Water operator math tutorial
Food-to-Microorganism (F/M) Ratio
Compare the BOD you feed each day to the biomass in the aeration tank.
The F/M ratio tells you whether the activated sludge has too much or too little food for its microorganisms, which drives settling and treatment quality.
Both the food (BOD) and the microorganisms (MLVSS) are converted to pounds with the pounds formula, then divided.
The formula
F/M = [flow (MGD) × BOD (mg/L) × 8.34] ÷ [tank volume (MG) × MLVSS (mg/L) × 8.34]
| Variable | Unit |
|---|---|
| Influent flow | MGD |
| Influent BOD | mg/L |
| Aeration tank volume | MG |
| MLVSS | mg/L |
| Result: F/M ratio | lb BOD/day per lb MLVSS |
Conversion factors used: 8.34 lb per gallon of water (cancels in the ratio).
Applies to: Wastewater Treatment.
Common exam tip: Both 8.34 factors cancel, but write them anyway so you do not drop a unit conversion.
Calculator
F/M ratio
0.25 lb BOD/day per lb MLVSS
- 1. Food = 4 × 180 × 8.34 = 6,004.8 lb BOD/day
- 2. Microorganisms = 1.2 × 2,400 × 8.34 = 24,019.2 lb MLVSS
- 3. F/M = 6,004.8 ÷ 24,019.2 = 0.25
For instructional use. Results assume the standard conversion factors shown on this page. Actual field calculations may need chemical strength, density, temperature, and other adjustments, so check your plant’s procedures.
Worked examples
Example 1
4 MGD at 180 mg/L BOD into a 1.2 MG tank with 2,400 mg/L MLVSS.
- Food = 4 × 180 × 8.34 = 6,004.8 lb BOD/day
- Microorganisms = 1.2 × 2,400 × 8.34 = 24,019.2 lb MLVSS
- F/M = 6,004.8 ÷ 24,019.2 = 0.25
Answer: 0.25 lb BOD/day per lb MLVSS
Example 2
2.5 MGD at 200 mg/L BOD into a 0.9 MG tank with 2,500 mg/L MLVSS.
- Food = 2.5 × 200 × 8.34 = 4,170 lb BOD/day
- Microorganisms = 0.9 × 2,500 × 8.34 = 18,765 lb MLVSS
- F/M = 4,170 ÷ 18,765 = 0.22
Answer: 0.22 lb BOD/day per lb MLVSS
Common mistakes
Wrong result: 4 lb BOD/day per lb MLVSS
putting the microorganisms on top (upside down)
Wrong result: 0.6 lb BOD/day per lb MLVSS
using the wrong biomass number
Practice questions
1. 6 MGD, 150 mg/L BOD, 1.5 MG tank, 3,000 mg/L MLVSS. F/M?
- A5 lb BOD/day per lb MLVSS
- B0.6 lb BOD/day per lb MLVSS
- C2 lb BOD/day per lb MLVSS
- D0.2 lb BOD/day per lb MLVSS
Show answer and explanation
Correct answer: D. Food = 6 × 150 × 8.34 = 7,506 lb BOD/day Microorganisms = 1.5 × 3,000 × 8.34 = 37,530 lb MLVSS F/M = 7,506 ÷ 37,530 = 0.2 Answer: 0.2 lb BOD/day per lb MLVSS. A common slip is putting the microorganisms on top (upside down), which gives 5 lb BOD/day per lb MLVSS.
2. 3.2 MGD, 220 mg/L BOD, 1.1 MG tank, 2,000 mg/L MLVSS. F/M?
- A0.32 lb BOD/day per lb MLVSS
- B3.13 lb BOD/day per lb MLVSS
- C0.64 lb BOD/day per lb MLVSS
- D3.2 lb BOD/day per lb MLVSS
Show answer and explanation
Correct answer: A. Food = 3.2 × 220 × 8.34 = 5,871.4 lb BOD/day Microorganisms = 1.1 × 2,000 × 8.34 = 18,348 lb MLVSS F/M = 5,871.4 ÷ 18,348 = 0.32 Answer: 0.32 lb BOD/day per lb MLVSS. A common slip is putting the microorganisms on top (upside down), which gives 3.13 lb BOD/day per lb MLVSS.
3. 1.8 MGD, 160 mg/L BOD, 0.6 MG tank, 2,800 mg/L MLVSS. F/M?
- A5.83 lb BOD/day per lb MLVSS
- B0.17 lb BOD/day per lb MLVSS
- C0.48 lb BOD/day per lb MLVSS
- D1.7 lb BOD/day per lb MLVSS
Show answer and explanation
Correct answer: B. Food = 1.8 × 160 × 8.34 = 2,401.9 lb BOD/day Microorganisms = 0.6 × 2,800 × 8.34 = 14,011.2 lb MLVSS F/M = 2,401.9 ÷ 14,011.2 = 0.17 Answer: 0.17 lb BOD/day per lb MLVSS. A common slip is putting the microorganisms on top (upside down), which gives 5.83 lb BOD/day per lb MLVSS.
Where this formula applies
This calculation can appear in the disciplines below. Exact exam coverage depends on your state, certification level, and official exam outline.
Wastewater Treatment
Study guides
Keep going with the full guide
Water Operator Math Made Simple walks through operator math step by step and includes printable references plus a full practice exam.
Water Operator Math Made Simple
A Step-by-Step Guide to the Math Every Water Operator Needs to Know