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Set an improbable target of 515, Bangladesh were bowled out for 234 in their second innings, with first-inning centurion Ashwin (6/88) doing the bulk of the damage at the his home ground at the Chepauk.
Ravindra Jadeja picked up 3/58.
Skipper Najmul Hossain Shanto top-scored for Bangladesh with an 82 off 127 balls.
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India declared their second innings at 287 for 4 riding on twin hundreds from Rishabh Pant (109) and Shubman Gill (119) for an overall lead of 514.
Brief Scores:
India 376 and 287/4 in 64 overs (Shubman Gill 119 not out, Rishabh Pant 109).
Bangladesh 149 and 234 all out in 62.1 overs (Najmul Hossain Shanto 82, Shadman Islam 35, Zakir Hasan 33; Ravichandran Ashwin 6/88).