Part-to-Part Ratio
Fractions & DecimalsA part-to-part ratio compares one part of a group directly to another part of the same group, not to the total.
Formula
\text{part-to-part ratio} = \text{part A} : \text{part B}
Definition
A part-to-part ratio compares one part of a group to another part of the same group, not to the total. It tells you how the two parts relate to each other.
Example
A bag has $4$ red and $6$ blue marbles. Part-to-part ratio of red to blue $= 4:6 = 2:3$. For every $2$ red marbles, there are $3$ blue marbles. (Not to the total, which would be $4:10$.)
Key Insight
Part-to-part ratios compare pieces to pieces. Part-to-whole compares a piece to everything. Both come from the same situation but answer different questions: "how do the parts compare?" vs. "what fraction of the total is this?"
Definition
A part-to-part ratio $a:b$ compares two distinct categories within a whole. Given a part-to-part ratio $a:b$, the part-to-whole ratios are $a/(a+b)$ and $b/(a+b)$. Conversely, given part-to-whole ratios $p$ and $q$ with $p+q=1$, the part-to-part ratio is $p:q$. Both representations carry the same information but suit different contexts.
Example
Boys:girls ratio $= 3:7$ (part-to-part). Total parts $= 10$. Boys are $3/10 = 30\%$ of the class; girls are $7/10 = 70\%$. In a class of $30$: $3/10 \times 30 = 9$ boys, $7/10 \times 30 = 21$ girls. Part-to-part tells you relative numbers; part-to-whole tells you proportions.
Key Insight
Part-to-part ratios scale more naturally when you change the total. A $3:7$ ratio could describe $9$ boys and $21$ girls, or $30$ boys and $70$ girls. The ratio stays the same regardless of scale. This flexibility makes part-to-part ratios useful in chemistry (mole ratios in compounds), cooking, and engineering.
Definition
A part-to-part ratio $a:b$ defines the same equivalence class as its associated fraction $a/b$. In odds (probability), the part-to-part ratio of favorable to unfavorable outcomes converts to probability: $P = a/(a+b)$. Likelihood ratios in statistics compare the probability of data under two hypotheses, analogous to a part-to-part comparison. In chemistry, stoichiometric coefficients express molar ratios (part-to-part) between reactants and products.
Example
In $\text{H}_2\text{O}$, hydrogen:oxygen $= 2:1$ (part-to-part mole ratio). To produce $36$g of water ($2$ moles): need $4$g $\text{H}_2$ ($2$ mol) and $32$g $\text{O}_2$ ($1$ mol). The mole ratio is preserved regardless of scale - an application of part-to-part ratios to quantitative chemistry.
Key Insight
Odds ratios in epidemiology and clinical medicine are part-to-part ratios: odds of disease in exposed group vs. unexposed group. An odds ratio of $3:1$ means exposed individuals are $3$ times as likely to have the disease - a comparison of two parts (exposed diseased : exposed non-diseased) to two other parts. This is distinct from (and more nuanced than) a simple relative risk (part-to-whole ratio comparison).