Question

When a player in a certain game tossed a coin a number of times, 4 more heads than tails resulted. Heads or tails resulted each time the player tossed the coin. How many times did heads result?
(1) The player tossed the coin 24 times.
(2) The player received 3 points each time heads resulted and 1 point each time tails resulted, for a total of 52 points.

• 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 together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (D)

Each statement alone is sufficient.

Let h represent the number of heads that resulted and t represent the number of tails obtained by the player. Then the information given can be expressed as h = t +4.

1. The additional information can be expressed as h + t = 24. When this equation is paired with the given information, h = t + 4, there are two linear equations in two unknowns. One way to conclude that we can determine the number of heads is to solve the equations simultaneously, thereby obtaining the number of heads and the number of tails: Solving h = t + 4 for t, which gives t = h - 4, and substituting the result in h + t = 24 gives h + (h - 4) = 24, which clearly can be solved for h. Another way to conclude that we can determine the number of heads is to note that the pair of equations represents two non-parallel lines in the coordinate plane; SUFFICIENT.
2. The additional information provided can be expressed as 3h + t = 52. The same comments in (1) apply here as well. For example, solving h = t + 4 for t, which gives t = h - 4, and substituting the result in 3h + t = 52 gives 3h + (h - 4) = 52, which clearly can be solved for h; SUFFICIENT.

• The correct answer is D; each statement alone is sufficient.