Challenging P6 2009 SA1 P2 RGS Q17

17. The cuboid below is made of 2-cm cubes. All 6 faces of the cuboid are painted.

Find the ratio of the number of cubes with 3 painted faces to the number of cubes with 2 painted faces to the number of cubes without any painted faces in its simples form. [4]

Number of cubes with 3 painted faces

[4 at Top corners + 4 at Bottom corners]

4 x 2 = 8 cubes

Number of cubes with 2 painted faces

[4 vertical on each edge x 4 vertical edges]

[4 horizontal front x 4 (front top, back top, front bottom, back bottom)]

[3 horizontal side x 4 (right top, left top, right bottom, left bottom)]

4 x 4 = 16

3 x 4 = 12

16 + 12 + 16 = 44

Number of cubes without any painted faces

The cuboid is 6 x 5 x 6. Remove all cubes by the sides, the cuboid left is 4 x 3 x 4, which gives the number of cubes not painted.

4 x 3 x 4 = 48 cubes

8 : 44 : 48 = 2 : 11 : 12

The ratio is 2 : 11 : 12.