Add the following rational numbers - i -2-5 and 4-5 ii -6-11 and -4-11 iii -11-8 and 5-8 iv -7-3 and 1-3 v 5-6 and -1-6 iv -17-15 and -1-15

(i) 2/5 + 4/5 = 2 + 4/5 = 2/5 (ii) 6/11 + 4/11 = 6 4/11 = 10/11 (iii) 11/8 + 5/8 = 11 + 5/8 = 6/8 = 6/8 = 6 ÷ 2/8 ÷ 2 = 3/4 (iv) 7/3 + 1/3 = 7 + ...

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Standard VII

Mathematics

Addition of Rational Numbers

Add the follo...

Question

Add the following rational numbers :

(i) $\frac{- 2}{5}$ and $\frac{4}{5}$ (ii) $\frac{- 6}{11}$ and $\frac{- 4}{11}$ (iii) $\frac{- 11}{8}$ and $\frac{5}{8}$

(iv) $\frac{- 7}{3}$ and $\frac{1}{3}$ (v) $\frac{5}{6}$ and $\frac{- 1}{6}$ (iv) $\frac{- 17}{15}$ and $\frac{- 1}{15}$

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Solution

(i) $\frac{- 2}{5} + \frac{4}{5} = \frac{- 2 + 4}{5} = \frac{2}{5}$

(ii) $\frac{- 6}{11} + \frac{- 4}{11} = \frac{- 6 - 4}{11} = \frac{- 10}{11}$

(iii) $\frac{- 11}{8} + \frac{5}{8} = \frac{- 11 + 5}{8} = \frac{- 6}{8} = \frac{- 6}{8} = \frac{- 6 \div 2}{8 \div 2} = \frac{- 3}{4}$

(iv) $\frac{- 7}{3} + \frac{1}{3} = \frac{- 7 + 1}{3} = \frac{- 6}{3} \frac{- 6 \div 3}{3 \div 3} = \frac{- 2}{1} = - 2$

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

(vi) $\frac{- 17}{15} + \frac{- 1}{15} = \frac{- 17 - 1}{15} = \frac{- 18}{15} \frac{- 18 \div 3}{15 \div 3} = \frac{- 6}{5}$

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Add the following rational numbers:
(i) $\frac{3}{4}$ and $\frac{- 3}{5}$
(ii) $\frac{5}{8}$ and $\frac{- 7}{12}$
(iii) $\frac{- 8}{9}$ and $\frac{11}{6}$
(iv) $\frac{- 5}{16}$ and $\frac{7}{24}$
(v) $\frac{7}{- 18}$ and $\frac{8}{27}$
(vi) $\frac{1}{- 12}$ and $\frac{2}{- 15}$
(vii) $- 1$ and $\frac{3}{4}$
(viii) $2$ and $\frac{- 5}{4}$
(ix) $0$ and $\frac{- 2}{5}$

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

Verify the following :

(i) $\frac{3}{7} \times (\frac{5}{6} + \frac{12}{13}) = (\frac{3}{7} \times \frac{5}{6}) + (\frac{3}{7} \times \frac{12}{13})$
(ii) $\frac{- 15}{4} \times (\frac{3}{7} + \frac{- 12}{5}) = (\frac{- 15}{4} \times \frac{3}{7}) + (\frac{- 15}{4} \times \frac{- 12}{5})$
(iii) $(\frac{- 8}{3} + \frac{- 13}{12}) \times \frac{5}{6} = (\frac{- 8}{3} \times \frac{5}{6}) + (\frac{- 13}{12} \times \frac{5}{6})$
(iv) $\frac{- 16}{7} \times (\frac{- 8}{9} + \frac{- 7}{6}) = (\frac{- 16}{7} \times \frac{- 8}{9}) + (\frac{- 16}{7} \times \frac{- 7}{6})$

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

Prove that:
(i) $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
(ii) $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$

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Add the following rational numbers : (i) −25 and 45 (ii) −611 and −411 (iii) −118 and 58 (iv) −73 and 13 (v) 56 and −16 (iv) −1715 and −115

(i) −25+45=−2+45=25 (ii) −611+−411=−6−411=−1011 (iii) −118+58=−11+58=−68=−68=−6÷28÷2=−34 (iv) −73+13=−7+13=−63−6÷33÷3=−21=−2 (v) 56+−16=5−16=46=4÷26÷2=23 (vi) −1715+−115=−17−115=−1815−18÷315÷3=−65

Add the following rational numbers :

(i) $\frac{- 2}{5}$ and $\frac{4}{5}$ (ii) $\frac{- 6}{11}$ and $\frac{- 4}{11}$ (iii) $\frac{- 11}{8}$ and $\frac{5}{8}$

(iv) $\frac{- 7}{3}$ and $\frac{1}{3}$ (v) $\frac{5}{6}$ and $\frac{- 1}{6}$ (iv) $\frac{- 17}{15}$ and $\frac{- 1}{15}$