PK 2020 Q6
6. Mary had ten pieces of paper. Some of them were squares, and the rest were triangles. She cut three squares diagonally from corner to corner. She then found that the total number of vertices of the 13 pieces of paper was 42.
How many triangles did she have before making the cuts?
- A 8
- B 7
- C 6
- D 5
- E 4
Pink Kangaroo 2020, Question 6. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 4
Let \(s\) be the number of squares, and \(t\) the number of triangles that Mary started with. The number of vertices was \(4s + 3t\). Also, there were 10 pieces of paper, so
\[s + t = 10. \quad [1]\]
When she cuts a square diagonally to create two triangles, she increases the number of vertices by 2 (from 4 to 6). Hence before she cut the three squares, she had \(42 - 3 \times 2 = 36\) vertices. Therefore
\[4s + 3t = 36. \quad [2]\]
Subtracting three times equation [1] from equation [2] gives \(s = 6\). Hence \(t = 4\).
Solution © UK Mathematics Trust, from the PK 2020 solutions.