Question

In a general body election, 3 candidates, p, q and r were contesting for a membership of the board. How many votes did each receive?
(I) p received 17 votes more than q and 103 votes more than r.
(II) Total votes cast were 1703.

• A. Statement I alone is sufficient to answer the question.
• B. Statement II alone is sufficient to answer the question.
• C. Both statement I and II together are necessary to answer the question.
• D. Both statements I and II together are not sufficient to answer the question.

Hint:

Count unknowns (3) versus equations given by each statement; only combined do they match.

Answer 1

The Correct Option is C

Step 1: Set up variables.
Let the votes received by p, q and r be p, q and r respectively.
Step 2: Examine statement (I) alone.
Statement (I) gives p = q + 17 and p = r + 103. This provides two equations relating p, q and r, but there are three unknowns and only two equations, so infinitely many solutions are possible. For example, p = 200, q = 183, r = 97 fits the ratios, and so does p = 300, q = 283, r = 197. Statement (I) alone is not sufficient.
Step 3: Examine statement (II) alone.
Statement (II) only gives the total votes cast, p + q + r = 1703. This is a single equation in three unknowns, so it cannot pin down individual values of p, q and r. Statement (II) alone is not sufficient.
Step 4: Combine both statements.
From statement (I): q = p - 17 and r = p - 103. Substitute these into the total from statement (II): p + (p - 17) + (p - 103) = 1703.
Step 5: Solve the combined equation.
This simplifies to 3p - 120 = 1703, so 3p = 1823, giving a definite value of p. Once p is known, q = p - 17 and r = p - 103 can both be computed directly. So combining the two statements gives exactly one equation in one unknown (p), which can be solved uniquely, and then q and r follow immediately.
Step 6: Conclude.
Since statement (I) alone is not sufficient, statement (II) alone is not sufficient, but both statements together pin down p, q and r uniquely, the correct option is (3): both statements together are necessary to answer the question.