1,6,9 are figures which are half shaded by slanting lines.

2,4,7 are all divided into equal parts (either 3 or four parts) by straight lines and also have a black circle at the centre.

3,5,8 have similar designs and have their four corners shaded black.

In each row, the third figure is a combination of the first and second figure.

Clearly, there are 4 columns containing 1 cube each, 4 columns containing 2 cubs each and 1 column containing 3 cubes.

Hence, there are (4 x 1) + (4 x 2) + (1 x 3) = 4 + 8 + 3 = 15 cubes in the given figure.

The figure may be labelled as shown

Rectangles :

The simplest rectangles are CVSR, VETS, RSWM and STKW i.e 4 in number.

The rectangles composed of two components each are CETR, VEKW, RTKM and CVWM i.e 4 in number.

The rectangles composed of three components each are ACRP, PRMO, EGHT and THIK i.e 4 in number.

The rectangles composed of four components each are CEKM, AVSP, PSWO,VGHS and SHIW i.e 5 in number.

The rectangles composed of five components each are AETP, PTKO, CGHR and RHIM i.e 4 in number.

The rectangles composed of six components each are ACMO and EGIK i.e 2 in number.

The rectangles composed of eight components each are AGHP, PHIO, AVWO and VGIW i.e 4 in number.

The rectangles composed of ten components each are AEKO and CGIM i.e 2 in number.

AGIO is the only rectangle having sixteen components

Total number of rectangles in the given figure = 4 + 4 + 4 + 5 + 4 + 2 + 4 + 2 + 1 = 30.

Hexagons :

The hexagons in the given figure are CDEKLM, CEUKMQ, CFHJMQ, BEUKNP and BFHJNP. So, there are 5 hexagons in the given figure.

Let father's age be 3x and son's age be x years

Father's age after 5 years = 3x + 5 = 26 years

Son's age after 5 years = x + 5 = 12 years

 Ratio of father's and son's ages = 26 : 12 = 13 : 6

Total age of 10 students = 150 years

Total age of 15 students = 240 years

Total age of 5 new students = 240 - 150 = 90 years

 Average age of 5 new students = 90/5 = 18 years.

As the ratio of their present ages is 3 : 4 , let their present ages be 3X and 4X.

So, 5 years ago, as the ratio of their ages was 5 : 7, we can write  (3x-5) : (4x-5) = 5 : 7. Solving, we get X = 10. Hence, their present ages are 3X = 30 and 4X = 40

Given that, A+B = 60  and A = 2B.

Solving, We get A = 40, B = 20.

Five years their ages will be 45 and 25, so the sum of their ages will be 70.