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Mathematics

The arrangement of rational numbers −710,5−8,2−3\dfrac{-7}{10}, \dfrac{5}{-8}, \dfrac{2}{-3}, in ascending order is

  1. 2−3,5−8,−710\dfrac{2}{-3}, \dfrac{5}{-8}, \dfrac{-7}{10}

  2. −710,2−3,5−8\dfrac{-7}{10}, \dfrac{2}{-3}, \dfrac{5}{-8}

  3. 5−8,−710,2−3\dfrac{5}{-8}, \dfrac{-7}{10}, \dfrac{2}{-3}

  4. −710,5−8,2−3\dfrac{-7}{10}, \dfrac{5}{-8}, \dfrac{2}{-3}

Rational Numbers

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Answer

First write each rational number with a positive denominator.

⇒5−8=5×(−1)−8×(−1)=−58⇒2−3=2×(−1)−3×(−1)=−23\Rightarrow \dfrac{5}{-8} = \dfrac{5 \times (-1)}{-8 \times (-1)} = \dfrac{-5}{8} \\[1em] \Rightarrow \dfrac{2}{-3} = \dfrac{2 \times (-1)}{-3 \times (-1)} = \dfrac{-2}{3}

Now write each rational number with the LCM of the denominators as the common denominator.

[ LCM of 10,8 and 3=120]⇒−710=(−7)×1210×12=−84120⇒−58=(−5)×158×15=−75120⇒−23=(−2)×403×40=−80120[\text{ LCM of } 10, 8 \text{ and } 3 = 120] \\[1em] \Rightarrow \dfrac{-7}{10} = \dfrac{(-7) \times 12}{10 \times 12} = \dfrac{-84}{120} \\[1em] \Rightarrow \dfrac{-5}{8} = \dfrac{(-5) \times 15}{8 \times 15} = \dfrac{-75}{120} \\[1em] \Rightarrow \dfrac{-2}{3} = \dfrac{(-2) \times 40}{3 \times 40} = \dfrac{-80}{120}

As -84 < -80 < -75, so −84120<−80120<−75120\dfrac{-84}{120} \lt \dfrac{-80}{120} \lt \dfrac{-75}{120}.

⇒−710<2−3<5−8\Rightarrow \dfrac{-7}{10} \lt \dfrac{2}{-3} \lt \dfrac{5}{-8}

Hence, Option 2 is the correct option.

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