Least Common Denominator
Fractions & DecimalsThe least common denominator (LCD) is the smallest number that is a multiple of two or more denominators, used to add or compare fractions efficiently.
Formula
\text{LCD}(b, d) = \text{LCM}(b, d) = \frac{bd}{\gcd(b, d)}
Definition
The least common denominator (LCD) is the smallest number that can be used as the bottom number for two fractions at the same time. It is the smallest common denominator you can use.
Example
For $1/4$ and $1/6$, list multiples: $4, 8, 12, 16, \ldots$ and $6, 12, 18, \ldots$ The first number on both lists is $12$. The LCD is $12$. So $1/4 = 3/12$ and $1/6 = 2/12$.
Key Insight
Using the LCD (instead of just any common denominator) keeps your numbers as small and simple as possible - less work and fewer chances for mistakes.
Definition
The least common denominator of fractions $a/b$ and $c/d$ is $\text{LCM}(b, d)$, the least common multiple of the denominators. It can be computed as $\text{LCM}(b, d) = bd/\gcd(b, d)$. For three or more fractions, take the LCM of all denominators.
Example
LCD of $5/12$ and $7/18$: $\gcd(12, 18) = 6$. $\text{LCM} = 12 \times 18/6 = 36$. Convert: $5/12 = 15/36$; $7/18 = 14/36$. Subtract: $15/36 - 14/36 = 1/36$.
Key Insight
The LCD is not just for addition and subtraction - it also appears when comparing fractions. Rewriting fractions over their LCD makes size comparisons trivial: the fraction with the larger numerator is larger.
Definition
The LCD of a finite set of rationals $\{a_i/b_i\}$ is $\text{LCM}(b_1, b_2, \ldots, b_n)$, computed via prime factorization: LCM = product of each prime to its maximum exponent across all denominators. This equals $b_1 \times b_2 \times \ldots \times b_n$ / (all pairwise and higher-order GCDs via inclusion-exclusion).
Example
LCD of $1/12$, $1/18$, $1/20$: $12 = 2^2 \times 3$, $18 = 2 \times 3^2$, $20 = 2^2 \times 5$. $\text{LCM} = 2^2 \times 3^2 \times 5 = 180$. Each fraction converts: $1/12 = 15/180$, $1/18 = 10/180$, $1/20 = 9/180$. Sum $= 34/180 = 17/90$.
Key Insight
The LCD computation via prime factorization is an instance of the lattice join in the divisibility poset of positive integers: $\text{LCM}(a, b)$ is the join (least upper bound) of $a$ and $b$, while $\gcd(a, b)$ is their meet (greatest lower bound).