Biology math practice

Every skill in this set

Chi-square goodness of fitHeredity

χ² = Σ (observed − expected)² / expected

The exam asks you to compute χ² and compare it with a critical value at the stated degrees of freedom. Degrees of freedom is categories − 1, and the null hypothesis is rejected when χ² EXCEEDS the critical value — the direction students most often reverse.

Hardy-Weinberg: allele frequency from phenotypeNatural Selection

p + q = 1 · p² + 2pq + q² = 1

The recessive phenotype is the only genotype you can count directly, so every Hardy-Weinberg problem starts there: q² is the recessive frequency, q is its square root. Starting anywhere else is the standard wrong turn.

Water potentialCell Structure & Function

Ψ = Ψp + Ψs · Ψs = −iCRT

Water moves from higher to LOWER water potential — a more negative number. Solute potential is always negative, and the minus sign in −iCRT is where most of the lost marks are.

Surface area to volume ratioCell Structure & Function

Cube: SA = 6s² · V = s³ · ratio = 6/s

The whole argument for why cells stay small. As a cell grows, volume rises with the cube of length and surface area only with the square, so the ratio falls and exchange across the membrane cannot keep up.

Exponential population growth rateEcology

dN/dt = rmax N

The exam distinguishes exponential from logistic growth by whether carrying capacity appears. This is the exponential case — unlimited resources — and the answer is a rate per unit time, not a population size.

Logistic population growthEcology

dN/dt = rmax N (K − N) / K

Growth is fastest at K/2 and falls to zero at K. If your answer is negative, the population is above carrying capacity — which is a real result, not an error.

Standard error and error barsNatural Selection

SE = s / √n · error bars = mean ± 2 SE

The exam asks whether two means "differ significantly", and the answer is decided by whether their 2 SE error bars overlap. Overlapping bars mean you cannot claim a difference — a conclusion students state far too confidently.

Gibbs free energyCellular Energetics

ΔG = ΔH − TΔS

A negative ΔG means the reaction is spontaneous — energetically favorable — and that sign is the whole answer to most questions using it. Temperature must be in kelvin.

DilutionChemistry of Life

C₁V₁ = C₂V₂

Every enzyme and osmosis lab starts here. The quantity you almost always want is how much stock to take — and then how much water to add, which is the difference, not the total.

Rate of reaction from a graphCellular Energetics

rate = Δamount / Δtime

Enzyme questions ask for a rate from two points on a curve. Take the change in the measured quantity over the change in time, and keep the units the question used — a rate with no unit rarely earns the point.

Magnification and actual sizeCell Structure & Function

magnification = image size / actual size

A micrograph question gives you two of the three quantities. The trap is unit mismatch: convert millimeters to micrometers (× 1000) before dividing, or the answer is off by three orders of magnitude.

Net and gross primary productivityEcology

NPP = GPP − R

Only net primary productivity is available to the next trophic level, because respiration has already spent the rest. Questions about energy passing up a food chain are asking about NPP.

Energy transfer between trophic levelsEcology

roughly 10% of energy passes to the next trophic level

The 10% rule explains why food chains are short. Each step multiplies by about 0.1, so three steps up leaves a thousandth of the original energy.

Common questions