SOLUTION: If n(A∪B)=40, n(A∩B)=10 and n(A)=15, find n(B)

By Venn diagram:



n(A∪B) = x + y + z = 40, 

n(A∩B) = y = 10 

n(A) = x + y = 15

n(B) = y + z = ?

So the system of equations is:

(eq. 1)              x + y + z = 40
(eq. 2)                  y     = 10
(eq. 3)              x + y     = 15

Subtract (eq. 3) minus (eq. 2)

 (eq. 3)              x + y     =  15
-(eq. 2)                 -y     = -10
----------------------------------------
                      x         =  5

Subtract (eq. 1) minus (eq. 3)

 (eq. 1)              x + y + z =  40

-(eq. 3)             -x - y     = -15
----------------------------------------
                              z =  25

Substitute x = 5 in (eq. 3)

 (eq. 3)              5 + y     =  15  
                          y     =  10



n(B) = y + z = 10 + 25 = 35

----------------------------------

By formula

n(A∪B) = n(A) + n(B) - n(A∩B) 
  40   =  15  + n(B) -  10
  40   =   5  + n(B)
  35   =  n(B)

It's easier to use the formula, but you don't learn 
what's going on unless you understand the Venn diagram
method.

Edwin