GK 2018 Q10
10. A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew coloured all the squares in the middle row. How many squares did he not colour?
- A 20
- B 30
- C 32
- D 35
- E 39
Grey Kangaroo 2018, Question 10. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 32
The possible rectangles containing 40 equal squares have dimensions (length × height) 1 × 40, 2 × 20, 4 × 10, 5 × 8, 8 × 5, 10 × 4, 20 × 2 and 40 × 1. The question tells us that when it is divided into squares, the rectangle contains more than one row and also that it has a middle row and hence has an odd number of rows. The number of rows of squares the rectangle contains is related to its height which must be an odd value. Therefore the rectangle has dimensions 8 × 5. Hence, when Andrew coloured the middle row, he coloured eight squares. Therefore the number of squares he did not colour is 40 – 8 = 32.
Solution © UK Mathematics Trust, from the GK 2018 solutions.