Calculate the number of girls next year and advise the head teacher on saving funds for a new dormitory.

Current student population = 500.
Annual increase rate = $8\%$.
Increase in students = $0.08 \times 500 = 40$ students.
Total student population next year = $500 + 40 = 540$ students.

The ratio of boys to girls is $4:5$.
Total parts in the ratio = $4 + 5 = 9$.
Number of girls next year = $\frac{5}{9} \times 540 = 5 \times 60 = 300$ girls.

The head teacher plans to build a new girls' dormitory if the number of girls will exceed 250.
Since $300 > 250$, the number of girls next year will indeed exceed 250.
Therefore, the head teacher should be certain to start saving funds now for the new girls' dormitory.

The quantities of materials are: 360 bags of cement, $240 \text{ kg}$ of binding wire, and 300 steel rods.
To ensure each trip carries equal quantities of each item with no remainder, the number of trips must be a common divisor of these quantities. To minimize the number of trips (and thus the cost), we find the Greatest Common Divisor (GCD).
Prime factorization of the quantities:

The GCD is found by taking the lowest power of each common prime factor:

The number of trips required is equal to the GCD of the quantities.
Number of trips = 60.

Cost per trip = Shs 55,000.
Total cost of transportation = Number of trips $\times$ Cost per trip
Total cost = $60 \times \text{Shs } 55,000 = \text{Shs } 3,300,000$.
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