Let's break down the problem:

1. The bakery has 72 half-gallons of milk.
2. It takes 1 2/3 cups (which equals $\frac{5}{3}$ cups) of milk to make one cake.
3. It is given that 1 cup = $\frac{1}{8}$ half-gallon = 0.125 half-gallons.

Step-by-step calculation:

1. First, let's convert the milk required for one cake (1 2/3 cups) into half-gallons: $\frac{5}{3} \text{ cups} \times 0.125 \text{ half-gallons per cup} = \frac{5}{3} \times 0.125 = 0.2083 \text{ half-gallons}$ So, it takes 0.2083 half-gallons of milk to make one cake.

2. Now, let's figure out how many cakes the bakery can make with 72 half-gallons of milk: $\frac{72 \text{ half-gallons}}{0.2083 \text{ half-gallons per cake}} \approx 345.6$ Since the bakery can only make whole cakes, we round down to 345 cakes.

Final answer:

The bakery can make 345 cakes.

Let me know if you'd like further explanation or have any questions!

Related Questions:

1. How many cups of milk are there in 72 half-gallons?
2. What if each cake required 2 cups of milk? How many cakes could be made?
3. If the bakery had 100 half-gallons of milk, how many cakes could it make?
4. How would the result change if the milk requirement was 1 1/4 cups per cake?
5. How many cakes can be made with 50 half-gallons of milk?

Tip:

Always make sure to convert units consistently, especially when mixing fractions and decimals!