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

$x = 10$ here

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@Arjun sir

how $x=10?$ i got $x=5$

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Basically the $25$ you marked must include the 'x' also as the question says "25 play both cricket and hockey" and not "25 play only cricket and hockey". Like wise for all groups of two you have to include group of 3 also.

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Yes, sir I will correct it.

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@Lakshman Patel RJIT plz calculate no of people participate in all three. I am not able to calculate it from given data.

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@Shubhm

number of people participated in all three games $x = 300 - 295 = 5$

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thanks @Lakshman Patel RJIT

I considered it like, a + b = 25, but in actual there is, a = 25

 

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$a+b = 25$ is given in the question

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But how u get it when u took a+b=25 can u plz describe by the help of ven diagram how to find the no of player who play all 3 game only.

@Lakshman Patel RJIT

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Here, $a + x = 25\rightarrow(1)$

$b + x = 30\rightarrow(2)$

$c + x = 15\rightarrow(3)$

 

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Please clear my doubt.

with reference to the above venn diagram:

if x (number of people who played all three sports) = 5, then a = 20, b = 25, c = 10

a+b+c+x + C + H + F = 5+20+25+10+105+50+70 = 285

Please explain where i am doing wrong. Why is the summation not equal to 300?

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All the data in the question correspond to exclusive parts of the Venn diagram. So you can just add up everything and get 295. Remaining 5 plays all 3 games.

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Sir can u tell me is this question is ambigous bcz after adding we cant get  300

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@Lakshman Patel RJIT

as per the Venn diagram you given .

lets take the #peoples who play all three as X. 

then those who play C and H = 25 – X

those who play H and F = 15 – X

those who play C and F = 30 – X

Now, as the total #peoples = 300

Then  105 + 70 + 50 + 25 – X + 15 – X + 30 – X  + X = 300

               295 – 2X = 300.

getting X as negative.

where I am going wrong?

@Abhrajyoti00

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@Pranavpurkar Why are you taking those who play C and H = 25 – X ? It should be those who play C and H = 25 (as given in the question). Similarly for other two.

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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?