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Question Number 52741 by Otchere Abdullai last updated on 12/Jan/19
   please help me solve this question sir.  In a survey of  100 out-patients who   reported at the hospital one day, It   was found out that 70 coplained of   fever, 50 complained of stomach   ache and 30 were  injured. All    patients had at least one of the   complaints and 44 had exactly two   of the complaints. How many had all   complaints?
$$\: \\ $$$${please}\:{help}\:{me}\:{solve}\:{this}\:{question}\:{sir}. \\ $$$$\mathrm{I}{n}\:{a}\:{survey}\:{of}\:\:\mathrm{100}\:{out}-{patients}\:{who}\: \\ $$$${reported}\:{at}\:{the}\:{hospital}\:{one}\:{day},\:{It}\: \\ $$$${was}\:{found}\:{out}\:{that}\:\mathrm{70}\:{coplained}\:{of}\: \\ $$$${fever},\:\mathrm{50}\:{complained}\:{of}\:{stomach}\: \\ $$$${ache}\:{and}\:\mathrm{30}\:{were}\:\:{injured}.\:{All}\:\: \\ $$$${patients}\:{had}\:{at}\:{least}\:{one}\:{of}\:{the}\: \\ $$$${complaints}\:{and}\:\mathrm{44}\:{had}\:{exactly}\:{two}\: \\ $$$${of}\:{the}\:{complaints}.\:{How}\:{many}\:{had}\:{all}\: \\ $$$${complaints}? \\ $$
Answered by mr W last updated on 12/Jan/19
A...=patients with exact one complaint  B...=patients with exact two complaints  C=patients with all three complaints  A_1 +B_1 +B_3 +C=70   ...(1)  A_2 +B_1 +B_2 +C=50   ...(2)  A_3 +B_2 +B_3 +C=30   ...(3)  (1)+(2)+(3):  A_1 +A_2 +A_3 +2(B_1 +B_2 +B_3 )+3C=150   ...(4)  A_1 +A_2 +A_3 +B_1 +B_2 +B_3 +C=100   ...(5)  (4)−(5):  B_1 +B_2 +B_3 +2C=50    ...(6)  B_1 +B_2 +B_3 =44    ...(7)  (6)−(7):  2C=6  ⇒C=3  i.e. three patients have all complaints.
$${A}…={patients}\:{with}\:{exact}\:{one}\:{complaint} \\ $$$${B}…={patients}\:{with}\:{exact}\:{two}\:{complaints} \\ $$$${C}={patients}\:{with}\:{all}\:{three}\:{complaints} \\ $$$${A}_{\mathrm{1}} +{B}_{\mathrm{1}} +{B}_{\mathrm{3}} +{C}=\mathrm{70}\:\:\:…\left(\mathrm{1}\right) \\ $$$${A}_{\mathrm{2}} +{B}_{\mathrm{1}} +{B}_{\mathrm{2}} +{C}=\mathrm{50}\:\:\:…\left(\mathrm{2}\right) \\ $$$${A}_{\mathrm{3}} +{B}_{\mathrm{2}} +{B}_{\mathrm{3}} +{C}=\mathrm{30}\:\:\:…\left(\mathrm{3}\right) \\ $$$$\left(\mathrm{1}\right)+\left(\mathrm{2}\right)+\left(\mathrm{3}\right): \\ $$$${A}_{\mathrm{1}} +{A}_{\mathrm{2}} +{A}_{\mathrm{3}} +\mathrm{2}\left({B}_{\mathrm{1}} +{B}_{\mathrm{2}} +{B}_{\mathrm{3}} \right)+\mathrm{3}{C}=\mathrm{150}\:\:\:…\left(\mathrm{4}\right) \\ $$$${A}_{\mathrm{1}} +{A}_{\mathrm{2}} +{A}_{\mathrm{3}} +{B}_{\mathrm{1}} +{B}_{\mathrm{2}} +{B}_{\mathrm{3}} +{C}=\mathrm{100}\:\:\:…\left(\mathrm{5}\right) \\ $$$$\left(\mathrm{4}\right)−\left(\mathrm{5}\right): \\ $$$${B}_{\mathrm{1}} +{B}_{\mathrm{2}} +{B}_{\mathrm{3}} +\mathrm{2}{C}=\mathrm{50}\:\:\:\:…\left(\mathrm{6}\right) \\ $$$${B}_{\mathrm{1}} +{B}_{\mathrm{2}} +{B}_{\mathrm{3}} =\mathrm{44}\:\:\:\:…\left(\mathrm{7}\right) \\ $$$$\left(\mathrm{6}\right)−\left(\mathrm{7}\right): \\ $$$$\mathrm{2}{C}=\mathrm{6} \\ $$$$\Rightarrow{C}=\mathrm{3} \\ $$$${i}.{e}.\:{three}\:{patients}\:{have}\:{all}\:{complaints}. \\ $$
Commented by mr W last updated on 12/Jan/19
Commented by Otchere Abdullai last updated on 12/Jan/19
God richly bless you sir infact if   there is something greater than   blessing thus what i pray for you sir
$${God}\:{richly}\:{bless}\:{you}\:{sir}\:{infact}\:{if}\: \\ $$$${there}\:{is}\:{something}\:{greater}\:{than}\: \\ $$$${blessing}\:{thus}\:{what}\:{i}\:{pray}\:{for}\:{you}\:{sir} \\ $$
Commented by mr W last updated on 12/Jan/19
thank you sir!
$${thank}\:{you}\:{sir}! \\ $$

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