The score on a trivia game is obtained by subtracting the number of incorrect answers from twice the number of...

Math Problem: Test Scoring System

A student answers 40 questions on a test. For each correct answer, the student gets 2 points, and for each incorrect answer, 1 point is deducted. If the student's total score is 50 points, how many questions did the student answer correctly?

Solution Process

Step 1: Define Variables

Let's define:

• \(x\) = number of correct answers
• \(y\) = number of incorrect answers

Step 2: Set Up the System of Equations

From the problem, we can write two equations:

• Total questions: \(x + y = 40\)
• Total score: \(2x - y = 50\)

Step 3: Solve the System

Method: Addition/Elimination

Add the two equations together:

\((x + y) + (2x - y) = 40 + 50\)

\(x + y + 2x - y = 90\)

\(3x = 90\)

\(x = 90 \div 3 = 30\)

Step 4: Find the Number of Incorrect Answers

Substitute \(x = 30\) into the first equation:

\(30 + y = 40\)

\(y = 40 - 30 = 10\)

Step 5: Verify the Solution

Let's check our answer:

• Total questions: \(30 + 10 = 40\) ✓
• Total score: \(2(30) - 10 = 60 - 10 = 50\) ✓

Answer

The student answered 30 questions correctly.

Common Error Paths

Error 1: Incorrect Score Formula

Some students might incorrectly write the score equation as \(2x + y = 50\) (adding points for wrong answers instead of subtracting).

• This would lead to \(x = 10\) and \(y = 30\)
• Verification would fail: \(2(10) + 30 = 50\) but this doesn't match the problem's penalty system

Error 2: Arithmetic Mistakes

Common calculation errors:

• Incorrectly simplifying \(3x = 90\) to \(x = 27\) or \(x = 33\)
• Sign errors when combining equations
• Forgetting to distribute negative signs