Independent stage ratios commute for the final product, but not for the middle budget. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One order has a twenty-unit intermediate budget

The 4/5 stage first gives intermediate budget 20 and final budget 4.

254515=425\cdot {4\over 5}\cdot {1\over 5}=4
Large intermediate firstStage order changes the middle value.ffcratioinoutffcratioinoutlowpasslowpass

The reversed order has a five-unit intermediate budget

The final budget is still 4, but the intermediate budget is only 5.

251545=425\cdot {1\over 5}\cdot {4\over 5}=4
Small intermediate firstThe same product closes with a different middle value.ffcratioinoutffcratioinoutlowpasslowpass