Benchmark Fraction
Benchmark fractions are common reference fractions like 0, 1/4, 1/2, 3/4, and 1 used to estimate and compare other fractions.
Definition
Benchmark fractions are friendly, well-known fractions, typically $0$, $1/4$, $1/2$, $3/4$, and $1$, that help you estimate and compare other fractions without doing exact arithmetic. A fraction $a/b$ is compared to $1/2$ by checking whether $2a < b$ (less than half), $2a = b$ (equal to half), or $2a > b$ (more than half), a test that avoids finding a common denominator entirely. More broadly, benchmarks formalize approximating rational numbers by elements of a fixed finite set, a discretization of $[0,1]$; the Stern-Brocot tree and Farey sequences provide systematic frameworks for finding the "best rational approximations" using this idea.
Example
Is $5/8$ closer to $1/2$ or to $1$? Since $4/8 = 1/2$ and $8/8 = 1$, and $5/8$ is just $1/8$ above $1/2$, it is closer to $1/2$, a benchmark comparison made without calculating exactly. Comparing $7/12$ and $11/20$ using the $1/2$-test: $2 \times 7 = 14 > 12$, so $7/12 > 1/2$, and $2 \times 11 = 22 > 20$, so $11/20 > 1/2$ too; for precision, convert both to sixtieths, $7/12 = 35/60$ and $11/20 = 33/60$, confirming $7/12 > 11/20$. The Farey sequence $F_4 = \{0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1\}$ lists all fractions with denominator $\le 4$ in order, and any fraction not in $F_4$ can be bounded between two consecutive Farey fractions for a precise benchmark comparison.
Key Insight
Benchmarks are mental shortcuts: instead of computing every comparison exactly, you ask "is this fraction less than $1/2$ or more than $1/2$?" and narrow it down quickly, just like knowing landmarks helps you navigate a city. The mediant property of Farey sequences, that if $a/b$ and $c/d$ are adjacent in $F_n$, then $(a+c)/(b+d)$ is the next fraction to appear between them, is used in the Stern-Brocot tree and underpins the theory of continued fractions and best rational approximations.