The Tigers also removed three West Indies batsmen early at Sher-e-Bangla National Cricket Stadium on Saturday.
At the end of day three, the West Indies are leading by 154 runs with seven wickets in hand. They lost three early wickets, which helped the Tigers bring parity in the game.
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Bangladesh managed to score 296 in the first innings – on the back of Mushfiqur (54), Liton (71) and Mehidy (57). But Rahkeem Cornwall removed both overnight batsmen – Mithun and Mushfiqur.
Cornwall, the right-arm spinner, registered a five-wicket haul for the tourists conceding 74 runs while Shannon Gabriel took three and Alzarri Joseph scalped two.
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The West Indies secured a handy lead of 113 runs at the end of Bangladesh's first innings.
In the last session of day three, the West Indies batted for 21 overs and scored 41 runs losing three wickets.
Nayeem Hasan took the first wicket while Mehidy and Taijul Islam took the other two wickets.
And Mehidy became the fastest Bangladeshi bowler to take 100 Test wickets in 24 innings. He surpassed Taijul who needed 25 innings.
Mehidy is now the fourth Bangladeshi bowler to grab 100 or more wickets in Tests.
Also read: Mithun, Mushfiq fall as Bangladesh fight to avoid follow-on
Dhaka Test score after day three
West Indies (first innings) 409/10 in 142.2 overs (Joshua 92, Bonner 90, Joseph 82); Abu Jayed 4/98, Taijul 4/108, Soumya 1/48
Bangladesh (first innings) 296/10 in 96.5 overs (Liton 71, Mehidy 57, Mushfiqur 54); Cornwall 74/5, Gabriel 3/70
West Indies (second innings) 41/3 in 21 overs (Campbell 18, Nkrumah Bonner 8*, Nkrumah Bonner 2*); Taijul 1/13, Nayeem 1/14, Mehidy 1/14