Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.
Key knowledge
The ratio a:b compares a with b and can be expressed as a/b when b ≠ 0. A rate compares quantities with different units; dividing both amounts by the denominator's numerical value gives a rate per one denominator unit.
Values and equations
average rate = Δq/Δt
Δq
Change in the measured quantity
Δt
Nonzero elapsed time s when time is expressed in SI
Conditions and scope
Convert quantities of the same kind to the same unit before forming their numerical ratio.
An amount divided by elapsed time gives an average rate over that interval, not necessarily an instantaneous rate.
The quotient is averaged across the specified interval.
Sources · 9
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Ratio notation, compatible units and unit-rate meaning.
2026-10-10 · Checked quantity order and units; excluded the textbook's medical and regulatory examples.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Scope or convention differs
Ratios, per-unit rates and finite-interval average speed.
The Portuguese derivation distinguishes the average from a pointwise derivative.
Australian Mathematical Sciences Institute and University of Melbourne
A bijection of ℝⁿ uniformly scaling all distances by s>0 is a similarity and has form T(x)=sQx+b with Q orthogonal. It preserves angles and ratios of nonzero distances; distances are multiplied by s. Similarities form a group containing the isometries s=1.
For compatible near-infrared and red surface-reflectance bands, NDVI=(ρ_NIR−ρ_R)/(ρ_NIR+ρ_R). With nonnegative reflectances and positive sum it lies between−1 and1. The ratio can help characterize vegetation greenness, but does not uniquely measure biomass, plant health or a species. Common multiplicative scaling cancels; additive atmospheric effects do not generally cancel.
p% means p/100. For a positive original amount, signed percent change is one hundred times the new-minus-original difference divided by the original amount; a negative result denotes a decrease.
Percent expresses a quotient relative to an explicitly chosen base per hundred.
For positive x and y, direct proportion means y=kx with fixed k>0, so y/x stays constant. Inverse proportion means y=k/x, so xy stays constant. The constants in the two relationships are separately defined.
Direct proportion fixes a ratio; inverse proportion fixes a product.
Learning guidance · 5 / 5
Preserved learning suggestions. They are not proof, empirical evidence or mandatory prerequisites.