Part-to-Part Ratio
A 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 directly to another part of the same group, not to the total, telling you how the two parts relate to each other. Formally, a part-to-part ratio $a:b$ compares two distinct categories within a whole; given $a:b$, the part-to-whole ratios are $a/(a+b)$ and $b/(a+b)$, and conversely, part-to-whole ratios $p, q$ with $p+q=1$ convert back to the part-to-part ratio $p:q$. This same idea underlies odds in probability, where the ratio of favorable to unfavorable outcomes converts to probability $P = a/(a+b)$, likelihood ratios in statistics, and stoichiometric mole ratios between reactants and products in chemistry.
Example
A bag with $4$ red and $6$ blue marbles has a part-to-part ratio of red to blue of $4:6 = 2:3$, for every $2$ red marbles there are $3$ blue (not $4:10$, which would be part-to-whole). A boys:girls ratio of $3:7$ means boys are $3/10 = 30\%$ and girls $7/10 = 70\%$ of the class, so a class of $30$ has $9$ boys and $21$ girls; the same $3:7$ ratio could just as easily describe $30$ boys and $70$ girls, since part-to-part ratios scale naturally with the total. In $\text{H}_2\text{O}$, hydrogen to oxygen is $2:1$ by mole ratio, so producing $36$g of water ($2$ moles) requires $4$g of $\text{H}_2$ and $32$g of $\text{O}_2$, the ratio preserved regardless of scale.
Key Insight
Part-to-part ratios compare pieces to pieces, while part-to-whole compares a piece to everything, both come from the same situation but answer different questions: "how do the parts compare?" versus "what fraction of the total is this?" Because part-to-part ratios stay the same regardless of scale, they are especially useful in chemistry (mole ratios), cooking, and engineering. Odds ratios in epidemiology are a real-world part-to-part comparison: an odds ratio of $3:1$ means exposed individuals are $3$ times as likely to have a disease as unexposed individuals, distinct from (and more nuanced than) a simple relative-risk, part-to-whole comparison.