A car travels from city P to city Q along a highway that is 700 km long. The car's fuel tank holds at most 30 litres and the car always runs 12 km per litre, so a full tank lasts 360 km. The car starts at P with an empty tank and must reach Q; there is no requirement to have fuel left on arrival. Fuel can be bought in any quantity (including fractions of a litre) at the following fuel stations, the first of which is at P itself:
| Station | Distance from P (km) | Price (₹ per litre) |
|---|---|---|
| S0 (at P) | 0 | 102 |
| S1 | 120 | 96 |
| S2 | 250 | 104 |
| S3 | 340 | 92 |
| S4 | 480 | 98 |
| S5 | 590 | 110 |
A *refuelling stop* is any station other than S0 at which the car buys fuel.
| Station | Distance from P (km) | Price (₹ per litre) |
|---|---|---|
| S0 (at P) | 0 | 102 |
| S1 | 120 | 96 |
| S2 | 250 | 104 |
| S3 | 340 | 92 |
| S4 | 480 | 98 |
| S5 | 590 | 110 |
A *refuelling stop* is any station other than S0 at which the car buys fuel.
Question 1
Single correct option
What is the minimum number of refuelling stops the car needs to make to reach Q?
- A. 1
- B. 2
- C. 3
- D. 4
Show answer and solution
Answer: (A) 1
At least one stop is needed because 700 km exceeds the 360 km range.
One stop suffices only if some station lies within 360 km of P and within 360 km of Q, i.e., at a distance between 340 and 360 km from P. S3 at 340 km qualifies: fill 340/12 litres at P, reach S3 empty, fill 30 litres and cover exactly the remaining 360 km.
Minimum = 1 stop (at S3).
One stop suffices only if some station lies within 360 km of P and within 360 km of Q, i.e., at a distance between 340 and 360 km from P. S3 at 340 km qualifies: fill 340/12 litres at P, reach S3 empty, fill 30 litres and cover exactly the remaining 360 km.
Minimum = 1 stop (at S3).
Question 2
Numerical answer
What is the minimum possible total amount (in ₹) that must be spent on fuel for the trip?
Show answer and solution
Answer: 5540
Greedy rule for cheapest refuelling: at each station, if a cheaper station lies within range, buy just enough to reach it; otherwise fill as much as needed (up to a full tank).
- At S0 (₹102): S1 (₹96) is cheaper and 120 km away → buy 10 L (₹1,020).
- At S1 (₹96): the next cheaper station S3 (₹92) is 220 km away → buy 220/12 L (₹1,760).
- At S3 (₹92): no cheaper station ahead, and Q is exactly 360 km away → buy 30 L (₹2,760).
Total = 1,020 + 1,760 + 2,760 = ₹5,540.
Skipping S1 (buying all 340 km of fuel at ₹102) costs ₹130 more.
- At S0 (₹102): S1 (₹96) is cheaper and 120 km away → buy 10 L (₹1,020).
- At S1 (₹96): the next cheaper station S3 (₹92) is 220 km away → buy 220/12 L (₹1,760).
- At S3 (₹92): no cheaper station ahead, and Q is exactly 360 km away → buy 30 L (₹2,760).
Total = 1,020 + 1,760 + 2,760 = ₹5,540.
Skipping S1 (buying all 340 km of fuel at ₹102) costs ₹130 more.
Question 3
Single correct option
If the car must make exactly one refuelling stop, what is the minimum total amount spent on fuel?
- A. ₹5,720
- B. ₹5,540
- C. ₹5,600
- D. ₹5,650
Show answer and solution
Answer: (D) ₹5,650
With one stop, the stop must be within 360 km of both P and Q, so it must be S3 (340 km).
- At S0: buy fuel for 340 km = 340/12 L at ₹102 = ₹2,890.
- At S3: buy fuel for 360 km = 30 L at ₹92 = ₹2,760.
Total = ₹5,650.
This is ₹110 more than the unrestricted minimum of ₹5,540, the difference coming from buying 220 km of fuel at ₹102 instead of ₹96 (220/12 × 6 = 110).
- At S0: buy fuel for 340 km = 340/12 L at ₹102 = ₹2,890.
- At S3: buy fuel for 360 km = 30 L at ₹92 = ₹2,760.
Total = ₹5,650.
This is ₹110 more than the unrestricted minimum of ₹5,540, the difference coming from buying 220 km of fuel at ₹102 instead of ₹96 (220/12 × 6 = 110).
Question 4
Numerical answer
In the cheapest possible plan for the trip, how many litres of fuel are bought at S0?
Show answer and solution
Answer: 10
Station S1, 120 km ahead, sells fuel at ₹96, cheaper than S0's ₹102, and is within range. So the car should buy at S0 only what is needed to reach S1: 120/12 = 10 litres.
Buying more at S0 would only replace fuel that could be bought more cheaply at S1 (or even more cheaply at S3). Buying less is impossible since the car must reach S1.
The complete cheapest plan: 10 L at S0, 18⅓ L at S1, 30 L at S3.
Buying more at S0 would only replace fuel that could be bought more cheaply at S1 (or even more cheaply at S3). Buying less is impossible since the car must reach S1.
The complete cheapest plan: 10 L at S0, 18⅓ L at S1, 30 L at S3.
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