GATE2019 CE-1: GA-7

Question

In a sports academy of $300$ peoples, $105$ play only cricket, $70$ play only hockey, $50$ play only football, $25$ play both cricket and hockey, $15$ play both hockey and football and $30$ play both cricket and football. The rest of them play all three sports. What is the percentage of people who play at least two sports?

• A. $23.30$
• B. $25.00$
• C. $28.00$
• D. $50.00$ 

Answer 1

Since everyone plays at least one sports, to get the number of people who plays at least two sports we can do

$\text{Total} - \text{No. of people playing only one game}$

$\qquad=300 - (105+70+50)$

$ \qquad= 75.$

So, percentage of people who play at least $2$ sports $ = \frac{75}{300} \times 100\% = 25\%.$

Correct Answer: Option B.

Answer 2

In the sports academy, the total number of people $n(U)=300$

Number of people who play only cricket $n(C)=105$

Number of people who play only hockey $n(H)=70$

Number of people who play only football $n(F)=50$

Number of people, who play both cricket and hockey $n(C\cap H)=25$

Number of people, who play both hockey and football $n(H\cap F)=15$

Number of people, who play both cricket and football $n(C\cap F)=30$

The number of people who play only one sport $n(C)+n(H)+n(F)=105+70+50=225$

The number of people, who pay at least two sport $($it mean the total number of people $-$ the number of people who play only one sport$)=300-225=75$  $($at least two sports means two or more sports$)$

Percentage $=\dfrac{75}{300}\times 100=25\%$

So,correct answer is $(B).$

Comments

Abhrajyoti

in the question they have mentioned 

 25 play both cricket and hockey

not like 25 play both cricket and hockey and not football

also see this comment of @Arjun Sir

 https://gateoverflow.in/313440/gate2019-ce-1-ga-7?show=314328#c314328

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@Pranavpurkar Exactly what I'm saying. So why aren't you taking x? Why are you subtracting X? 

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See 25 is including X and if i want just a(See the Venn diag in my comment) then it will be 25-X na.

as X is the #people playing all three. and 25-X is #people playing cricket and Hockey only. So we have to exclude X from here right?