13.2 Laboratory Mathematics
Key Takeaways
- Antibody working solutions use C1V1 = C2V2, which for a simple 1:N dilution is V_concentrate = V_working / N; convert milliliters and microliters before you pipette (1 mL = 1000 µL).
- Serial dilutions multiply: a 1:50 step followed by a 1:2 step is a final 1:100, not 1:52 and not 1:25.
- "10% NBF" is a 1-plus-9 (about 1:10) volume dilution of ~37% formaldehyde stock, which yields about 3.7% formaldehyde; 100 mL of 37% stock plus 900 mL buffer makes 1 L.
- Slides from a concentrate vial equal working volume divided by volume dispensed per slide; leftover dead volume in the vial and on the stainer is not free extra slides.
- Percent solutions are defined (w/v or v/v) before the arithmetic; a 1% w/v solution is 1 g in 100 mL, not 1 mL of mystery stock in 100 mL unless the stock is already 100% liquid of known density.
13.2 Laboratory Mathematics
Quick Answer: For a 1:N antibody dilution, volume of concentrate = working volume ÷ N. A 1:50 dilution followed by a 1:2 dilution is a 1:100 final. Ten percent NBF is about 100 mL of 37% formaldehyde stock plus 900 mL buffer per liter, not 10% formaldehyde by weight. Convert 1000 µL = 1 mL before you pipette.
This OpenExamPrep section is independent teaching on IHC laboratory mathematics covering published QIHC laboratory-operations topic areas. It is not an ASCP publication and does not claim Board approval. QIHC tests the arithmetic, so every example below shows the algebra.
Units: microliters versus milliliters
Pipettes speak microliters. Graduated cylinders speak milliliters. The conversion is exact:
| From | To | Factor |
|---|---|---|
| 1 mL | µL | × 1000 |
| 1 µL | mL | ÷ 1000 |
| 100 µL | mL | 0.10 mL |
| 50 µL | mL | 0.05 mL |
| 10 µL | mL | 0.01 mL |
| 0.2 mL | µL | 200 µL |
A stem that mixes "200 µL per slide" with "10 mL working solution" is inviting a 1000-fold error. Convert one side first, then divide.
C1V1 = C2V2 for antibody dilutions
When a concentrate is treated as concentration 1 (undiluted) and the working reagent is a 1:N dilution, the working concentration is 1/N. Then:
C1 × V1 = C2 × V2 → 1 × V1 = (1/N) × V2 → V1 = V2 / N
Diluent volume = V2 − V1.
Worked example A — 1:200, need 10.0 mL working solution
- V2 = 10.0 mL
- N = 200
- V1 = 10.0 / 200 = 0.050 mL = 50 µL of concentrate
- Diluent = 10.0 − 0.050 = 9.95 mL
Wrong answers you will see on items: 200 µL (that is 1:50 into 10 mL), 10 µL (that is 1:1000), or 2.0 mL (that is 1:5). The algebra is division by 200, not by 50 and not "200 because the titer is 200."
Worked example B — 1:50, need 500 µL for a titration rack
Convert first: 500 µL = 0.500 mL.
- V1 = 0.500 / 50 = 0.010 mL = 10 µL concentrate
- Diluent = 500 − 10 = 490 µL
If you forget to convert and compute 500 / 50 thinking the 500 is already milliliters, you pipette 10 mL of concentrate into a microtube and destroy the experiment.
Worked example C — ready-to-use is not a 1:1 math trick
A labeled RTU vial is already the working concentration. You do not dilute it 1:100 again unless the laboratory's validated method says to further dilute an RTU (unusual, and that would be a method change). Math items that hand you an RTU and a 1:N factor are testing whether you notice the bottle is already working strength.
Serial dilutions: 1:50 then 1:2
Serial means each step uses the previous dilution as the new "concentrate." Factors multiply.
Final dilution factor = 50 × 2 = 100, so the product is 1:100.
Worked example D — build 1.0 mL at 1:50, then 1.0 mL at 1:2 from that intermediate
Step 1 (1:50), V2 = 1000 µL:
- V1 = 1000 / 50 = 20 µL concentrate
- Diluent = 980 µL
- Intermediate = 1:50
Step 2 (1:2 of the intermediate), V2 = 1000 µL:
- Volume of 1:50 = 1000 / 2 = 500 µL
- Diluent = 500 µL
- Final = 1:(50 × 2) = 1:100
Common traps:
- Adding 50 + 2 = 1:52 (addition is not how serial dilutions work)
- Dividing 50 / 2 = 1:25 (that would be a less dilute product)
- Reporting 1:500 (extra zero)
If a stem says "dilute 1:50, then dilute that product 1:2," the answer is 1:100 unless it also changes the volumes in a way that is not a true 1:2 (always check the actual volumes).
How many slides from a concentrated vial
Working volume available = vial volume × dilution factor N (if you use the entire vial at 1:N).
Number of slides = working volume / volume per slide.
Keep units identical. Convert everything to microliters when the per-slide volume is in microliters.
Worked example E — 1.0 mL concentrate, titer 1:100, 200 µL working solution per slide
- Working volume = 1.0 mL × 100 = 100 mL = 100,000 µL
- Per slide = 200 µL
- Slides = 100,000 / 200 = 500 slides
Worked example F — 0.50 mL concentrate, titer 1:50, 250 µL per slide
- Working volume = 0.50 × 50 = 25 mL = 25,000 µL
- Slides = 25,000 / 250 = 100 slides
Reality check for the bench (and for application items): autostainers have dead volume, predilute bulk may not empty the last 50 µL, and you should not plan the last patient slide on a theoretical 500 if the SOP requires leftover for a repeat. Exam stems that ignore dead volume want the clean division. Stems that mention residual volume want you to subtract it first.
Worked example G — leftover adjustment
Same as E, but 1.0 mL vial with 50 µL unusable residue: usable concentrate = 0.95 mL; working = 95 mL = 95,000 µL; slides = 95,000 / 200 = 475 slides.
37% formalin to about 10% NBF
10% formalin in histology language is 10 mL of ~37–40% formaldehyde stock per 100 mL of solution (a 1 + 9 volume recipe), buffered to make NBF. It is not 10 grams of formaldehyde gas in 100 mL, and it is not 10% w/w formaldehyde. Because stock is about 37% formaldehyde w/w, 10% formalin contains about 3.7% formaldehyde.
Use C1V1 = C2V2 on the formaldehyde percentage:
- C1 = 37% (stock)
- C2 = 3.7% (the formaldehyde content of 10% formalin)
- V2 = 1000 mL
- V1 = (3.7 × 1000) / 37 = 100 mL of 37% stock
- Diluent (water plus buffer salts as the SOP specifies) = 900 mL
Same answer from the 1 + 9 rule: 100 mL stock + 900 mL diluent = 1 L of 10% formalin.
If a stem asks for 500 mL of 10% NBF from 37% stock: V1 = 50 mL stock + 450 mL buffer. If a stem tricks you into making "10% formaldehyde" as 10% w/v from 37% stock, that would be a different (much stronger) solution and is not routine NBF. QIHC wants the histology convention unless the stem explicitly asks for a true 10% w/v formaldehyde solution.
Percent solutions
State the basis before you calculate:
- % w/v = grams of solute per 100 mL of solution (1% = 1 g/100 mL = 10 g/L)
- % v/v = milliliters of liquid solute per 100 mL of solution
- % w/w = grams per 100 g
Worked example H — 0.1% w/v sodium azide in 100 mL diluent (illustrative preservative math): 0.1 g per 100 mL = 100 mg in 100 mL. That is 1 mg/mL. Do not take 0.1 mL of a mystery powder.
Worked example I — 5% v/v goat serum in 20 mL antibody diluent: V1 = 0.05 × 20 = 1.0 mL serum + 19 mL diluent.
Hydrogen peroxide blocking solutions are often percent v/v of a stock (for example, diluting 30% H2O2). If C1 = 30% and you want 3% in 10 mL: V1 = (3 × 10) / 30 = 1.0 mL of 30% stock + 9.0 mL diluent. Always read whether the stock is 30% or already 3%.
Dilution example table (work these until they are automatic)
| Goal | Given | Algebra shown | Pipette |
|---|---|---|---|
| 1:100 working, 5.0 mL | Concentrate = 1 | V1 = 5.0 / 100 = 0.050 mL | 50 µL Ab + 4.95 mL diluent |
| 1:200 working, 10.0 mL | Concentrate = 1 | V1 = 10.0 / 200 = 0.050 mL | 50 µL Ab + 9.95 mL diluent |
| 1:50 working, 500 µL | Convert 500 µL = 0.50 mL | V1 = 0.50 / 50 = 0.010 mL | 10 µL Ab + 490 µL diluent |
| Serial 1:50 then 1:2 | Factors multiply | 50 × 2 = 100 | Final 1:100 |
| Slides from 1.0 mL at 1:100, 200 µL/slide | Working = 100 mL = 100,000 µL | 100,000 / 200 | 500 slides |
| Slides from 0.50 mL at 1:50, 250 µL/slide | Working = 25 mL = 25,000 µL | 25,000 / 250 | 100 slides |
| 1 L of 10% NBF from 37% stock | 37 × V1 = 3.7 × 1000 | V1 = 100 mL | 100 mL stock + 900 mL buffer |
| 500 mL of 10% NBF from 37% stock | Half of the liter recipe | V1 = 50 mL | 50 mL stock + 450 mL buffer |
| 3% H2O2 from 30% stock, 10 mL | C1V1 = C2V2 | V1 = 1.0 mL | 1.0 mL of 30% + 9.0 mL diluent |
Write V1 = V2 / N on the scrap paper, convert µL ↔ mL in the same line, and only then pick an option. Most missed IHC math items are unit errors, adding serial factors, or treating 10% NBF as 10% formaldehyde by weight.
A concentrated primary antibody is used at 1:200. You need 10.0 mL of working solution. Using C1V1 = C2V2, how much concentrate and diluent do you combine?
An antibody is diluted 1:50, and that product is then diluted 1:2. What is the final dilution?