Liza is playing a game in which she earns 3 points for each correct response but loses 2 points for each incorrect response. She missed the first 3 questions. Right now she just hopes to break even. Which statement is true?

Question

To break even, Liza must answer 3 questions correctly to cancel out the 3 she answered incorrectly.
To break even, Liza must answer 2 questions correctly to give her 6 points, because she now has –6 points and the result would be a score of 0.
To break even, Liza must answer 6 questions correctly to give her 6 points, because she now has –6 points and the result would be a score of 0.
To break even, Liza must answer 1 question correctly, because she needs 3 points to cancel out the 3 questions she answered incorrectly.

Answer 1

Let's analyze Liza's situation step by step.

1. Liza missed the first 3 questions. Since she loses 2 points for each incorrect response, we calculate the points lost:

   • Points lost = 3 incorrect responses × 2 points lost per response = 6 points.
   • Therefore, Liza currently has a score of -6 points.

2. To break even (reach a score of 0), Liza needs to earn enough points to cancel out the -6 points. She earns 3 points for each correct response.

Now, we can calculate how many correct responses she needs:

Let \( x \) be the number of correct responses needed.

The points from correct answers would be:

• 3 points per correct answer × \( x \) correct answers = 3x points.

To break even, she needs her total points to equal 0: \[ 3x - 6 = 0 \]

Solving for \( x \): \[ 3x = 6 \] \[ x = 2 \]

So, Liza needs to answer 2 questions correctly to break even.

Now let's evaluate the provided statements:

1. To break even, Liza must answer 3 questions correctly to cancel out the 3 she answered incorrectly.

   • This statement is false. She needs 2 correct answers, not 3.

2. To break even, Liza must answer 2 questions correctly to give her 6 points, because she now has –6 points and the result would be a score of 0.

   • This statement is true. She needs to answer 2 correctly to offset her -6 points.

3. To break even, Liza must answer 6 questions correctly to give her 6 points, because she now has –6 points and the result would be a score of 0.

   • This statement is false. She only needs 2 correct answers, not 6.

4. To break even, Liza must answer 1 question correctly, because she needs 3 points to cancel out the 3 questions she answered incorrectly.

   • This statement is false. She needs to answer 2 questions correctly.

Based on this analysis, the correct statement is: To break even, Liza must answer 2 questions correctly to give her 6 points, because she now has –6 points and the result would be a score of 0.