Distance travelled by body $A$ in ${5^{th}}$sec and distance travelled by body $B$ in ${3^{rd}}$ sec. of its motion are equal.

$0 + \frac{{{a_1}}}{2}(2 \times 5 - 1) = 0 + \frac{{{a_2}}}{2}[2 \times 3 - 1]$

$9{a_1} = 5{a_2} \Rightarrow \frac{{{a_1}}}{{{a_2}}} = \frac{5}{9}$