Let d = distance around their track.
Let L = speed of ligtning lady
Let S = speed of stirling monarch
When Ligthning Lady (LL) traveled 500m, SM traveled (d-500) m, and that is for the same time t1.
t1 = 500/L = (d -500)/S
L = 500S /(d-500)
When LL continued and traveled [(d-500) +200]m, SM traveled (500 -200)m, and that is also in the same t2.
t2 = [(d-500) +200] /L = (500-200) /S
(d-300)/L = 300/S
L = (d-300)S /300
L = L
500S /(d-500) = (d-300)S /300
500(300) = (d-500)(d-300)
150,000 = d^2 -1600d +150,000
0 = d^2 -1600d
0 = d -1600
d = 1600 meters ---------answer.