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

A cleaner way to see this is to count independent pieces of information (equations) needed versus given, without solving statement I by itself.

1. There are three unknowns to find: the votes for $p$, $q$, and $r$. To find three unknowns uniquely, three independent linear equations are generally needed.
2. Statement (I) alone supplies two relations: $p = q + 17$ and $p = r + 103$. That is two equations for three unknowns, one short of what is needed, so it cannot fix all three values.
3. Statement (II) alone supplies just one relation: $p + q + r = 1703$. That is only one equation for three unknowns, far short of sufficient.
4. Combining both statements gives a total of three independent equations:\[ p = q + 17, \quad p = r + 103, \quad p + q + r = 1703 \]
5. Three independent linear equations in three unknowns generally have a unique solution. Substituting $q = p-17$ and $r = p-103$ into the third equation gives $3p - 120 = 1703$, so $p$ is fixed uniquely, and then $q$ and $r$ follow directly.
6. Since three equations are available only when both statements are used together, both statements together are necessary, matching option (3).

\[\boxed{\text{Both statements I and II together are necessary (option 3)}}\]