Question

the denominator of a rational number is greater than the numerator by 5. if 1 is subtracted from the numerator and 3 is added ro the denominator,the new number become¼. find the original number.​

Answer 1

Answer:

hi..............................

Answer 2

$\sf Given \begin{cases} & \sf{Denominator = \bf{x + Numerator}} \\ \\ & \sf{\dfrac{Numerator - 1}{Denominator + 3} = \bf{ \dfrac{1}{4}}} \end{cases}$

To find: The original number.

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☯ Let Numerator and Denominator of a rational number be x and y.

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Given that,

The denominator of a rational number is greater than the numerator by 5.

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Therefore,

$:\implies\sf y = x + 5\qquad\qquad\bigg\lgroup\bf eq\: (1)\bigg\rgroup$

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

If 1 is subtracted from the numerator and 3 is added to the denominator, the new number become 1/4.

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$:\implies\sf \dfrac{x - 1}{y + 3} = \dfrac{1}{4}\\ \\ \\ :\implies\sf \dfrac{x - 1}{(x + 5) + 3} = \dfrac{1}{4}\\ \\ \\ :\implies\sf \dfrac{x - 1}{x + 8} = \dfrac{1}{4}\\ \\ \\ :\implies\sf (x + 8) = 4(x - 1)\\ \\ \\ :\implies\sf (x + 8) = 4x - 4\\ \\ \\ :\implies\sf 8 + 4 = 4x - x\\ \\ \\ :\implies\sf 12 = 3x\\ \\ \\ :\implies\sf x = \cancel{\dfrac{12}{3}}\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{x = 4}}}}}\;\bigstar$

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Therefore,

$\sf Numerator\:of\:rational\:number\:,x = \bf{4}$

$\sf Denominator\:of\:rational\:number\:,(x + 5) = 4 + 5 = \bf{9}$

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$\therefore\:{\underline{\sf{The\: original\:number\;is\: \bf{ \dfrac{4}{9}}.}}}$