Question

If 2 is added to the numerator of a
fraction it reduces to 1/2 and if 1 is subtracted from the denominater it reduces to 1/3. Find the fraction.​

Answer 1

❍ Let's say, that the numerator of the fraction be x and denominator of the fraction be y.

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$\underline{\sf{Case\;}\it{1}:}$

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• If 2 is added to the numerator of the fraction, it reduces to 1/2.

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Therefore,

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$\begin{gathered}:\implies\sf{\dfrac{x+2}{y}=\dfrac{1}{2}}\\\\\\:\implies\sf{2x+4=y}\\\\\\:\implies\sf{2x=y-4}\qquad\qquad\bigg\lgroup\sf{eq^n\;1}\bigg\rgroup\end{gathered}$

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$\underline{\sf{Case\;}\it{2}:}$

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• If 1 is subtracted from the denominator of the fraction, it reduces to 1/3.

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Therefore,

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$\begin{gathered}:\implies\sf{\dfrac{x}{y-1}=\dfrac{1}{3}}\\\\\\:\implies\sf{3x=y-1}\qquad\qquad\bigg\lgroup\sf{eq^n\;2}\bigg\rgroup\end{gathered}$

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$\qquad\qquad\footnotesize{\underline{\bf{\dag\;}\frak{Subtracting\;eq^n\;2\;from\;eq^n\;1:}}}$

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$\begin{gathered}:\implies\sf{3x-2x=y-1-y-4}\\\\\\:\implies\underline{\boxed{\pink{\frak{\pmb{x=3}}}}}\;\bigstar\end{gathered}$

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$\qquad\qquad\footnotesize{\underline{\bf{\dag\;}\frak{Substituting\;value\;of\;x\;in\;eq^n\;1:}}}$

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$\begin{gathered}:\implies\sf{2x=y-4}\\\\\\:\implies\sf{2(3)=y-4}\\\\\\:\implies\sf{6=y-4}\\\\\\:\implies\sf{6=y-4}\\\\\\:\implies\sf{y=-4-6}\\\\\\:\implies\underline{\boxed{\purple{\frak{\pmb{y=10}}}}}\;\bigstar\end{gathered}$

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Hence,

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• Numerator = x = 3
• Denominator = y = 10

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$\therefore\;{\underline{\sf{Hence,\;the\;fraction\;is\;\frak{\dfrac{3}{10}}}.}}$⠀⠀