Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers- i -11-5-and-4-7 ii4-9-and-7-12 iii -3-5-and-2-15 iv 2-7-and-12-35 v 4-and-3-5 vi -4-and-4-7

(i) 11/5 and 4/7 11/5+4/7 ∴ 11/5= 11 × 7/5 × 7= 77/35 and 4/7=4 × 5/7 × 5= 20/35 ∴ 11/5+4/7= 77/35+20/35 = 77+20/35= 57/35 and4/7+ 11/5=20/35 77/35 =20 7 ...

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Standard VIII

Mathematics

Commutative Property of Rational Numbers

Verify commut...

Question

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) $- 4 a n d \frac{4}{- 7}$

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Solution

(i) $\frac{- 11}{5} a n d \frac{4}{7} \frac{- 11}{5} + \frac{4}{7} ∴ \frac{- 11}{5} = \frac{- 11 \times 7}{5 \times 7} = \frac{- 77}{35} a n d \frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35} ∴ \frac{- 11}{5} + \frac{4}{7} = \frac{- 77}{35} + \frac{20}{35} = \frac{- 77 + 20}{35} = \frac{- 57}{35} a n d \frac{4}{7} + \frac{- 11}{5} = \frac{20}{35} - \frac{77}{35} = \frac{20 - 77}{35} = \frac{- 57}{35} ∴ \frac{- 11}{5} + \frac{4}{7} = \frac{4}{7} + \frac{- 11}{5}$

(ii) $\frac{4}{9} a n d \frac{7}{- 12} \frac{7 \times (- 1)}{- 12 \times (- 1)} = \frac{- 7}{12} N o w \frac{4}{9} + \frac{- 7}{12} L C M o f 9, 12 = 36 ∴ \frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36} \frac{- 7}{12} = \frac{- 7 \times 3}{12 \times 3} = \frac{- 21}{36} ∴ \frac{4}{9} + \frac{- 7}{12} + \frac{4}{9}$

(iii) $\frac{- 3}{5} a n d \frac{- 2}{- 15} \frac{- 2}{- 15} = \frac{- 2 \times (- 1)}{- 15 \times (- 1)} = \frac{2}{15} L C M o f 5 a n d 15 = 15 ∴ \frac{- 3}{5} = \frac{- 3 \times 3}{5 \times 3} = \frac{- 9}{15} N o w \frac{- 3}{5} + \frac{2}{15} = \frac{- 9}{15} + \frac{2}{15} \frac{- 9 + 2}{15} = \frac{- 7}{15} a n d \frac{2}{15} + \frac{- 3}{5} = \frac{2}{15} + \frac{- 9}{15} = \frac{2 - 9}{15} = \frac{- 7}{15} ∴ \frac{- 3}{5} + \frac{2}{15} = \frac{2}{15} + \frac{- 3}{5}$

(iv) $\frac{2}{- 7} a n d \frac{12}{- 35} L C M o f 7, 35 = 35 \frac{2}{- 7} = \frac{2 \times (- 5)}{- 7 \times (- 5)} = \frac{- 10}{35} \frac{12}{- 35} = \frac{12 \times (- 1)}{- 35 \times (- 1)} = \frac{- 12}{35} N o w \frac{- 2}{7} + \frac{- 12}{35} = \frac{- 10}{35} + \frac{- 12}{35} = \frac{- 10 - 12}{35} = \frac{- 22}{35} a n d \frac{- 12}{35} + \frac{- 2}{7} = \frac{- 12}{35} + \frac{- 10}{35} = \frac{- 12 - 10}{35} = \frac{- 22}{35} ∴ \frac{- 2}{7} + \frac{- 12}{35} = \frac{- 12}{35} + \frac{- 2}{7}$

(v) $4 a n d \frac{- 3}{5} L C M o f 1, 5 = 5 ∴ \frac{4}{1} = \frac{4 \times 5}{1 \times 5} = \frac{20}{5} N o w \frac{4}{1} + \frac{- 3}{5} = \frac{20}{5} + \frac{- 3}{5} = \frac{20 - 3}{5} = \frac{17}{5} a n d \frac{- 3}{5} + \frac{4}{1} = \frac{- 3}{5} + \frac{20}{5} = \frac{- 3 + 20}{5} = \frac{17}{5} ∴ 4 + \frac{- 3}{5} = \frac{- 3}{5} + 4$

(vi) $- 4 a n d \frac{4}{- 7} \frac{- 4}{1} a n d \frac{4}{- 7} (∵ \frac{4}{- 7} = \frac{4 \times (- 1)}{- 7 \times (- 1)} = \frac{- 4}{7}) ∴ \frac{- 4}{1} = \frac{- 4 \times 7}{1 \times 7} = \frac{- 28}{7} a n d \frac{- 4}{7} = \frac{- 28 - 4}{7} = \frac{- 32}{7} a n d \frac{- 4}{7} + (- 4) = \frac{- 4}{7} + \frac{- 28}{7} = \frac{- 4 - 28}{7} = \frac{- 32}{7} ∴ - 4 + \frac{- 4}{7} = \frac{- 4}{7} + (- 4)$

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Similar questions

Q. Verify commutativty of addition of rational numbers for each of the following pairs of rotional numbers:
(i)

\frac{- 11}{5} and \frac{4}{7}

(ii)

\frac{4}{9} and \frac{7}{- 12}

(iii)

\frac{- 3}{5} and \frac{- 2}{- 15}

(iv)

\frac{2}{- 7} and \frac{12}{- 35}

(v)

4 and \frac{- 3}{5}

(vi)

- 4 and \frac{4}{- 7}

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

Q. Verify whether commutative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.
(i)

4

and

\frac{2}{5}

(ii)

\frac{- 3}{7}

and

\frac{- 2}{7}

For each set of rational numbers, given below, verify the associative property of addition of rational numbers:
$(i) \frac{1}{2}, \frac{2}{3} and - \frac{1}{6} (i i) \frac{- 2}{5}, \frac{4}{15} and \frac{- 7}{10} (i i i) \frac{- 7}{9}, \frac{2}{- 3} and \frac{- 5}{18} (i v) - 1, \frac{5}{6} ~ a n d \frac{- 2}{3}$

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 $\frac{- 2}{5}$

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Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers: (i) −115 and 47 (ii)49 and 7−12 (iii) −35 and −2−15 (iv) 2−7 and 12−35 (v) 4 and −35 (vi) −4 and 4−7