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Mathematics

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

−45\dfrac{-4}{5} and −13−15\dfrac{-13}{-15}

Rational Numbers

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Answer

To prove:

−45+−13−15=−13−15+−45\dfrac{-4}{5} + \dfrac{-13}{-15} = \dfrac{-13}{-15} + \dfrac{-4}{5}

Taking LHS: −45+−13−15=−45+1315\dfrac{-4}{5} + \dfrac{-13}{-15} \\[1em] = \dfrac{-4}{5} + \dfrac{13}{15} \\[1em]

LCM of 5 and 15 is 3 x 5 = 15

=−4×35×3+13×115×1=−1215+1315=−12+1315=115= \dfrac{-4 \times 3}{5 \times 3} + \dfrac{13 \times 1}{15 \times 1} \\[1em] = \dfrac{-12}{15} + \dfrac{13}{15} \\[1em] = \dfrac{-12 + 13}{15} \\[1em] = \dfrac{1}{15} \\[1em]

Taking RHS: −13−15+−45=1315+−45\dfrac{-13}{-15} + \dfrac{-4}{5} \\[1em] = \dfrac{13}{15} + \dfrac{-4}{5} \\[1em]

LCM of 15 and 5 is 3 x 5 = 15

=13×115×1+−4×35×3=1315+−1215=13+(−12)15=115= \dfrac{13 \times 1}{15 \times 1} + \dfrac{-4 \times 3}{5 \times 3} \\[1em] = \dfrac{13}{15} + \dfrac{-12}{15} \\[1em] = \dfrac{13 + (-12)}{15} \\[1em] = \dfrac{1}{15} \\[1em]

∴ LHS = RHS

Hence, −45+−13−15=−13−15+−45\dfrac{-4}{5} + \dfrac{-13}{-15} = \dfrac{-13}{-15} + \dfrac{-4}{5}

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

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