Question

Carry out the following additions of rational numbers.
\[\dfrac{11}{17}+\dfrac{13}{19}\]

Answer 1

Hint: To carry out this problem, try to make the denominator of both the rational numbers same and then proceed with the problem.

Complete step by step solution:
The addition of rational numbers is done in the same way as that of addition of fractions. But there are few things to be kept in mind which are if the two rational numbers have the same denominator, then we can add the numerators directly keeping the denominator the same.
For example,
 \[\dfrac{2}{3}+\dfrac{8}{3}=\dfrac{10}{3}\]
If the two rational numbers have different denominators, first we will have to make the denominators of both the rational numbers same by either taking the LCM of denominators or multiplying the numerator and denominator of both rational numbers by a certain number so that the denominator of both rational numbers become same and then we can simply add the numerators.
For example,
\[\begin{align}
  & \dfrac{3}{2}+\dfrac{7}{3}=\dfrac{3\times 3}{2\times 3}+\dfrac{7\times 2}{3\times 2}=\dfrac{23}{6} \\
 & \\
\end{align}\]
The LCM of the denominators is 6. Now, we express \[\dfrac{3}{2}\] and \[\dfrac{7}{3}\] into forms in which both of them have the same denominator\[6\].
Now , we have understood the concept, let’s solve the question.
LCM of \[17\] and \[19\] is \[343\]. LCM of two prime numbers is the multiplication of both the numbers as they don’t have any common factor other than \[1\]. \[17\] and \[19\]are both prime numbers. Therefore, the LCM is \[17\times 19=343\]
\[\dfrac{11}{17}+\dfrac{13}{19}=\dfrac{\left( 11\times 19 \right)}{\left( 17\times 19 \right)}+\dfrac{\left( 13\times 17 \right)}{\left( 19\times 17 \right)}=\dfrac{430}{323}\]

Therefore, the addition of \[\dfrac{11}{17}+\dfrac{13}{19}\] = $\dfrac{430}{323}$.

Note:
LCM stands for least common multiple. The LCM of a group is the smallest non-zero number that is multiple of all numbers. If the numbers are not prime, then you would have to find the L.C.M by prime factorisation method or any other method.