Problem 5 · 2020 Math Kangaroo
Medium
Algebra & Patterns
sum-constraintsubstitution
Maria has ten pieces of paper. Some are squares and the others are triangles. She cuts each square along one of its diagonals. She then counts the total number of vertices of the pieces she has now and gets 48. How many squares did she have before making the cuts?
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Answer: D — 6
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Hint 1 of 2
A whole square has 4 corners, but a cut square becomes two triangles — count their corners.
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Hint 2 of 2
Set up one equation for the 10 pieces and one for the 48 vertices.
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Approach: count vertices of cut squares vs triangles and solve
- Cutting a square along a diagonal gives two triangles, 6 vertices in all.
- An uncut triangle has 3 vertices.
- With s squares and (10−s) triangles: 6s + 3(10−s) = 48.
- This gives 3s = 18, so s = 6.
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