There are N stations on a railroad. After adding x stations 46 additional tickets have to be printed. Find N and X.

a)x=5 and N=15
b)x=9 and N=13
c)x=2 and N=11

Answer is x=2 and N=11

• Let before adding X stations, total number of tickets
  
  t = N(N-1)
  
  After adding X stations total number of tickets are
  
  t + 46 = (N+X)(N+X-1)
  
  Subtracting 1st from 2nd
  
  46 = (N+X)(N+X-1) - N(N-1)
  
  46 = N2 + NX - N + NX + X2 - X - N2 + N
  
  46 = 2NX + X2 - X
  
  46 = (2N - 1)X + X2
  
  X2 + (2N - 1)X - 46 = 0
  
  Now there are only two possible factors of 46. They are
  (46,1) and (23,2)
  
  Case I: (46,1)
  
  2N - 1 = 45
  
  2N = 46
  
  N = 23
  
  And X = 1
  
  Case II: (23,2)
  
  2N - 1 = 21
  
  2N = 22
  
  N = 11
  
  And X = 2
  
  Hence, there are 2 possible answers.
  
• 12 years agoHelpfull: Yes(1) No(3)