IMC 2010 Q12
12. A cuboid is cut away from a cube of side 10 cm as shown.
By what fraction does the total surface area of the solid decrease as a result?
- A \(\dfrac{1}{4}\)
- B \(\dfrac{1}{6}\)
- C \(\dfrac{1}{10}\)
- D \(\dfrac{1}{12}\)
- E \(\dfrac{1}{18}\)
Intermediate Mathematical Challenge 2010, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\dfrac{1}{12}\)
When the cuboid is cut away, the surface area of the solid ‘loses’ two rectangles measuring 10 cm × 5 cm and two squares of side 5 cm. However, it also ‘gains’ two rectangles measuring 10 cm × 5 cm. So the surface area decreases by an area equal to one half of the area of one of the faces of the original cube, that is one twelfth of its original surface area.
Solution © UK Mathematics Trust, from the IMC 2010 solutions.