Problem 24 · 2014 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Jan and Eva take on a challenge to solve mathematics questions. They each get an identical list of 100 questions. For each question, the first to solve it gets 4 points while the slower person gets 1 point. Jan solved 60 questions and Eva also solved 60 questions. Together they scored 312 points. How many questions were solved by both Jan and Eva?
Show answer
Answer: D — 56
Show hints
Hint 1 of 2
Let x be the number of questions both solved; a shared question scores 4 + 1 = 5 points.
Still stuck? Show hint 2 →
Hint 2 of 2
Write the total score in terms of x using how many questions were solved by one person versus both.
Show solution
Approach: count points by 'both' versus 'one only'
- Let x questions be solved by both. Each such question gives 4 + 1 = 5 points total.
- Jan-only and Eva-only questions number (60−x) + (60−x) = 120 − 2x, each worth 4 points.
- Total points: 5x + 4(120 − 2x) = 480 − 3x = 312, so 3x = 168 and x = 56.
Mark:
· log in to save