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 the 5th day?

Let:

• $B$ = initial backlog of orders

• $r$ = new orders received per day

• $c$ = capacity of one truck per day

From the given cases:

1. 7 trucks in 4 days clear the backlog plus 4 days of incoming orders:

   $$7 \times 4 \times c = B + 4r \implies 28c = B + 4r$$

2. 3 trucks in 10 days clear the backlog plus 10 days of incoming orders:

   $$3 \times 10 \times c = B + 10r \implies 30c = B + 10r$$

Subtracting the first equation from the second:

Substitute $r$ into the first equation to find $B$:

Let $n$ be the number of trucks required to clear the orders by the end of the 5th day:

Since the number of trucks must be an integer, the minimum number of trucks required is $\lceil 5.67 \rceil = 6$.