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Question Number 57513 by pete last updated on 06/Apr/19
There are 128 players in the first round  of a knockout competition. Half of the  players were knocked out in each round.  How many players took part in the  fourth round? How many rounds were there  in this competion?
$$\mathrm{There}\:\mathrm{are}\:\mathrm{128}\:\mathrm{players}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first}\:\mathrm{round} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{knockout}\:\mathrm{competition}.\:\mathrm{Half}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{players}\:\mathrm{were}\:\mathrm{knocked}\:\mathrm{out}\:\mathrm{in}\:\mathrm{each}\:\mathrm{round}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{players}\:\mathrm{took}\:\mathrm{part}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{fourth}\:\mathrm{round}?\:\mathrm{How}\:\mathrm{many}\:\mathrm{rounds}\:\mathrm{were}\:\mathrm{there} \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{competion}? \\ $$
Commented by mr W last updated on 06/Apr/19
1st round: 128→64  2nd round: 64→32  3rd round: 32→16  4th round: 16→8  5th round: 8→4  6th round: 4→2  7th round: 2→1    ⇒16 players took part in 4th round,  there were 7 rounds.
$$\mathrm{1}{st}\:{round}:\:\mathrm{128}\rightarrow\mathrm{64} \\ $$$$\mathrm{2}{nd}\:{round}:\:\mathrm{64}\rightarrow\mathrm{32} \\ $$$$\mathrm{3}{rd}\:{round}:\:\mathrm{32}\rightarrow\mathrm{16} \\ $$$$\mathrm{4}{th}\:{round}:\:\mathrm{16}\rightarrow\mathrm{8} \\ $$$$\mathrm{5}{th}\:{round}:\:\mathrm{8}\rightarrow\mathrm{4} \\ $$$$\mathrm{6}{th}\:{round}:\:\mathrm{4}\rightarrow\mathrm{2} \\ $$$$\mathrm{7}{th}\:{round}:\:\mathrm{2}\rightarrow\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow\mathrm{16}\:{players}\:{took}\:{part}\:{in}\:\mathrm{4}{th}\:{round}, \\ $$$${there}\:{were}\:\mathrm{7}\:{rounds}. \\ $$
Commented by pete last updated on 06/Apr/19
Thank you sir
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by rahul 19 last updated on 06/Apr/19
(b) 2^x =128 ⇒x=7 rounds.
$$\left({b}\right)\:\mathrm{2}^{{x}} =\mathrm{128}\:\Rightarrow{x}=\mathrm{7}\:{rounds}. \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 06/Apr/19
Rahul ...you pls feed problems in the forum..  we are starved ...feedind is less so forum is getting  slowed down...
$${Rahul}\:…{you}\:{pls}\:{feed}\:{problems}\:{in}\:{the}\:{forum}.. \\ $$$${we}\:{are}\:{starved}\:…{feedind}\:{is}\:{less}\:{so}\:{forum}\:{is}\:{getting} \\ $$$${slowed}\:{down}… \\ $$

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