GCF and LCM of Fractions | Number Properties

Let’s take a look at how to find GCF and LCM (Least Common Multiple) of fractions.

LCM of two or more fractions is given by: $\frac{LCM\ of\ numerators}{GCF\ of\ denominators}$

GCF of two or more fractions is given by: $\frac{GCF\ of\ numerators}{LCM\ of\ denominators}$

Why do we calculate LCM and GCF of fractions in this way? Let’s look at the algebraic explanation first. Then we will look at a more intuitive reason.

Algebraic Approach:

Consider 2 fractions $\frac{a}{b}$ and $\frac{c}{d}$ in their lowest form, their LCM is $\frac{Ln}{Ld}$ and GCF is $\frac{Gn}{Gd}$ , also in their lowest forms.

Let’s work on figuring out the LCM first.

LCM is a multiple of both the numbers so $\frac{Ln}{Ld}$ must be divisible by $\frac{a}{b}$ . This implies $\frac{(\frac{Ln}{Ld})}{(\frac{a}{b})}$ is an integer. We can re-write this as:

$\frac{Ln \times b}{Ld \times a}$ is an integer.

Since $\frac{a}{b}$ and $\frac{Ln}{Ld}$ are in their lowest forms, $Ln$ must be divisible by $a$ ; also, $b$ must be divisible by $Ld$ (because $a$ and $b$ have no common factors and $Ln$ and $Ld$ have no common factors).

Using the same logic, $Ln$ must be divisible by $c$ ; also $d$ must be divisible by $Ld$ .

$Ln$ , the numerator of LCM, must be divisible by both $a$ and $c$ and hence should be the LCM of $a$ and $c$ , the numerators. $Ln$ cannot be just any multiple of $a$ and $c$ ; it must be the lowest common multiple so that $\frac{Ln}{Ld}$ is the lowest multiple of the two fractions.

$b$ and $d$ both must be divisible by $Ld$ , the denominator of LCM, and hence $Ld$ must be their highest common factor. Mind you, it cannot be just any common factor; it needs to be the highest common factor so that $\frac{Ln}{Ld}$ is the lowest multiple possible.

This is why LCM of two or more fractions is given by: $\frac{LCM\ of\ numerators}{GCF\ of\ denominators}$ .

Using similar reasoning, you can figure out why we find GCF of fractions the way we do.

Now let me give you some feelers. They are more important than the algebraic explanation above. They build intuition.

Intuitive Approach:

Let me remind you first that LCM is the lowest common multiple. It is that smallest number which is a multiple of both the given numbers.

Say, we have two fractions: $\frac{1}{4}$ and $\frac{1}{2}$ . What is their LCM? It’s $\frac{1}{2}$ because $\frac{1}{2}$ is the smallest fraction which is a multiple of both $\frac{1}{2}$ and $\frac{1}{4}$ . It will be easier to understand in this way:

$\frac{1}{2} = \frac{2}{4}$ . (Fractions with the same denominator are comparable.)

LCM of $\frac{2}{4}$ and $\frac{1}{4}$ will obviously be $\frac{2}{4}$ .

If this is still tricky to see, think about their equivalents in decimal form:

$\frac{1}{2}$ = 0.50 and $\frac{1}{4}$ = 0.25. You can see that 0.50 is the lowest common multiple they have.

Let’s look at GCF now.

What is GCF of two fractions? It is that greatest factor which is common between the two fractions. Again, let’s take $\frac{1}{2}$ and $\frac{1}{4}$ . What is the greatest common factor between them?

Think of the numbers as $\frac{2}{4}$ and $\frac{1}{4}$ . The greatest common factor between them is $\frac{1}{4}$ .

(Note that $\frac{1}{2}$ and $\frac{1}{4}$ are both divisible by other factors too e.g. $\frac{1}{8}$ , $\frac{1}{24}$ etc but $\frac{1}{4}$ is the greatest such common factor)

Now think, what will be the LCM of $\frac{2}{3}$ and $\frac{1}{8}$ ?

We know that $\frac{2}{3} = \frac{16}{24}$ and $\frac{1}{8} = \frac{3}{24}$ .

LCM = $\frac{16\times 3}{24} = \frac{48}{24} = 2$ .

LCM is a fraction greater than both the fractions or equal to one or both of them (when both fractions are equal). When you take the LCM of the numerator and GCF of the denominator, you are making a fraction greater than (or equal to) the numbers.

Also, what will be the GCF of $\frac{2}{3}$ and $\frac{1}{8}$ ?

We know that $\frac{2}{3} = \frac{16}{24}$ and $\frac{1}{8} = \frac{3}{24}$ .

GCF = $\frac{1}{24}$ .

GCF is a fraction smaller than both the fractions or equal to one or both of them (when both fractions are equal). When you take the GCF of the numerator and LCM of the denominator, you are making a fraction smaller than (or equal to) the numbers.

We hope the concept of GCF and LCM of fractions makes sense to you now.

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