The ratio between the number of sides of two regular polygons is 3 : 4 and ratio between the sum of their interior angles is 2 : 3. Find the number of sides in each polygon.

Let the number of sides of two regular polygons be 3a and 4a.

Sum of interior angles of regular polygons = (n - 2) x 180°

$⇒ \dfrac{(3a - 2) \times 180°}{(4a - 2) \times 180°} = \dfrac{2}{3}$

⇒ 3[(3a - 2) x 180°] = 2[(4a - 2) x 180°]

⇒ (9a - 6) x 180° = (8a - 4) x 180°

⇒ 9a - 6 = 8a - 4

⇒ 9a - 8a = 6 - 4

⇒ a = 2

Thus, the number of sides of the polygons are 3a = 3 x 2 = 6 and 4a = 4 x 2 = 8.

Hence, the number of sides in the two polygons are 6 and 8.