# Express each of the following as a rational number of the form

Question:

Express each of the following as a rational number of the form $\frac{p}{q}$ :

(i) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3$

(ii) $\frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7}$

(iii) $\frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6}$

(iv) $\frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14}$

(v) $\frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2$

Solution:

(i) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3$

$=\frac{-64}{24}+\frac{-6}{24}+\frac{-44}{24}+\frac{9}{24}-\frac{72}{24}$

$=\frac{(-64)+(-6)+(-44)+9+(-72)}{24}$

$=\frac{-64-6-44+9-72}{24}$

$=\frac{-177}{24}$

$=\frac{-59}{8}$

(ii) $\frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7}$

$=\frac{54}{63}+\frac{63}{63}+\frac{-49}{63}+\frac{57}{63}+\frac{-108}{63}$

$=\frac{54+63+(-49)+57+(-108)}{63}$

$=\frac{54+63-49+57-108}{63}$

$=\frac{17}{63}$

(iii) $\frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6}$

$=\frac{180}{24}+\frac{27}{24}+\frac{-88}{24}+\frac{144}{24}+\frac{-28}{24}$

$=\frac{180+27+(-88)+144+(-28)}{24}$

$=\frac{180+27-88+144-28}{24}$

$=\frac{235}{24}$

(iv) $\frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14}$

$=\frac{-245}{140}+\frac{-252}{140}+\frac{266}{140}+\frac{110}{140}$

$=\frac{(-245)+(-252)+266+110}{140}$

$=\frac{-245-252+266+110}{140}$

$=\frac{-121}{140}$

(v) $\frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2$

$=\frac{-21}{12}+\frac{20}{12}+\frac{-6}{12}+\frac{-10}{12}+\frac{24}{12}$

$=\frac{-21+20-6-10+24}{12}$

$=\frac{7}{12}$