Question Number 198321 by necx122 last updated on 17/Oct/23
$${A}\:{certain}\:{preliminary}\:{science}\:{class} \\ $$$${contains}\:\mathrm{50}\:{students}\:{all}\:{of}\:{whom}\:{take} \\ $$$${Mathematics}.\:\mathrm{18}\:{study}\:{Chemistry},\:\mathrm{17} \\ $$$${study}\:{Biology},\:\mathrm{24}\:{study}\:{Physics}.\:{Of} \\ $$$${those}\:{taking}\:{three}\:{subjects},\:\mathrm{5}\:{study} \\ $$$${Physics}\:{and}\:{Chemistry},\:\mathrm{7}\:{study}\:{Physics} \\ $$$${and}\:{Biology}\:{and}\:\mathrm{6}\:{study}\:{Chemistry} \\ $$$${and}\:{Biology}\:{while}\:\mathrm{2}\:{take}\:{all}\:{four}\:{subjects}. \\ $$$${How}\:{many}\:{students}\:{study}\:{only}\:{Mathematics}? \\ $$
Commented by necx122 last updated on 17/Oct/23
This is another question that troubled me. please help me with the Venn diagram and the solution . Thanks in advance.
Answered by mr W last updated on 17/Oct/23
Commented by mr W last updated on 17/Oct/23
$$\mathrm{13}\:{study}\:{only}\:{mathematics}. \\ $$
Commented by necx122 last updated on 17/Oct/23
Thank you sir. I really understand this but the textbook says the answer is 7 students.
Commented by mr W last updated on 17/Oct/23
$${then}\:{textbook}\:{is}\:{wrong}. \\ $$
Commented by necx122 last updated on 17/Oct/23
Ok sir. I actually believe your solving than my textbooks however. After seeing your Venn diagram and the answer you got I resubstituted the values and got 7.
I substituted it such that the intersection between each of biology and physics only, biology and chemistry only and chemistry and physics only to reduce by 2. Thus giving us 5 , 4 and 3.
Yet, I'll want to agree with you sir. Thank you.
Commented by AST last updated on 18/Oct/23
$${The}\:{answer}\:{should}\:{be}\:\mathrm{7}. \\ $$
Commented by mr W last updated on 18/Oct/23
$${can}\:{you}\:{please}\:{show}\:{the}\:{corresponding} \\ $$$${venn}\:{diagram}? \\ $$
Commented by AST last updated on 18/Oct/23
Commented by AST last updated on 18/Oct/23
$${We}\:{can}\:{also}\:{use}\:{the}\:{formula}\:{in}\:{the}\:{answer}\: \\ $$$${below}. \\ $$
Commented by mr W last updated on 18/Oct/23
$${according}\:{to}\:{question}: \\ $$$$\underline{{of}\:{those}\:{taking}\:{three}\:{subjects}},\:\mathrm{5}\:{study} \\ $$$${physics}\:{and}\:{chemistry}. \\ $$$${but}\:{according}\:{to}\:{your}\:{venn}\:{diagram} \\ $$$${only}\:\mathrm{3}\:{study}\:{physics}\:{and}\:{chemistry}. \\ $$
Commented by AST last updated on 18/Oct/23
$$\mathrm{3}+\mathrm{2}=\mathrm{5},\:{we}\:{add}\:{the}\:\mathrm{2}\:{from}\:{P},{C}\:{and}\:{B}. \\ $$$$\mathrm{3}\:{take}\:{Physics}\:{and}\:{Chemistry}\:{only},\:{but}\:\mathrm{3}+\mathrm{2}=\mathrm{5} \\ $$$${take}\:{Physics}\:{and}\:{Chemistry}. \\ $$
Commented by mr W last updated on 18/Oct/23
$${but}\:{the}\:\mathrm{2}\:{students}\:{take}\:{all}\:{four}\:{subjects}, \\ $$$${so}\:{they}\:{are}\:{not}\:{those}\:{taking}\:{three}\: \\ $$$${subjects}. \\ $$
Commented by mr W last updated on 18/Oct/23
$${i}\:{only}\:{accept}\:{the}\:{answer}\:{of}\:{the}\:{textbook}, \\ $$$${if}\:{in}\:{the}\:{question}\:{the}\:{underlined} \\ $$$${sentence}\:{doesn}'{t}\:{exist},\:{or}\:{if}\:{it}\:{is} \\ $$$$\underline{{of}\:{those}\:{taking}\:{at}\:{least}\:{three}\:{subjects}.} \\ $$
Commented by AST last updated on 18/Oct/23
$${My}\:{solution}\:{below}\:{is}\:{also}\:{another}\:{way}\:{to}\:{look} \\ $$$${at}\:{it}. \\ $$
Commented by mr W last updated on 18/Oct/23
$${at}\:{first}\:{i}\:{solved}\:{exactly}\:{like}\:{you},\:{then} \\ $$$${i}\:{changed}\:{my}\:{mind}\:{due}\:{to}\:{the} \\ $$$${sentence}\:“\underline{{of}\:{those}\:{taking}\:{three}\:} \\ $$$$\underline{{subjects}},\:….''.\: \\ $$$${anyway}\:{i}\:{think}\:{the}\:{answer}\:\mathrm{7}\:{in}\:{textbook}\: \\ $$$${is}\:{wrong},\:{if}\:{we}\:{strictly}\:{follow}\:{the}\: \\ $$$${text}\:{in}\:{the}\:{question}. \\ $$
Commented by AST last updated on 18/Oct/23
$$\mathrm{2}\:{take}\:{all}\:{subjects}\:{imply}\:\mathrm{2}\:{take}\:{Physics}, \\ $$$${Chemistry}\:{and}\:{Biology}. \\ $$
Commented by mr W last updated on 18/Oct/23
$${and}\:{mathematics}\:{as}\:{well},\:{since}\:{all} \\ $$$${students}\:{in}\:{the}\:{class}\:{study}\: \\ $$$${mathematics}. \\ $$
Commented by AST last updated on 18/Oct/23
$${Yes},\:{but}\:{we}\:{are}\:{considering}\:{P},{B}\:{and}\:{C}\:{like}\:{in} \\ $$$${the}\:{answer}\:{below}. \\ $$$${We}\:{find}\:{the}\:{total}\:{number}\:{that}\:{take}\:{P},{B}\:{or}\:{C}, \\ $$$${which}\:{is}\:\mathrm{43}.\:{This}\:{means}\:{there}\:{are}\:\mathrm{50}−\mathrm{43}=\mathrm{7}\:{that} \\ $$$${do}\:{no}\:{take}\:{any}\:{of}\:{P},{B}\:{or}\:{C}.{So},{the}\:{remaining} \\ $$$${take}\:{Maths}\:{only}. \\ $$
Commented by necx122 last updated on 18/Oct/23
Thank you very much sir. These comments have helped me understand the question properly. Thank you.
Answered by AST last updated on 18/Oct/23
$$\mid{PuBuC}\mid=\mid{P}\mid+\mid{B}\mid+\mid{C}\mid−\mid{PnB}\mid−\mid{BnC}\mid−\mid{PnC}\mid \\ $$$$+\mid{PnBnC}\mid\Rightarrow\mid{PuBuC}\mid=\mathrm{18}+\mathrm{17}+\mathrm{24}−\mathrm{5}−\mathrm{7}−\mathrm{6}+\mathrm{2} \\ $$$$=\mathrm{43} \\ $$$$\Rightarrow\mid{Maths}\:{only}\mid=\mathrm{50}−\mathrm{43}=\mathrm{7} \\ $$