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Mathematics

If we multiply a number by 52\dfrac{5}{2} and add 23\dfrac{2}{3} to the product, we get −712\dfrac{-7}{12}. Find the number.

Polynomials

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Answer

Let the number be x.

According to the question:

52x+23=−712\dfrac{5}{2}x + \dfrac{2}{3} = \dfrac{-7}{12}

⇒52x=−712−23⇒52x=−712−2×43×4[Taking LCM of 12 and 3 as 12]⇒52x=−712−812⇒52x=−7−812⇒52x=−1512⇒x=−1512×25⇒x=−15×212×5⇒x=−3060⇒x=−12\Rightarrow \dfrac{5}{2}x = \dfrac{-7}{12} - \dfrac{2}{3} \\[1em] \Rightarrow \dfrac{5}{2}x = \dfrac{-7}{12} - \dfrac{2 \times 4}{3 \times 4} \quad\text{[Taking LCM of 12 and 3 as 12]}\\[1em] \Rightarrow \dfrac{5}{2}x = \dfrac{-7}{12} - \dfrac{8}{12} \\[1em] \Rightarrow \dfrac{5}{2}x = \dfrac{-7 - 8}{12} \\[1em] \Rightarrow \dfrac{5}{2}x = \dfrac{-15}{12} \\[1em] \Rightarrow x = \dfrac{-15}{12} \times \dfrac{2}{5} \\[1em] \Rightarrow x = \dfrac{-15 \times 2}{12 \times 5} \\[1em] \Rightarrow x = \dfrac{-30}{60} \\[1em] \Rightarrow x = -\dfrac{1}{2}

Hence, the number is −12-\dfrac{1}{2}.

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