Let time taken to cover the distance = t

Therefore, Speed = \( \Large \left(2t + 1\right) \)

= \( \Large t \left(2t+1\right)=1830 = 2t^{2}+t=1830 \)

= \( \Large 2t^{2}+t-1830 = 0 \)

Therefore \( \Large t = \frac{-1 \pm \sqrt{ \left(1\right)^{2}-4 \times 2 \times \left(-1830\right) }}{2 \times 2} \)

= \( \Large \frac{-1\pm \sqrt{1+14640}}{4} = \frac{-1\pm 121}{4} \)

Taking '+' sign, \( \Large t = \frac{-1+121}{4}=\frac{120}{4}=30 \)

Therefore, Required ratio = \( \Large \left(2 \times 30+1\right) : 30 = 61 : 30 \)