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Mathematics

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

59\dfrac{5}{9} and 5−12\dfrac{5}{-12}

Rational Numbers

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Answer

To prove:

59+5−12=5−12+59\dfrac{5}{9} + \dfrac{5}{-12} = \dfrac{5}{-12} + \dfrac{5}{9}

Taking LHS: 59+5−12=59+−512\dfrac{5}{9} + \dfrac{5}{-12} \\[1em] = \dfrac{5}{9} + \dfrac{-5}{12} \\[1em]

LCM of 9 and 12 is 2 x 2 x 3 x 3 = 36

=5×49×4+−5×312×3=2036+−1536=20+(−15)36=536= \dfrac{5 \times 4}{9 \times 4} + \dfrac{-5 \times 3}{12 \times 3} \\[1em] = \dfrac{20}{36} + \dfrac{-15}{36} \\[1em] = \dfrac{20 + (-15)}{36} \\[1em] = \dfrac{5}{36} \\[1em]

Taking RHS: 5−12+59=−512+59\dfrac{5}{-12} + \dfrac{5}{9} \\[1em] = \dfrac{-5}{12} + \dfrac{5}{9} \\[1em]

LCM of 12 and 9 is 2 x 2 x 3 x 3 = 36

=−5×312×3+5×49×4=−1536+2036=(−15)+2036=536= \dfrac{-5 \times 3}{12 \times 3} + \dfrac{5 \times 4}{9 \times 4} \\[1em] = \dfrac{-15}{36} + \dfrac{20}{36} \\[1em] = \dfrac{(-15) + 20}{36} \\[1em] = \dfrac{5}{36} \\[1em]

∴ LHS = RHS

Hence, 59+5−12=5−12+59\dfrac{5}{9} + \dfrac{5}{-12} = \dfrac{5}{-12} + \dfrac{5}{9}

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

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