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 : How many possible routes are there if the journey takes exactly 7 hours?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
11
From the chart, the route times are: P–Q: 2, 3, 4 hours (3 routes); Q–R: 1, 1.5, 2, 2 hours (4 routes); R–S: 2, 2, 2.5, 3, 3.5 hours (5 routes). We need the number of route combinations (one from each stage) whose times add up to exactly 7 hours.
If P–Q = 2 hours, we need Q–R + R–S = 5: (1.5, 3.5) gives 1 way; (2, 3), using either of the two Q–R routes of time 2, gives 2 ways. Subtotal = 3.
If P–Q = 3 hours, we need Q–R + R–S = 4: (1, 3) gives 1 way; (1.5, 2.5) gives 1 way; (2, 2), using either of the two Q–R routes of time 2 matched with either of the two R–S routes of time 2, gives 2×2 = 4 ways. Subtotal = 1+1+4 = 6.
If P–Q = 4 hours, we need Q–R + R–S = 3: (1, 2), matched with either of the two R–S routes of time 2, gives 2 ways. Subtotal = 2.
Total number of routes = 3 + 6 + 2 = 11.
Similar Questions
There is a question followed by two statements I and II :
Question :
If x and y are integers, can we find |x-y| uniquely?
I. x2 + y2 = 25
II. x2 - y2 = 7
Which one of the following options is correct?
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