To solve this problem, we first determine the number of boxes that fit per meter of shelf space. Then, we use this value to find out how many boxes can fit in a shelf length of $20.4$ meters.

Step 1: Boxes per meter

We know: $68 \text{ boxes} \text{ occupy } 13.6 \text{ meters}.$

Boxes per meter: $\frac{68}{13.6} = 5 \text{ boxes per meter}.$

Step 2: Boxes for $20.4$ meters

Using the rate of $5 \text{ boxes per meter}$, the number of boxes for $20.4 \text{ meters}$ is: $20.4 \times 5 = 102 \text{ boxes}.$

Final Answer:

102 boxes would occupy a shelf length of $20.4$ meters.

Would you like further explanation, or do you have any questions?

5 Related Questions:

1. How would the solution change if each box required double the space on the shelf?
2. If $68$ boxes take up $13.6$ meters, how much space does one box occupy individually?
3. What is the maximum shelf length required to fit $150$ boxes of the same commodity?
4. How many boxes would occupy a shelf length of $50$ meters at this rate?
5. What happens if there are $2\%$ of boxes that are damaged and cannot be stored?

Tip:

When solving proportion problems, always find the unit rate first—it simplifies the calculations for any adjustments.