P, Q, R and S are four places. To travel from P to R, one travels via Q and to travel from P to S, one travels via both Q and R. Between P and Q, there are 3 routes; between Q and R, there are 4 routes; and between R and S, there are 5 routes. The time taken to travel through these routes is exhibited on the following chart in hours for a specific vehicle :

Routes	Between P and Q	Between Q and R	Between R and S
1	2	1	2
2	3	1.5	2
3	4	2	2.5
4	-	2	3
5	-	-	3.5

How many possible routes are there if the journey takes exactly 7 hours?

To find the total number of possible routes taking exactly 7 hours, we analyze the combinations of travel times between P-Q, Q-R, and R-S such that their sum is 7.

• Case 1: PQ = 2 hours (Route 1)
  QR + RS must be 5 hours.
  - QR = 1.5 (R2), RS = 3.5 (R5) → 1 route
  - QR = 2 (R3 or R4), RS = 3 (R4) → 2 routes
  Subtotal: 3 routes
• Case 2: PQ = 3 hours (Route 2)
  QR + RS must be 4 hours.
  - QR = 1 (R1), RS = 3 (R4) → 1 route
  - QR = 1.5 (R2), RS = 2.5 (R3) → 1 route
  - QR = 2 (R3 or R4), RS = 2 (R1 or R2) → 2 × 2 = 4 routes
  Subtotal: 6 routes
• Case 3: PQ = 4 hours (Route 3)
  QR + RS must be 3 hours.
  - QR = 1 (R1), RS = 2 (R1 or R2) → 1 × 2 = 2 routes
  Subtotal: 2 routes

Total possible routes = 3 + 6 + 2 = 11.