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Mathematics

What should be subtracted from 23\dfrac{-2}{3} to get 1 ?

Rational Numbers

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Answer

Let the required number be x. Then,

23x=123=1+xx=231=23+(additive inverse of 1)=23+11\dfrac{-2}{3} - x = 1 \\[1em] \Rightarrow \dfrac{-2}{3} = 1 + x \\[1em] \Rightarrow x = \dfrac{-2}{3} - 1 \\[1em] = \dfrac{-2}{3} + (\text{additive inverse of } 1) \\[1em] = \dfrac{-2}{3} + \dfrac{-1}{1}

L.C.M. of denominators 3 and 1 is 3.

Now, expressing each fraction with denominator 3:

2×13×1+1×31×3=23+33=2+(3)3=53\dfrac{-2 \times 1}{3 \times 1} + \dfrac{-1 \times 3}{1 \times 3} \\[1em] = \dfrac{-2}{3} + \dfrac{-3}{3} \\[1em] = \dfrac{-2 + (-3)}{3} = \dfrac{-5}{3}

The required number is 53\dfrac{-5}{3}.

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