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Mathematics

The rational number that must be added to the sum of 37,−821 and −314\dfrac{3}{7}, -\dfrac{8}{21} \text{ and } -\dfrac{3}{14} to get the result −514\dfrac{-5}{14} is:

  1. −421\dfrac{-4}{21}

  2. 421\dfrac{4}{21}

  3. −521\dfrac{-5}{21}

  4. −16\dfrac{-1}{6}

Rational Numbers

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Answer

First find the sum of the three given rational numbers.

LCM of 7, 21 and 14 = 42

37+(−821)+(−314)=3×6+(−8)×2+(−3)×342=18−16−942=−742=−16\dfrac{3}{7} + \left(\dfrac{-8}{21}\right) + \left(\dfrac{-3}{14}\right) = \dfrac{3 \times 6 + (-8) \times 2 + (-3) \times 3}{42} \\[1em] = \dfrac{18 - 16 - 9}{42} \\[1em] = \dfrac{-7}{42} = \dfrac{-1}{6}

Let the required rational number be xx.

According to the given condition,

LCM of 14 and 6 = 42

⇒−16+x=−514⇒x=−514−(−16)⇒x=−514+16⇒x=(−5)×3+1×742⇒x=−15+742⇒x=−842=−421\Rightarrow \dfrac{-1}{6} + x = \dfrac{-5}{14} \\[1em] \Rightarrow x = \dfrac{-5}{14} - \left(\dfrac{-1}{6}\right) \\[1em] \Rightarrow x = \dfrac{-5}{14} + \dfrac{1}{6} \\[1em] \Rightarrow x = \dfrac{(-5) \times 3 + 1 \times 7}{42} \\[1em] \Rightarrow x = \dfrac{-15 + 7}{42} \\[1em] \Rightarrow x = \dfrac{-8}{42} = \dfrac{-4}{21}

Hence, Option 1 is the correct option.

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