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Some students are playing some sports on a playground. 35% of the students are playing cricket, 25% of the students are playing football, and 10% of the students only play both cricket and football. 48 Students are playing any other sport while 10% of students do not play any sport. So how many students only play football on the field?

Some students are playing some sports on a playground. 35% of the students are playing cricket, 25% of the students are playing football, and 10% of the students only play both cricket and football. 48 Students are playing any other sport while 10% of students do not play any sport. So how many students only play football on the field? Correct Answer 18

Given:

Playing cricket = 35%

Playing football = 25%

Play both cricket and football = 10%

10% of students do not play any sport

48 Students are playing any other sport

Formula used:

Total number ⇒ n (A ∪ B) = n (A) + n (B) - n (A∩ B)

Where n (A) = the number of elements in set A.

Calculation:

Let the total number of students be x.

35% of the students are playing cricket

Then, the number of students who playing cricket = 35% of x

⇒ (35 / 100) × x = 7x / 20

25% of the students are playing football

Then, the number of students who playing football = 25% of x

⇒ (25 / 100) × x = x / 4

10% of the students only play both cricket and football

Then, the number of students who only play both cricket and football = 10% of x

⇒ (10 / 100) × x = x / 10

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Total number of students who play cricket and football

By formula,

⇒ (7x / 20) + (x / 4) - (x / 10) = x / 2

10% of students do not play any sport

Then, the number of students who do not play any sport = 10% of x

⇒ (10 / 100) × x = x / 10

Now the number of remaining students who play other sports

⇒ x - (x / 2) - (x / 10) = 2x / 5       ---(i)

But by the question, the number of remaining students who play other sports = 48       ---(i)

Now by equation (i) and (ii), we get

⇒ 2x / 5 = 48

⇒ x = 120

Now the number of students who play only football

⇒ (the number of students who playing football) - (the number of students who only play both cricket and football)

⇒ (x / 4) - (x / 10) = 3x / 20

Now put the value of x and we get

⇒ (3 / 20) × 120 = 18

∴ The number of students who play only football is 18.

Alternate method:

By percentage:

Let the total students be 100%.

By n (A ∪ B) = n (A) + n (B) - n (A∩ B)

Total number of students who play cricket and football

⇒ 35 + 25 - 10 = 50%

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10% students do not play any sport

Now the number of remaining student who play other sports

⇒ 100 - 50 - 10 = 40%

But by question, then

⇒ 40% = 48 then 100% = (48 / 40) × 100 = 120 students

Now the number of student who play only football

⇒ (the number of students who playing football) - (the number of students who only play both cricket and football)

⇒ 25 - 10 = 15%

⇒ 15% of 120

⇒ (15 / 100) × 120 = 18

∴ The number of student who play only football is 18.

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