Water operator math tutorial
Dilution (C₁V₁ = C₂V₂)
Find the diluted concentration when a solution is made up to a larger volume.
Dilution keeps the amount of chemical the same while the volume grows, so concentration drops in proportion.
C₁V₁ = C₂V₂ works with any matching units. Just make sure both volumes use the same unit and both concentrations use the same unit.
The formula
C₂ = (C₁ × V₁) ÷ V₂
| Variable | Unit |
|---|---|
| Starting concentration (C₁) | mg/L or % |
| Starting volume (V₁) | any volume unit |
| Final volume (V₂) | same unit as V₁ |
| Result: Final concentration | same unit as C₁ |
Applies to: Water Treatment, Wastewater Treatment.
Common exam tip: The answer must be smaller than the starting concentration. If it is larger, you flipped the volumes.
Calculator
Final concentration
1,000 same unit as C₁
- 1. C₂ = (50,000 × 10) ÷ 500 = 1,000
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
10 gal of a 50,000 mg/L solution is diluted to 500 gal.
- C₂ = (50,000 × 10) ÷ 500 = 1,000
Answer: 1,000 same unit as C₁
Example 2
40 gal of a 12.5% solution is diluted to 100 gal.
- C₂ = (12.5 × 40) ÷ 100 = 5
Answer: 5 same unit as C₁
Common mistakes
Wrong result: 2,500,000 same unit as C₁
swapping the two volumes
Wrong result: 49,510 same unit as C₁
adding and subtracting instead of using the ratio
Practice questions
1. 25 L of 100 mg/L is diluted to 250 L. Final concentration?
- A10 same unit as C₁
- B1,000 same unit as C₁
- C100 same unit as C₁
- D1 same unit as C₁
Show answer and explanation
Correct answer: A. C₂ = (100 × 25) ÷ 250 = 10 Answer: 10 same unit as C₁. A common slip is swapping the two volumes, which gives 1,000 same unit as C₁.
2. 60 gal of 15% is diluted to 300 gal. Final percent?
- A75 same unit as C₁
- B3 same unit as C₁
- C30 same unit as C₁
- D0.3 same unit as C₁
Show answer and explanation
Correct answer: B. C₂ = (15 × 60) ÷ 300 = 3 Answer: 3 same unit as C₁. A common slip is swapping the two volumes, which gives 75 same unit as C₁.
3. 4 L of 800 mg/L is diluted to 64 L. Final concentration?
- A12,800 same unit as C₁
- B740 same unit as C₁
- C50 same unit as C₁
- D500 same unit as C₁
Show answer and explanation
Correct answer: C. C₂ = (800 × 4) ÷ 64 = 50 Answer: 50 same unit as C₁. A common slip is swapping the two volumes, which gives 12,800 same unit as C₁.
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.
Water Treatment
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