Using commutativity and associativity of addition of rational numbers, express each of the following as a rational number:

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

(ii) $\frac{3}{7} + \frac{- 4}{9} + \frac{- 11}{7} + \frac{7}{9}$

(iii) $\frac{2}{5} + \frac{8}{3} + \frac{- 11}{15} + \frac{4}{5} + \frac{- 2}{3}$

(iv) $\frac{4}{7} + 0 + \frac{- 8}{9} + \frac{- 13}{7} + \frac{17}{21}$

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Solution

(i) $\frac{2}{5} + \frac{7}{3} + \frac{- 4}{5} + \frac{- 1}{3} = (\frac{2}{5} + \frac{- 4}{5}) + (\frac{7}{3} + \frac{- 1}{3}) = (\frac{2 - 4}{5}) + (\frac{7 - 1}{3}) = \frac{- 2}{5} + \frac{6}{3} = \frac{- 2}{5} + \frac{2}{1} = \frac{2}{1} - \frac{2}{5} = \frac{10 - 2}{5} = \frac{8}{5}$

(ii) $= (\frac{3}{7} + \frac{- 11}{7}) + (\frac{- 4}{9} + \frac{7}{9}) = \frac{3 - 11}{7} + \frac{- 4 + 7}{9} = \frac{- 8}{7} + \frac{3}{9} = \frac{- 8}{7} + \frac{1}{3} (∵ \frac{3}{9} = \frac{3 \div 3}{9 \div 3} = \frac{1}{3}) = \frac{- 24 + 7}{21} = \frac{- 17}{21}$

(iii) $\frac{2}{5} + \frac{8}{3} + \frac{- 11}{15} + \frac{4}{5} + \frac{- 2}{3} = (\frac{2}{5} + \frac{4}{5}) + (\frac{8}{3} + \frac{- 2}{3}) + \frac{- 11}{15} = \frac{2 + 4}{5} + \frac{8 - 2}{3} + \frac{- 11}{15} = \frac{6}{5} + \frac{6}{3} + \frac{- 11}{15} = \frac{18 + 30 - 11}{15} = \frac{48 - 11}{15} = \frac{37}{15}$

(iv) $\frac{4}{7} + 0 + \frac{- 8}{9} + \frac{- 13}{7} + \frac{17}{21} = (\frac{4}{7} + \frac{- 13}{7}) + (\frac{- 8}{9} + \frac{17}{21}) + 0 = \frac{4 - 13}{7} + \frac{- 56 + 51}{63} = \frac{- 9}{7} + \frac{- 5}{63} = \frac{- 81 - 5}{63} = \frac{- 86}{63}$

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Similar questions

Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers: (i) $\frac{- 11}{5} a n d \frac{4}{7}$ (ii) $\frac{4}{9} a n d \frac{7}{- 12}$ (iii) $\frac{- 3}{5} a n d \frac{- 2}{- 15}$ (iv) $\frac{2}{- 7} a n d \frac{12}{- 35}$ (v) $4 a n d \frac{- 3}{5}$ (vi) $- 4 a n d \frac{4}{- 7}$

Re-arrange suitably and find the sum in each of the following :

(i) $\frac{11}{12} + \frac{- 17}{3} + \frac{11}{2} + \frac{- 25}{2}$

(ii) $\frac{- 6}{7} + \frac{- 5}{6} + \frac{- 4}{9} + \frac{- 15}{7}$

(iii) $\frac{3}{5} + \frac{7}{3} + \frac{9}{5} + \frac{- 13}{15} + \frac{- 7}{3}$

(iv) $\frac{4}{13} + \frac{- 5}{8} + \frac{- 8}{13} + \frac{9}{13}$

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

(vi) $\frac{1}{8} + \frac{5}{12} + \frac{2}{7} + \frac{7}{12} + \frac{9}{7} + \frac{- 5}{16}$

In each of the following determine rational numbers a and b :

(i) $\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = a - b \sqrt{3}$

(ii) $\frac{4 + \sqrt{2}}{2 + \sqrt{2}} = a - \sqrt{b}$

(iii) $\frac{3 + \sqrt{2}}{3 - \sqrt{2}} = a + b \sqrt{2}$

(iv) $\frac{5 + 3 \sqrt{3}}{7 + 4 \sqrt{3}} = a + b \sqrt{3}$

(v) $\frac{\sqrt{11} - \sqrt{7}}{\sqrt{11} + \sqrt{7}} = a - b \sqrt{77}$

(vi) $\frac{4 + 3 \sqrt{5}}{4 - 3 \sqrt{5}} = a + b \sqrt{5}$

Q. Evaluate:

(i) \frac{3}{7} + \frac{- 4}{9} - \frac{- 11}{7} - \frac{7}{9}

(i i) \frac{2}{3} + \frac{- 4}{5} - \frac{1}{3} - \frac{2}{5}

(i i i) \frac{4}{7} - \frac{- 8}{9} - \frac{- 13}{7} + \frac{17}{9}

Evaluate :

$(i) \frac{3}{7} + \frac{- 4}{9} + \frac{- 11}{7} + \frac{7}{9} (i i) \frac{2}{3} + \frac{- 4}{5} + \frac{1}{3} + \frac{2}{5} (i i i) \frac{4}{7} + 0 + \frac{- 8}{9} + \frac{- 13}{7} + \frac{17}{9} (i v) \frac{3}{8} + \frac{- 5}{12} + \frac{3}{7} + \frac{3}{12} + \frac{- 5}{8} + \frac{- 2}{7}$

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Using commutativity and associativity of addition of rational numbers, express each of the following as a rational number: (i) 25+73+−45+−13 (ii) 37+−49+−117+79 (iii) 25+83+−1115+45+−23 (iv) 47+0+−89+−137+1721