Laboratory Mathematics

Dilutions And Dilution Factors

A dilution reduces concentration by adding diluent. A 1:10 dilution means 1 part specimen + 9 parts diluent for a total of 10 parts. The dilution factor (DF) is the reciprocal of the dilution (1:10 → DF = 10). To recover the original concentration, multiply the measured result by the DF.

Worked example: a serum glucose reads off-scale, so the tech makes a 1:5 dilution (1 mL serum + 4 mL saline) and the analyzer reports 90 mg/dL. The reported result is 90 × 5 = 450 mg/dL. Forgetting to multiply by the DF is the single most common dilution error.

Use the C1V1 = C2V2 equation to make a target solution: to prepare 100 mL of 0.1 M HCl from 1.0 M stock, V1 = (C2V2)/C1 = (0.1 × 100)/1.0 = 10 mL stock + 90 mL diluent.

A serial dilution repeats the same dilution stepwise; the factors multiply:

Three serial 1:10 steps therefore give a final 1:1000 dilution, not 1:30. In serology, the titer is the reciprocal of the highest dilution still showing reactivity (last reactive tube at 1:256 → titer = 256).

Concentrations, Conversions, And Precision Statistics

Know the concentration units that appear repeatedly on the exam:

• Molarity (M) = moles of solute per liter of solution.
• Normality (N) = equivalents per liter (N = M × valence).
• Percent solutions: %w/v = grams per 100 mL (5% albumin = 5 g/100 mL); %v/v = mL per 100 mL.
• mg/dL ↔ mmol/L: divide mg/dL by the molecular weight and adjust the factor of 10. For glucose (MW 180), mmol/L = mg/dL ÷ 18; a glucose of 90 mg/dL = 5.0 mmol/L.

Worked example: to make 500 mL of 0.9% NaCl, %w/v means 0.9 g per 100 mL, so 0.9 × 5 = 4.5 g NaCl dissolved and brought to 500 mL.

Precision statistics quantify reproducibility and recur in QC questions:

• Mean = sum of values ÷ number of values (central tendency).
• Standard deviation (SD) = spread about the mean.
• Coefficient of variation (CV) = (SD ÷ mean) × 100, expressed as a percent.

The CV normalizes the SD to the mean, so it lets you compare precision between a low and a high control or between two methods. Worked example: a control has a mean of 100 mg/dL and SD of 4 mg/dL, so CV = (4/100) × 100 = 4%. A lower CV means better precision. By contrast, accuracy (closeness to the true/target value) is assessed against reference materials or proficiency testing, while precision (CV) reflects only reproducibility. A method can be precise but inaccurate (tight cluster off-target) — a frequent distractor pairing on the exam.

Standard Curves, Sensitivity, And Specificity

Many calculations rest on a standard (calibration) curve of absorbance versus concentration. For a linear assay, an unknown's concentration is found by proportion: C(unknown) = C(standard) × A(unknown)/A(standard). Worked example: a 100 mg/dL protein standard reads 0.40 absorbance; a patient reads 0.30. C = 100 × (0.30/0.40) = 75 mg/dL. Always confirm the reading falls within the linear range of the curve before applying the proportion; readings beyond linearity require dilution.

Diagnostic math also appears. Build a 2×2 table (true disease vs test result) and apply:

Worked example: of 100 diseased patients, a test detects 90 (TP) and misses 10 (FN), giving sensitivity = 90/100 = 90%. PPV and NPV depend on disease prevalence; sensitivity and specificity do not — a classic distractor.

Enzyme Units, Osmolality, And Common Conversions

Enzyme activity is reported in International Units (IU): one IU = the amount of enzyme converting 1 micromole of substrate per minute under defined conditions. Activity is derived from the rate of absorbance change (ΔA/min) using the molar absorptivity of the product.

Calculated values that recur on the MLS exam:

• Anion gap = Na − (Cl + HCO₃); reference ~8–12 mmol/L. A patient with Na 140, Cl 100, HCO₃ 16 has a gap of 140 − 116 = 24, an elevated (high) anion gap metabolic acidosis.
• Corrected calcium (mg/dL) = measured Ca + 0.8 × (4.0 − albumin).
• Calculated osmolality = 2(Na) + glucose/18 + BUN/2.8; an osmolal gap (measured − calculated > 10) suggests an unmeasured solute such as ethanol or methanol.
• Creatinine clearance = (U × V) / P, where U is urine creatinine, V is urine volume in mL/min, and P is plasma creatinine; correct to body surface area when asked.

Watch unit conversions: 1 dL = 100 mL, 1 L = 1000 mL, 1 g = 1000 mg, 1 mg = 1000 µg, and 1 mole = 1000 mmol. A frequent trap mixes dL and mL — confirm the denominator before computing. As with every operations item, solve the math, then verify the chosen option actually answers the question the stem asked, since a correct number applied to the wrong quantity still produces the wrong best answer.