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?
2.0 mL concentrate plus 8.0 mL diluent
200 µL concentrate plus 9.80 mL diluent
10 µL concentrate plus 9.99 mL diluent
50 µL (0.050 mL) concentrate plus 9.95 mL diluent
An antibody is diluted 1:50, and that product is then diluted 1:2. What is the final dilution?
1:52, because serial steps are added
1:100, because serial dilution factors multiply (50 × 2)
1:25, because the second step halves the first factor
1:500, because an extra zero is required for two steps
Sections you finish are checked off in the contents.