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Mathematics

For each pair of rational number, verify commutative property of addition of rational numbers.

2−5\dfrac{2}{-5} and 11−15\dfrac{11}{-15}

Rational Numbers

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Answer

To prove:

2−5+11−15=11−15+2−5\dfrac{2}{-5} + \dfrac{11}{-15} = \dfrac{11}{-15} + \dfrac{2}{-5}

Taking LHS: 2−5+11−15=−25+−1115\dfrac{2}{-5} + \dfrac{11}{-15} \\[1em] = \dfrac{-2}{5} + \dfrac{-11}{15} \\[1em]

LCM of 5 and 15 is 3 x 5 = 15

=−2×35×3+−11×115×1=−615+−1115=−6+(−11)15=−1715= \dfrac{-2 \times 3}{5 \times 3} + \dfrac{-11 \times 1}{15 \times 1} \\[1em] = \dfrac{-6}{15} + \dfrac{-11}{15} \\[1em] = \dfrac{-6 + (-11)}{15} \\[1em] = \dfrac{-17}{15} \\[1em]

Taking RHS: 11−15+2−5=−1115+−25\dfrac{11}{-15} + \dfrac{2}{-5} \\[1em] = \dfrac{-11}{15} + \dfrac{-2}{5} \\[1em]

LCM of 15 and 5 is 3 x 5 = 15

=−11×115×1+−2×35×3=−1115+−615=(−11)+(−6)15=−1715= \dfrac{-11 \times 1}{15 \times 1} + \dfrac{-2 \times 3}{5 \times 3} \\[1em] = \dfrac{-11}{15} + \dfrac{-6}{15} \\[1em] = \dfrac{(-11) + (-6)}{15} \\[1em] = \dfrac{-17}{15} \\[1em]

∴ LHS = RHS

Hence, 2−5+11−15=11−15+2−5\dfrac{2}{-5} + \dfrac{11}{-15} = \dfrac{11}{-15} + \dfrac{2}{-5}

So, the commutative property for the addition of the rational number is verified.

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