A transporter receives the same number of orders each day. Currently, he has…
2011
A transporter receives the same number of orders each day. Currently, he has some pending orders (backlog) to be shipped. If he uses 7 trucks, then at the end of the 4th day he can clear all the orders. Alternatively, if he uses only 3 trucks, then all the orders are cleared at the end of the 10th day. What is the minimum number of trucks required so that there will be no pending order at the end of 5th day?
- A.
4
- B.
5
- C.
6
- D.
7
Attempted by 23 students.
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Correct answer: C
Let k be the number of orders one truck can deliver per day, I be the number of new orders received each day, and S be the initial backlog.
Using 7 trucks: total delivered in 4 days = 4 * 7k. This equals initial backlog plus 4 days of incoming orders: 4 * 7k = S + 4I, so S = 28k - 4I.
Using 3 trucks: total delivered in 10 days = 10 * 3k. This equals initial backlog plus 10 days of incoming orders: 10 * 3k = S + 10I, so S = 30k - 10I.
Equate the two expressions for S: 28k - 4I = 30k - 10I. Rearranging gives 6I = 2k, so I = k/3.
Substitute I = k/3 back to find S: S = 28k - 4*(k/3) = (80/3)k.
For n trucks to clear all orders by the end of 5 days: total delivered in 5 days = 5nk must be at least S plus 5 days of incoming orders: 5nk >= S + 5I = (80/3)k + 5*(k/3) = (85/3)k.
Divide both sides by k (k > 0): 5n >= 85/3, so n >= 85/15 = 17/3 ≈ 5.667. Therefore the minimum integer number of trucks needed is 6.
Answer: 6 (6 trucks is the smallest whole number that clears the backlog by the end of the 5th day).