In this chapter we will learn about the HCF and LCM of fractions. We will see what do we mean when we say HCF/LCM of two or more fractions.
We will begin by understanding what do we mean by $\unicode{0x201C}\text{HCF and LCM of fractions}\unicode{0x201D}$, followed by the steps to find these values.
II. LCM Of Fractions
Let us begin with $LCM$.
We learnt that $LCM$ of two or more integers is the smallest such number, which when divided by any of the given integers, will leave a remainder of zero, or in other words the quotient of the division will be an integer.
The definition of $LCM$ is exactly the same. Given a number of fractions, their $LCM$ is the smallest such number, which when divided by any of these fractions will give an integer quotient.
Let us say we have two fractions $\dfrac{n_1}{d_1}$ and $\dfrac{n_2}{d_2}$ where $n_1,\ n_2$ are the numerators and $d_1,\ d_2$ are the denominators.
If we calculate the $LCM$ of just the numerators, let's say $N$, and we divide by each of the fractions, what do we get?
$N \div \dfrac{n_1}{d_1} = N \times \dfrac{d_1}{n_1}$
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Since $N$ is the LCM of $n_1$ and $n_2$, it is divisible by $n_1$, hence $n_1$ will cancel out with $N$, and we will be left with an integer. The same will be true when we divide $N$ by
$\dfrac{n_2}{d_2}$
.
We can see that $N$, when divided by any of the given fractions will give an integer result, and therefore, is a
common multiple
of the given fractions. But it may not be the lowest such multiple. Let us see what we can do to reduce the number further.
What do we get when we divide $N$ by the $HCF$ of the denominators of the given fractions?
Let us say the $HCF$ of $d_1$ and $d_2$ is $D$. So our result is $\dfrac{N}{D}$.
Now let us divide this number by each of the given fractions.
Now we can see that $N$ being the the $LCM$ of $n_1$ and $n_2$ is divisible by $n_1$ and $D$ being the $HCF$ of $d_1$ and $d_2$ is a factor of $d_1$. Therefore, $n_1$ cancels out with $N$ and $D$ cancels out with $d_1$, and since both the denominators cancel out, we will be left with an integer. And this is the smallest number that is divisible by all given fractions.