Question

Carry out the following additions of rational numbers.

(i) $\frac{5}{36}+\frac{6}{42}$

(ii) $1\frac{2}{3}+2\frac{4}{5}$

(iii) $\frac{11}{17}+\frac{13}{19}$

(iv) $2\frac{3}{11}+1\frac{3}{77}$

Answer 1

$\left(\mathrm{i}\right)\frac{5}{36}+\frac{6}{42}$
At first, we will calculate the LCM of 36 and 42. The prime factorisation is 36 and 42 is,
36 = 2 × 2 × 3 × 3
42 = 2 × 3 × 7
Now, LCM of 36 and 42 = 2 × 2 × 3 × 3 × 7 = 252
$$\begin{gathered} \frac{5}{36}+\frac{6}{42} \\ =\frac{5\times 7}{36\times 7}+\frac{6\times 6}{42\times 6} \\ =\frac{35}{252}+\frac{36}{252} \\ =\frac{35+36}{252} \\ =\frac{71}{252} \end{gathered}$$
$$\begin{gathered} \left(\mathrm{ii}\right)1\frac{2}{3}+2\frac{4}{5} \\ =\frac{1\times 3+2}{3}+\frac{2\times 5+4}{5} \\ =\frac{5}{3}+\frac{14}{5} \end{gathered}$$
Now, LCM of 3 and 5 is 15.
$$\begin{gathered} \frac{5}{3}+\frac{14}{5} \\ =\frac{5\times 5}{3\times 5}+\frac{14\times 3}{5\times 3} \\ =\frac{25}{15}+\frac{42}{15} \\ =\frac{67}{15} \\ =4\frac{7}{15} \end{gathered}$$
$\left(\mathrm{iii}\right)\frac{11}{17}+\frac{13}{19}$
Now, LCM of 17 and 19 is 323.
$$\begin{gathered} \frac{11}{17}+\frac{13}{19} \\ =\frac{11\times 19}{17\times 19}+\frac{13\times 17}{19\times 17} \\ =\frac{209}{323}+\frac{221}{323} \\ =\frac{430}{323} \end{gathered}$$
$$\begin{gathered} \left(\mathrm{iv}\right)2\frac{3}{11}+1\frac{3}{77} \\ =\frac{2\times 11+3}{11}+\frac{1\times 77+3}{77} \\ =\frac{25}{11}+\frac{80}{77} \end{gathered}$$
Now, LCM of 11 and 77 is 77.
$$\begin{gathered} \frac{25}{11}+\frac{80}{77} \\ =\frac{25\times 7}{11\times 7}+\frac{80\times 1}{77\times 1} \\ =\frac{175}{77}+\frac{80}{77} \\ =\frac{255}{77}=3\frac{24}{77} \end{gathered}$$