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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12

B. 10

C. 11

D. 15

E. 14

Can someone set this up for me?

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,464

Draw three overlapping circles. There are 8 regions including the outside (= 18)

Label each circle like this H = 20 etc but do not put this number inside any of the regions. The number in a region should be the number who do just those sports. There's no easy way to fill in the individual numbers because, although enough information given, you cannot immediately calculate any region yet.

So put x for the number in H + C + F, the middle region. (Cannot find a symbol for intersection in the list above the posts)

Then write expressions for H + C + F' = 7 in terms of x and the other two sport regions.

Continue writing in expressions with x in them for all the regions and hence form an equation for x.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

Bob wrote:

Draw three overlapping circles. There are 8 regions including the outside (= 18)

Label each circle like this H = 20 etc but do not put this number inside any of the regions. The number in a region should be the number who do just those sports. There's no easy way to fill in the individual numbers because, although enough information given, you cannot immediately calculate any region yet.

So put x for the number in H + C + F, the middle region. (Cannot find a symbol for intersection in the list above the posts)

Then write expressions for H + C + F' = 7 in terms of x and the other two sport regions.

Continue writing in expressions with x in them for all the regions and hence form an equation for x.

Bob

I drew 3 circles as you suggested but was unable to form the needed equation.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,464

Ok. Here's a start:

The total in the H circle is 20. Don't write it inside the circle as the circle has 4 regions and they total 20 when you add up all four.

The overlap of all three (ie. those who do all three sports) is x.

We know that H and C is 7. H and C overlap in 2 regions that I'm calling H+C+F and H+C+F' (hockey, cricket but not football)

As H+C+F is x, that means that H+C+F' = 7-x

You have the information to similarly do H+C'+F (hockey and football but not cricket) and H'+C+F.

Finally you can do H+C'+F' (just hockey on its own) so that the four regions add up to 20 and the others similarly.

Take 18 from 50 to get the total of the 7 regions within the circles and make an eqaution using all those x expressions and solve for x. Then you can write in all the numbers for the regions. Check consistency with the initial information.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

Bob wrote:

Ok. Here's a start:

https://i.imgur.com/SrBFYZG.gif

The total in the H circle is 20. Don't write it inside the circle as the circle has 4 regions and they total 20 when you add up all four.

The overlap of all three (ie. those who do all three sports) is x.

We know that H and C is 7. H and C overlap in 2 regions that I'm calling H+C+F and H+C+F' (hockey, cricket but not football)

As H+C+F is x, that means that H+C+F' = 7-x

You have the information to similarly do H+C'+F (hockey and football but not cricket) and H'+C+F.

Finally you can do H+C'+F' (just hockey on its own) so that the four regions add up to 20 and the others similarly.

Take 18 from 50 to get the total of the 7 regions within the circles and make an eqaution using all those x expressions and solve for x. Then you can write in all the numbers for the regions. Check consistency with the initial information.

Bob

Thank you for the venn diagram photo but I still can't come up with the equation. What is the correct equation to solve for x?

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**amnkb****Member**- Registered: 2023-09-19
- Posts: 253

sologuitar wrote:

In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

Can someone set this up for me?

Bob wrote:

Ok. Here's a start:

sologuitar wrote:

I still can't come up with the equation. What is the correct equation to solve for x?

'set this up for me'

'what is correct eqn?'

*Last edited by amnkb (2023-11-22 12:20:31)*

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

amnkb wrote:

sologuitar wrote:In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

Can someone set this up for me?

Bob wrote:Ok. Here's a start:

https://i.imgur.com/SrBFYZG.gif

sologuitar wrote:I still can't come up with the equation. What is the correct equation to solve for x?

'set this up for me'

'what is correct eqn?'

Reply using LaTex form. I am tired of clicking on rectangles to see the answer.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,464

See my important comment here: http://www.mathisfunforum.com/viewtopic … 44#p436044

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**sologuitar****Member**- Registered: 2022-09-19
- Posts: 467

Moving on....

Thanks.

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