Question

Bags I, II and III together have ten balls. If each bag contains at least one ball, how many balls does each bag have? Decide whether the data given in the statements are sufficient to answer the question.
Statement (1): Bag I contains five balls more than bag III.
Statement (2): Bag II contains half as many balls as bag I

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

Answer 1

The Correct Option is C

To solve this problem, we need to determine the number of balls in each of the three bags, I, II, and III, given that together they contain a total of 10 balls. Let's define:
Let \( x \) be the number of balls in Bag I,
\( y \) be the number of balls in Bag II,
\( z \) be the number of balls in Bag III.
The relationship between these is \( x + y + z = 10 \).

Statement (1): Bag I contains five balls more than Bag III.
This can be expressed as \( x = z + 5 \).
Substituting into the total equation:
\( (z + 5) + y + z = 10 \)
Simplifying gives \( 2z + y + 5 = 10 \) or \( 2z + y = 5 \).
With this equation alone, we cannot uniquely determine the values of \( x \), \( y \), and \( z \) because different combinations of \( z \) and \( y \) will satisfy the equation.
Therefore, statement 1 alone is insufficient.

Statement (2): Bag II contains half as many balls as Bag I.
This relationship can be expressed as \( y = \frac{x}{2} \).
Substituting into the total equation:
\( x + \frac{x}{2} + z = 10 \)
This simplifies to \( \frac{3x}{2} + z = 10 \) or \( 3x + 2z = 20 \).
With this equation alone, similar to statement 1, we have multiple solutions because different pairs of \( x \) and \( z \) can satisfy this equation.
Thus, statement 2 alone is insufficient.

Combining Statements (1) and (2):
From statement (1): \( x = z + 5 \)
From statement (2): \( 3x + 2z = 20 \)
Substitute \( x = z + 5 \) from statement (1) into statement (2):
\( 3(z + 5) + 2z = 20 \)
Simplifying, \( 3z + 15 + 2z = 20 \)
This simplifies to \( 5z = 5 \), yielding \( z = 1 \).
Substitute back \( z = 1 \) into \( x = z + 5 \), thus \( x = 6 \).
Substituting \( x = 6 \) into \( y = \frac{x}{2} \), gives \( y = 3 \).

Hence, using both statements together, we can determine the exact number of balls in each bag as:
Bag I: 6 balls, Bag II: 3 balls, Bag III: 1 ball.

The correct option is: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.