Free Serial Dilution Calculator With Every Tube Worked Out

The dilution factor of a single tube

One tube's dilution factor is the total volume sitting in that tube divided by the volume of sample you moved into it. Transfer 1 mL into a tube already holding 9 mL of diluent and the tube contains 10 mL, of which 1 mL is sample: 10 ÷ 1 = 10. That is a 10-fold dilution, written 1:10.

The number on the bottom of that fraction is the total volume, not the diluent volume. 1 mL into 9 mL is 1:10, not 1:9. This is the single most common slip in the whole calculation, and it is dangerous precisely because being off by one tenth still produces an answer that looks reasonable.

Because the factor is a ratio, the unit cancels out. 1 mL into 9 mL and 100 µL into 900 µL are both 10-fold. The only rule is that the transfer volume and the diluent volume must be in the same unit before you divide — that is why the calculator above uses one unit selector for both.

• 1 mL into 9 mL → 10-fold (1:10)
• 0.1 mL into 9.9 mL → 100-fold (1:100)
• 100 µL into 100 µL → 2-fold (1:2), the standard antibody or antibiotic titration
• 1 mL into 4 mL → 5-fold (1:5)

Dilutions multiply, they never add

Each tube dilutes whatever came out of the tube before it, so the factors multiply. Three 10-fold tubes give 10 × 10 × 10 = 1,000-fold, not 30-fold. After n tubes at f-fold each, the cumulative dilution is f raised to the power n.

Worked through with a culture at 3.6 × 10⁸ cells/mL and 1 mL into 9 mL: tube 1 is 1:10 and holds 3.6 × 10⁷ cells/mL. Tube 2 is 10 × 10 = 100, so 3.6 × 10⁶. Tube 3 is 100 × 10 = 1,000, so 3.6 × 10⁵. By tube 6 the cumulative dilution is 10⁶ and the tube holds 360 cells/mL. The concentration in any tube is always the starting concentration divided by that tube's cumulative factor — never divided by the per-tube factor alone.

In a strictly 10-fold series the tube number is the exponent, which is why microbiologists talk about "the minus five plate": tube 5 is 10⁻⁵. That shorthand is a property of 10-fold series only. In a 100-fold series each tube moves the exponent by two, so tube 3 is 10⁻⁶, not 10⁻³. In a 2-fold or 5-fold series there is no power-of-ten shorthand at all, and tube 4 of a 5-fold series is 5⁴ = 625.

Reading CFU/mL back off a plate

Counting colonies tells you how many viable units were in the small volume you spread on that plate. To get back to the original sample you undo both the dilution and the plating volume at once:

CFU/mL = colonies ÷ (volume plated in mL × the dilution you plated)

Say you counted 150 colonies from 0.1 mL spread from tube 3 of a 10-fold series. Tube 3 is 1:1,000, which as a number is 10⁻³. The denominator is 0.1 × 10⁻³ = 10⁻⁴ mL. So CFU/mL = 150 ÷ 10⁻⁴ = 1.5 × 10⁶.

It helps to read that denominator physically. 10⁻⁴ mL is the actual slice of undiluted sample sitting on the agar, and 150 colonies grew out of it. Everything else is bookkeeping. Dividing by a fraction is the same as multiplying, so you can also write it as 150 × 1,000 ÷ 0.1, which lands on the same 1.5 × 10⁶.

One more thing the abbreviation is telling you: the unit is colony-forming units, not cells. A clump of ten cells or a chain of streptococci produces one colony, so CFU/mL is a count of things capable of founding a colony, which is usually lower than a microscope count of the same sample.

Where these calculations usually go wrong

Almost every wrong answer in a serial dilution problem comes from one of a short list of mistakes. None of them make the arithmetic fail loudly — they all produce a number that looks fine.

• Dividing by the diluent instead of the total volume, turning a 1:10 into a 1:9.
• Adding the per-tube factors instead of multiplying them, so six 10-fold tubes come out as 60-fold rather than 10⁶.
• Double-counting the plating volume. Spreading 0.1 mL is already handled by the volume term in the formula; multiplying the dilution by another 10 to account for it inflates the answer tenfold.
• Forgetting to convert before dividing. The formula returns CFU per millilitre, so 100 µL has to become 0.1 mL first.
• Reading the tube number as a power of ten in a series that is not 10-fold.
• Reporting a count from a plate outside the countable range. Below about 25 colonies the scatter between replicate plates is large; above about 300 they merge and the count runs low.
• Recording an empty plate as 0 CFU/mL. Zero colonies means the sample held fewer than the detection limit of that plate — report it as a "less than" value.

How to check the answer before you write it down

Multiply back up. Take the concentration you calculated for the last tube and multiply it by that tube's cumulative dilution factor. You should land on the concentration you started with. If you do not, the error is in the cumulative column.

Do an order-of-magnitude check. In a 10-fold series each tube should move the concentration exactly one decimal place. If tube 4 and tube 5 differ by anything other than a factor of ten, a transfer volume is wrong somewhere in the chain.

Check the plate you counted was countable. Standard plate counts report from plates holding 25 to 250 colonies, and many courses teach 30 to 300. Either way, a plate with 8 colonies and a plate with 600 both need a different dilution rather than a calculation.

Work out what one colony would have meant. For 0.1 mL plated from a 1:1,000 tube, a single colony equals 10⁴ CFU/mL — that is the detection limit of the plate, and it tells you how precise your answer can honestly be.

Finally, compare neighbouring plates. Two consecutive 10-fold plates should give CFU/mL figures in the same ballpark once each is back-calculated. A large disagreement between them points at a pipetting error, not a maths error.