the manfrenhensons were a young adventuresome couple...... word problem

Question

with a great deal of forsight and calculation, thehy made elaborate plans for a summer tour in their battered station wagon through the less-traveled parts of mexico and central america. mapping their intinerary in advance, they calculatd that their journy wopuld cover a total of twenty-seven thousand miles. on the advice of a friend, the manfrenjensons decided to buy special tires, each one guarenteed to last twelve thousand total miles provided the car was not overloaded. jettisoning the heavy equipment so they would not load the station wagon with more weight than was allowed by the guarentee, they now calculated how many tires they would need for the long distance trip. what is the least number of tires that would carry them through their journy, and how could they make the best use of them? (this problem is harder than it seems, if you arrive at an answer quickly, you've done it wrong)

Answer 1

Since the vehicle has four wheels the total mileage is 4*27,000=108,000. The tyre mileage of 12,000 divides into this 9 times, so the minimum number of tyres to provide the mileage is 9. It's just a question of how.

The journey can be split into 9 stages of 3,000 miles each. Label each tyre with numbers 1 to 9, so that each one is identified by its number. Label the journey stages A to I. Stage A uses tyre numbers 1-4; B 2-5; ... F 6-9; G 1,7,8,9; H 8,9,1,2; I 9,1,2,3. So every 3,000 miles, a tyre is swapped. Between stages A and B, 5 replaces 1; between B and C 6 replaces 2, and so on. Over the whole journey then, tyre #1 is used in stages A, G, H, I so its total mileage is 12,000; B is used in stages A, B, H, I, so its total mileage is also 12,000. Each tyre takes part in 4 stages only and is unused for 5 stages. This arrangement fully utilises all 9 tyres.

JOURNEY STAGES

A	B	C	D	E	F	G	H	I
1	2	3	4	5	6	7	8	9
2	3	4	5	6	7	8	9	1
3	4	5	6	7	8	9	1	2
4	5	6	7	8	9	1	2	3

The table above shows the tyre-switching arrangements. Each stage is 3,000 miles.