Samira is 2 years younger than Nasim.

The difference between one-sixth of Samira’s age and one-seventh of Nasim’s age is 1 year.

How old are Samira and Nasim?

ALL SQUARE

In the diagram, ABCD is a square.

E is the midpoint of CB. AE and BD cross at point F.

Prove that the area of the quadrilateral FDCE equals the sum of the areas of triangles ADF and FEB.

SOLUTIONS Age gap (ages 8-12) Samira is 54 and Nasim is 56. Samira’s age must be a multiple of 6, and Nasim’s a multiple of 7. Trying solutions systematically until we fulfil the condition of a 2-year gap: 60; 127; 1814; 2421; 3028; 3635; 4242; 4849; 5456, 6063.

All square (ages 14-16) Let the area of DFEB be a. Ds FEB and ADF are similar, so AF = 2FE, since AD = 2BE. Ds AFB and FEB have the same height, and the base AF is twice the base FE, so DAFB = 2a. DABE = DAFB + DFEB = 3a, and is one quarter of the area of the square ABCD, which is therefore 12a. DADB = DDBC half the area of square ABCD = 6a.

DADF = DADB - DAFB = 6a - 2a = 4a. DADF + DFEB = 4a + a = 5a. Quadrilateral FDCE = DDBC - DFEB = 6a - a = 5a.