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Question Number 57023 by otchereabdullai@gmail.com last updated on 28/Mar/19

There are 28 players in a national football team. 14 play midfield and defence, 15 play defence and attack and 3 play midfield only. the number of players who play attack only is twice those who play defence only, and the number who play defence is equal to those who play attack. If 18 play midfield represent the information on a venn diagram. Find i. how many play at most two games ii. how many play neither attack nor defence iii. how many play either midfield or attack.

$$\mathrm{There}\:\mathrm{are}\:\mathrm{28}\:\mathrm{players}\:\mathrm{in}\:\mathrm{a}\:\mathrm{national}\: \\ $$$$\mathrm{football}\:\mathrm{team}.\:\mathrm{14}\:\mathrm{play}\:\mathrm{midfield}\:\mathrm{and} \\ $$$$\mathrm{defence},\:\mathrm{15}\:\mathrm{play}\:\mathrm{defence}\:\mathrm{and}\:\mathrm{attack}\: \\ $$$$\mathrm{and}\:\mathrm{3}\:\mathrm{play}\:\mathrm{midfield}\:\mathrm{only}.\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{players}\:\mathrm{who}\:\mathrm{play}\:\mathrm{attack}\:\mathrm{only}\:\mathrm{is}\: \\ $$$$\mathrm{twice}\:\mathrm{those}\:\mathrm{who}\:\mathrm{play}\:\mathrm{defence}\:\mathrm{only},\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{who}\:\mathrm{play}\:\mathrm{defence} \\ $$$$\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{those}\:\mathrm{who}\:\mathrm{play}\:\mathrm{attack}.\:\mathrm{If}\: \\ $$$$\mathrm{18}\:\mathrm{play}\:\mathrm{midfield}\:\mathrm{represent}\:\mathrm{the}\: \\ $$$$\mathrm{information}\:\mathrm{on}\:\mathrm{a}\:\mathrm{venn}\:\mathrm{diagram}. \\ $$$$\mathrm{Find} \\ $$$$\mathrm{i}.\:\mathrm{how}\:\mathrm{many}\:\mathrm{play}\:\mathrm{at}\:\mathrm{most}\:\mathrm{two}\:\mathrm{games} \\ $$$$\mathrm{ii}.\:\mathrm{how}\:\mathrm{many}\:\mathrm{play}\:\mathrm{neither}\:\mathrm{attack}\:\mathrm{nor} \\ $$$$\mathrm{defence} \\ $$$$\mathrm{iii}.\:\mathrm{how}\:\mathrm{many}\:\mathrm{play}\:\mathrm{either}\:\mathrm{midfield}\:\mathrm{or}\: \\ $$$$\mathrm{attack}. \\ $$

Answered by MJS last updated on 29/Mar/19

the venn diagram has got 7 areas (1) m+a+d+ma+md+ad+mad=28 (2) md+mad=14 ⇒ md=14−mad (3) ad+mad=15 ⇒ ad=15−mad (4) m=3 (5) a=2d (6) d+md+ad+mad=a+ma+ad+mad ⇒ ⇒ d+md=a+ma (7) m+ma+md+mad=18 (1) 3+2d+d+ma+14−mad+15−mad+mad=28 ⇒ ⇒ 3d+ma−mad=−4 (6) d+14−mad=2d+ma ⇒ ⇒ d+ma+mad=14 (7) 3+ma+14−mad+mad=18 ⇒ ma=1 (1) 3d−mad=−5 (6) d+mad=13 ⇒ d=2; mad=11 m=3 a=4 d=2 ma=1 md=3 ad=4 mad=11 i. 28−mad=17 ii. m=3 iii. m+a+md+ad=14

$$\mathrm{the}\:\mathrm{venn}\:\mathrm{diagram}\:\mathrm{has}\:\mathrm{got}\:\mathrm{7}\:\mathrm{areas} \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\:{m}+{a}+{d}+{ma}+{md}+{ad}+{mad}=\mathrm{28} \\ $$$$\left(\mathrm{2}\right)\:\:{md}+{mad}=\mathrm{14}\:\Rightarrow\:{md}=\mathrm{14}−{mad} \\ $$$$\left(\mathrm{3}\right)\:\:{ad}+{mad}=\mathrm{15}\:\Rightarrow\:{ad}=\mathrm{15}−{mad} \\ $$$$\left(\mathrm{4}\right)\:\:{m}=\mathrm{3} \\ $$$$\left(\mathrm{5}\right)\:\:{a}=\mathrm{2}{d} \\ $$$$\left(\mathrm{6}\right)\:\:{d}+{md}+{ad}+{mad}={a}+{ma}+{ad}+{mad}\:\Rightarrow \\ $$$$\:\:\:\:\:\Rightarrow\:{d}+{md}={a}+{ma} \\ $$$$\left(\mathrm{7}\right)\:\:{m}+{ma}+{md}+{mad}=\mathrm{18} \\ $$$$ \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\:\mathrm{3}+\mathrm{2}{d}+{d}+{ma}+\mathrm{14}−{mad}+\mathrm{15}−{mad}+{mad}=\mathrm{28}\:\Rightarrow \\ $$$$\:\:\:\:\:\Rightarrow\:\mathrm{3}{d}+{ma}−{mad}=−\mathrm{4} \\ $$$$\left(\mathrm{6}\right)\:\:{d}+\mathrm{14}−{mad}=\mathrm{2}{d}+{ma}\:\Rightarrow \\ $$$$\:\:\:\:\:\Rightarrow\:{d}+{ma}+{mad}=\mathrm{14} \\ $$$$\left(\mathrm{7}\right)\:\:\mathrm{3}+{ma}+\mathrm{14}−{mad}+{mad}=\mathrm{18}\:\Rightarrow\:{ma}=\mathrm{1} \\ $$$$ \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\:\mathrm{3}{d}−{mad}=−\mathrm{5} \\ $$$$\left(\mathrm{6}\right)\:\:{d}+{mad}=\mathrm{13} \\ $$$$\:\:\:\:\:\Rightarrow\:{d}=\mathrm{2};\:{mad}=\mathrm{11} \\ $$$$ \\ $$$${m}=\mathrm{3} \\ $$$${a}=\mathrm{4} \\ $$$${d}=\mathrm{2} \\ $$$${ma}=\mathrm{1} \\ $$$${md}=\mathrm{3} \\ $$$${ad}=\mathrm{4} \\ $$$${mad}=\mathrm{11} \\ $$$$ \\ $$$$\mathrm{i}.\:\mathrm{28}−{mad}=\mathrm{17} \\ $$$$\mathrm{ii}.\:{m}=\mathrm{3} \\ $$$$\mathrm{iii}.\:{m}+{a}+{md}+{ad}=\mathrm{14} \\ $$

Commented by otchereabdullai@gmail.com last updated on 29/Mar/19

thanks prof

$$\mathrm{thanks}\:\mathrm{prof} \\ $$

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