In an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. School A starts with 165 points and School B starts with 65 points, giving no incorrect... In an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. School A starts with 165 points and School B starts with 65 points, giving no incorrect answers and has the same number of correct answers as School A. Which equation models the scoring in the final round and how many answers did each school get in the final round? Read more

Steps to Solve

1. Define the Variables Let $x$ be the number of correct answers given by each school.

2. Calculate School A's Score School A starts with 165 points, earns 12 points for each correct answer, and loses points for incorrect answers. The score calculation for School A is: $$ \text{Score of School A} = 165 + 12x - 5x = 165 + 7x $$

3. Calculate School B's Score School B starts with 65 points and only provides correct answers. The score calculation for School B is: $$ \text{Score of School B} = 65 + 12x $$

4. Set the Scores Equal Since the game ends in a tie, we set the scores equal: $$ 165 + 7x = 65 + 12x $$

5. Solve for x To isolate $x$, move the terms involving $x$ to one side and constants to the other side: $$ 165 - 65 = 12x - 7x $$ This simplifies to: $$ 100 = 5x $$ Divide both sides by 5: $$ x = 20 $$

6. Find Total Answers Since both schools provided the same number of answers and each school gave $x$ correct answers, the total number of answers per school is: $$ \text{Total Answers} = x + (\text{incorrect answers}) $$ For School A, the incorrect answers can be calculated by knowing each school has an equal number of correct and incorrect answers. Thus, the total answers for each school is: $$ \text{Total Answers} = x + x = 2x = 2(20) = 40 $$