving \((I,II,III) = (6,3,1)\), which fits. Try \(III = 2\): \(II = 1\), giving \((I,II,III) = (7,1,2)\), which also fits. Two different valid triples exist, so statement (1) alone does not pin down a unique answer.

Step 3: Test statement (2) alone.
Statement (2) says \(II = \frac{I}{2}\), so \(I\) must be even. Substituting:
\[ I + \frac{I}{2} + III = 10 \implies III = 10 - \frac{3I}{2} \]
Try \(I = 2\): \(II=1, III=7\). Try \(I=4\): \(II=2, III=4\). Try \(I=6\): \(II=3, III=1\). All three fit \(I,II,III \ge 1\), so again there are multiple valid triples, and statement (2) alone is not enough either.

Step 4: Use both statements together.
Now \(I = III + 5\) and \(II = \frac{I}{2}\). Substitute both into the total:
\[ I + \frac{I}{2} + (I - 5) = 10 \]
\[ \frac{5I}{2} = 15 \implies I = 6 \]
Then \(II = 3\) and \(III = 1\), a single valid triple with all bags having at least 1 ball.

Step 5: Conclusion.
Neither statement alone fixes a unique split of the balls, but combining both gives exactly one valid answer, \(I=6, II=3, III=1\).
\[ \boxed{\text{Both statements together are needed}} \]