Question

Find the LCM of \[\left( \dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{6},\dfrac{7}{12} \right)\].

Answer 1

Hint: In order to find the Least Common Multiple of the given fractions, firstly we will be considering the fractions. Then we will be finding the HCF i.e. Highest Common Factor of the denominators of the given fractions and then we will be finding the LCM of the numerators of the given fractions. Then we will be dividing the LCM of numerators divided by HCF of denominators and that would be the required answer.

Complete step-by-step solution:
Now let us learn more about LCM of fractions. The Least Common Multiple of fractions is nothing but the least fraction that is the multiple of given fractions. LCM of two numbers can be found in the following four methods. They are: Listing the multiples, Prime Factorization Method, Division method and GCD method. The general formula to find the LCM of fractions is \[\dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}\].
Now let us find the LCM of the given fractions \[\left( \dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{6},\dfrac{7}{12} \right)\]
Firstly, let us consider the denominators and find the HCF of denominators.
The denominators are \[3,9,6,12\]
Let us find the HCF by prime factorization method.
\[\begin{align}
  & 3=1\times 3 \\
 & 9=3\times 3 \\
 & 6=2\times 3 \\
 & 12=2\times 2\times 3 \\
\end{align}\]
We can observe that only \[3\] is the common factor in all of the cases.
\[\therefore HCF=3\]
Now let us consider the numerators and let us find the LCM of numerators.
The numerators are \[2,4,5,7\].
The LCM of the numerators \[2,4,5,7\] is \[140\].
Now let us substitute the LCM and HCF values in \[\dfrac{\text{LCM of numerators}}{\text{HCFof denominators}}\]
We get,
\[\Rightarrow \dfrac{\text{LCM of numerators}}{\text{HCFof denominators}}=\dfrac{140}{3}\]
\[\therefore \] The LCM of \[\left( \dfrac{2}{3},\dfrac{4}{9},\dfrac{5}{6},\dfrac{7}{12} \right)\] is \[\dfrac{140}{3}\].

Note: In the above problem, we have used the method of prime factorization in finding the LCM. We must note that the method of finding LCM of fractions and integers are different. The common error committed could be not following the general formula correctly.