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 $\frac{- 2}{7}$

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Solution

(i)Addition
$4 + \frac{2}{5} = \frac{20 + 2}{5} = \frac{22}{5}$
$\frac{2}{5} + 4 = \frac{2 + 20}{5} = \frac{22}{5}$
Commutative property is satisfied.
Subtraction
$4 - \frac{2}{5} = \frac{20 - 2}{5} = \frac{18}{5}$
$\frac{2}{5} - 4 = \frac{2 - 20}{5} = \frac{- 18}{5}$
Commutative property is not satisfied.
Multiplication
$\frac{2}{5} \times 4 = \frac{8}{5}$
$4 \times \frac{2}{5} = \frac{8}{5}$
Commutative property is satisfied.
Division
$4 \div \frac{2}{5} = 4 \times \frac{5}{2} = \frac{20}{2}$
$\frac{2}{5} \div 4 = \frac{2}{5} \times \frac{1}{4} = \frac{2}{20}$
Commutative property is not satisfied.
(ii)Addition
$\frac{- 3}{7} + (\frac{- 2}{7}) = \frac{- 3 - 2}{7} = \frac{- 5}{7}$
$\frac{- 2}{7} + (\frac{- 3}{7}) = \frac{- 2 - 3}{7} = \frac{- 5}{7}$
Commutative property is satisfied.
Subtraction
$\frac{- 3}{7} - (\frac{- 2}{7}) = \frac{- 3 + 2}{7} = \frac{- 1}{7}$
$\frac{- 2}{7} - (\frac{- 3}{7}) = \frac{- 2 + 3}{7} = \frac{1}{7}$
Commutative property is not satisfied.
Multiplication
$\frac{- 2}{7} \times \frac{- 3}{7} = \frac{6}{7}$
$\frac{- 3}{7} \times \frac{- 2}{7} = \frac{6}{7}$
Commutative property is satisfied.
Division
$\frac{- 3}{7} \div \frac{- 2}{7} = \frac{- 3}{7} \times \frac{7}{- 2} = \frac{3}{2}$
$\frac{- 2}{7} \div \frac{- 3}{7} = \frac{- 2}{7} \times \frac{7}{- 3} = \frac{2}{3}$
Commutative property is not satisfied.

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