If A : B = 2 : 3, B : C = 4 : 5 and C : D = 6 : 7, find A : D.

$\dfrac{A}{B} = \dfrac{2}{3} \\[0.5em] \Rightarrow B = \dfrac{3A}{2} \\[0.5em]$

Putting this value of B in B : C

$\dfrac{B}{C} = \dfrac{4}{5} \\[0.5em] \Rightarrow \dfrac{\dfrac{3A}{2}}{C} = \dfrac{4}{5} \\[0.5em] \Rightarrow \dfrac{3A}{2} = \dfrac{4C}{5} \\[0.5em] \Rightarrow C = \dfrac{15A}{8} \\[0.5em]$

Putting this value of C in C : D

$C : D = 6 : 7 \\[0.5em] \Rightarrow \dfrac{\dfrac{15A}{8}}{D} = \dfrac{6}{7} \\[0.5em] \Rightarrow \dfrac{15A}{8D} = \dfrac{6}{7} \\[0.5em] \Rightarrow \dfrac{A}{D} = \dfrac{48}{105} = \dfrac{16}{35} \\[0.5em] \Rightarrow A : D = 16 : 35.$

Hence, the value of A : D is 16 : 35.