1985 AJHSME Problem 25

Revision as of 20:10, 31 January 2021 by Coolmath34 (talk | contribs) (Created page with "== Problem == Five cards are lying on a table as shown. <cmath>\begin{matrix} & \qquad & \boxed{\tt{P}} & \qquad & \boxed{\tt{Q}} \\ \\ \boxed{\tt{3}} & \qquad & \boxed{\tt...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Five cards are lying on a table as shown.

\[\begin{matrix} & \qquad & \boxed{\tt{P}} & \qquad & \boxed{\tt{Q}} \\  \\ \boxed{\tt{3}} & \qquad & \boxed{\tt{4}} & \qquad & \boxed{\tt{6}} \end{matrix}\]

Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over?

$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ \text{P} \qquad \text{(E)}\ \text{Q}$

Solution

To show Jane was wrong, Mary would have to show a card with a vowel. However, this is not possible. If Jane is correct, then the contrapositive ("If an odd number is on one side of the card, then a consonant is on the other side.") is true. So, Mary picked the card with a $\boxed{\text{(A)} 3}.$

Invalid username
Login to AoPS