We have MN || BC,

So, ∠AMN = ∠B and ∠ANM = ∠C (Corresponding angles)

We know that if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar (AA criteria).

∴ ΔAMN ~ ΔABC.

We know that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

i.e. ar(ΔAMN)/ ar(ΔABC) = (AM/AB)²

Given that AM: MB = 1: 2.

Since AB = AM + MB,

AB = 1 + 2 = 3.

⇒ ar(ΔAMN)/ ar(ΔABC) = (1/3)²

⇒ ar(ΔAMN)/ ar(ΔABC) = 1/9

area(ΔAMN)/ area(ΔABC) = 1/9