Australia will be eager to tighten their grip on the match as they lead India by 196 runs, with Usman Khawaja batting on 13 along with Marnus Labuschagne on eight.
While Thakur lived dangerously, Rahane went about rescuing India's hopes with his mix of delectable shot selection in the 109-run partnership for the seventh wicket.
While he fell 11 runs short of scoring a century in the second over after lunch, the 35-year-old crossed 5000 test runs during his 129-ball knock which included 11 fours and a six.
Following Rahane's departure, Thakur steered India past the follow-on target before bringing up his 50 with a sumptuous four.
Blinking rapidly, he hurriedly grabbed his bat to make his way to the crease and made sure Australia reached tea without losing any further wickets.
Persons:
Pat Cummins, Usman Khawaja, Marnus Labuschagne, Ajinkya Rahane, Shardul Thakur, Scott Boland, Srikar, Bharat, Thakur, Cummins, Cameron Green, Rahane, India's, Alex Carey, Rohit Sharma's, Umesh Yadav, Mohammed Shami, Labuschagne, David Warner, napping, Warner, Pritha Sarkar, Ken Ferris, Toby Davis
Organizations:
India, Australia, DRS, Cummins, ICC, WTC, Thomson
Locations:
India, Australia, London