Lesson Nice problems to setup them using Venn diagram

Nice problems to setup them using Venn diagram

Problem 1

               water park only     BOTH      roller skating only

             |----------------|------------|-----------------------|

                  4y = 2x            x                 y



Let x = "How many students did both".  and let y = "roller skating only"   (see the Figure above)


From the Figure and from the condition, we have these equations


    x + y + 2x = 210     (1)

    4y = 2x              (2)


We simplify  equation (2) to  x = 2y,  and substitute this expression into equation (1).

We have then


    (2y) + y + 2*(2y) = 210

     2y  + y +  4y    = 210

          7y          = 210

           y          = 210/7 = 30.


Thus x = 30:   30 students did BOTH.    ANSWER

Problem 2

The figure below shows all the class, from A to D.



        P h y s i c s  (from  A  to  C)

     |---------------|------------------------|---------------------|

     A               B                        C                     D
                        Both (from B to C)

                           C h e m i s t r y  (from  B  to  D)



Part of the class (from A to C) learns Physics.

Part of the class (from B to D) learns Chemistry.

Those in the intersection, from B to C, learn BOTH.


Let x be the unknown number of those, who learn BOTH.

Let P be the number of those who learn Physics and C be the number of those who learn Chemistry.



The problem says that  0.6P = x;  hence, the number of those who learn Physics only is 0.4P.

    Since 0.6P = x,  it implies that 0.4P =  = .


Next, the problem says that   = x;  hence, the number of those who learn Chemistry only is .

    Since  = x,  it implies that  = 2x.


So, the parts are  AB = ;  BC = x;  and  CD = 2x.


Altogether, these 3 parts comprise 110 students.


It means that   + x + 2x = 110.


At this point, the setup is done (!)


To solve this equation and to find x, multiply both sides by 3


    2x + 3x + 6x = 330

        11x      = 330

          x      = 330/11 = 30.


ANSWER.  30 students study both subjects.