Formula

LCM and HCF of fractions

Arithmetic • Number System • LCM and HCF

Formula

LCM (\frac{a _{1}}{b _{1}}, \dots, \frac{a _{n}}{b _{n}}) = \frac{LCM ( a _{1} , \dots , a _{n} )}{HCF ( b _{1} , \dots , b _{n} )}

For fractions written in lowest terms,

LCM = \frac{LCM of the numerators}{HCF of the denominators}

and

HCF = \frac{HCF of the numerators}{LCM of the denominators}

How to use it

1Reduce every fraction to lowest terms first; the rule is false otherwise.
2Sanity filter: the LCM of a set of fractions is never smaller than the largest of them, and the HCF never larger than the smallest.
3Confirm an answer by dividing it by each fraction — every quotient must be a whole number.
4The two formulas are mirror images, so remembering one and swapping LCM for HCF throughout recovers the other.

Worked example

What is the LCM of

\frac{3}{4}

\frac{5}{6}

and

\frac{7}{8}

Step-by-step

1
$LCM (3, 5, 7) = 105$
Why: The three numerators are pairwise coprime, so their LCM is the product.
2
$HCF (4, 6, 8) = 2$
Why: The largest number dividing all three denominators.
3
$LCM = \frac{105}{2}$
Why: Numerator LCM over denominator HCF.

Answer

\frac{105}{2}

Watch out

Inverting the rule — LCM of the denominators instead of their HCF — is the standard trap and lands on a value smaller than some of the fractions. Fractions not in lowest terms corrupt both halves of the formula at once.

Builds on

LCM and HCF of several numbers from prime powers

Also known as

lcm of fractions, hcf of fractions, gcd of rationals, lcm of rational numbers

Appears in past papers

2020 (I)

Q65