FactorsHCF and LCM of fractions and decimals
HCF and LCM are first met with whole numbers, but exam questions also ask for them with fractions and decimals. The rules are short. This page gives them with worked examples and shows why they are true.
The two rules for fractions
HCF of fractions = HCF of the numerators ÷ LCM of the denominators
LCM of fractions = LCM of the numerators ÷ HCF of the denominators
Put the fractions in lowest terms before you start.
Example: 2/3, 4/9 and 8/15
- Numerators 2, 4, 8: HCF = 2, LCM = 8.
- Denominators 3, 9, 15: HCF = 3, LCM = 45.
HCF = 2/45. Check: 2/3 ÷ 2/45 = 15, 4/9 ÷ 2/45 = 10 and 8/15 ÷ 2/45 = 12. All are whole numbers, so 2/45 divides each fraction exactly.
LCM = 8/3. Check: 8/3 ÷ 2/3 = 4, 8/3 ÷ 4/9 = 6 and 8/3 ÷ 8/15 = 5. Again whole numbers, so 8/3 is a common multiple.
What HCF and LCM mean for fractions
The HCF of a set of fractions is the largest fraction that goes into each of them a whole number of times. The LCM is the smallest fraction that each of them goes into a whole number of times. That is the same idea as for whole numbers, with "divides exactly" meaning "the quotient is a whole number".
Why the rules work
Write the fractions with a common denominator. 2/3, 4/9 and 8/15 become 30/45, 20/45 and 24/45. Now they are just 30, 20 and 24 counted in forty-fifths. The HCF of 30, 20 and 24 is 2, so the HCF of the fractions is 2/45. The LCM of 30, 20 and 24 is 120, so the LCM is 120/45, which simplifies to 8/3. The two short rules give the same answers without the extra step.
Decimals
Make all the decimals have the same number of decimal places, drop the point, work with whole numbers, then put the point back.
HCF and LCM of 0.6, 1.8 and 0.24. Write them as 0.60, 1.80 and 0.24: that is 60, 180 and 24 hundredths. HCF(60, 180, 24) = 12 and LCM(60, 180, 24) = 360. So the HCF is 0.12 and the LCM is 3.60, or 3.6.
A word problem
Three taps drip once every 1/2 second, 3/4 second and 5/6 second. They drip together now. After how long will they drip together again?
We need the LCM of 1/2, 3/4 and 5/6. LCM of numerators 1, 3, 5 is 15. HCF of denominators 2, 4, 6 is 2. The answer is 15/2 seconds, or 7.5 seconds. Check: in 7.5 seconds the taps drip 15, 10 and 9 times.
Points to remember
- Simplify each fraction first. With 2/4 in place of 1/2 the rule can give a wrong answer.
- The HCF of fractions is never larger than the smallest fraction, and the LCM is never smaller than the largest.
- The LCM of fractions may be a whole number or a fraction. Both are fine.
- For mixed numbers, change to improper fractions first.
For the whole-number steps inside these sums, the HCF and LCM calculator shows the working.