Question

If we add 1 to the numerator and subtract 1 from the denominator , a fraction reduces to 1 . It becomes 1/2 if we add 1 to the denominator . what is the fraction ?​

Answer 1

Given: If we add 1 to the Numerator and Subtract 1 from the Denominator, the fraction reduces to 1. & it becomes ½ if we add 1 to the Denominator.

Need to find: The fraction?

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❍ Let's Consider that the Numerator and Denominator of the fraction be x and y respectively.

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$\underline{\bigstar\:\boldsymbol{According\;to\;the\; Question\; :}}$⠀⠀

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• If we add 1 to the Numerator and Subtract 1 from the Denominator, the fraction reduces to 1.

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$:\implies\sf \Bigg(\dfrac{Numerator + 1}{Denominator - 1}\Bigg) = 1$

$:\implies\sf \Bigg(\dfrac{x + 1}{y - 1}\Bigg)= 1$

$:\implies\sf x + 1 = y -1$

$:\implies\sf x - y = - 1 - 1$

$:\implies\sf x - y = -2\qquad\qquad\quad\sf\Bigg\lgroup eq^{n}\;(1)\Bigg\rgroup$

Also,

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• The fraction becomes ½ if we add 1 to the Denominator.

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$:\implies\sf \Bigg(\dfrac{Numerator}{Denominator + 1}\Bigg) = \Bigg(\dfrac{1}{2}\Bigg)$

$:\implies\sf \Bigg(\dfrac{x}{y + 1}\Bigg) = \Bigg(\dfrac{1}{2}\Bigg)$

$:\implies\sf 2x = y + 1$

$:\implies\sf 2x - y = 1\qquad\qquad\quad\sf\Bigg\lgroup eq^{n}\;(2)\Bigg\rgroup$

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$\underline{\bf{\dag} \:\mathfrak{So, Using\;eq^n\;(1)\;\&\;eq^{n}\:(2)\: :}}$

$:\implies\sf x - y = -2$

$:\implies\sf 2x - y = 1$

$:\implies\sf \cancel{-}\;x = \cancel{-}\;3$

$:\implies\underline{\boxed{\pmb{\frak{\pink{x = 3}}}}}\;\bigstar$

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$\underline{\bf{\dag} \:\mathfrak{Using\;eq^n\;(1)\: :}}$⠀⠀⠀⠀

$:\implies\sf x - y = - 2$

$:\implies\sf3 - y = -2$

$:\implies\sf 3 - y = -2$

$:\implies\sf -y = -2 - 3$

$:\implies\sf \cancel{-}\;y = \cancel{-}\;5$

$:\implies\underline{\boxed{\pmb{\frak{\pink{y = 5}}}}}\;\bigstar$

Therefore,

• Numerator of the fraction, x = 3
• Denominator of the fraction, y = 5

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$\therefore{\underline{\sf{Hence,\;the\; required\; fraction\;is\;{\purple{\pmb{\sf{\dfrac{3}{5}}}}.}}}}$

Answer 2

Answer:

$\sf \: \purple{ \clubs \:Given : }$

• If we add 1 to the numerator and subtract 1 from the denominator , a fraction reduces to 1 .
• It becomes 1/2 if we add 1 to the denominator .

$\sf \purple{ \clubs \:To \: find : }$

• The Fraction

$\sf \purple{ \clubs \:Solution : }$

Let numerator be x and denominator be y.

Fraction =x/y

According to question

$\underline{Case \: 1 }\\ \boxed{\frac{x + 1}{y - 1} = 1 }\\ x + 1 = y - 1 \\ x - y = - 1 - 1 \\ x - y = - 2 \\ x = - 2 + y \longrightarrow(1) \\ \underline{Case \: 2 } \\ \boxed{\frac{x}{y + 1} = \frac{1}{2} }\\ 2x = y + 1 \\ \sf \: Putting \: value \: of \: x \: from \: equation \: (1) \\ 2( - 2 + y) = y + 1 \\ - 4 + 2y - y = 1 \\ y = 1 + 4 \\ \purple{\boxed{y = 5}} \\ \sf \: Putting \: value \: of \: y \: in \: equation \: (1) \\ x = - 2 + y \\ x = - 2 + 5 \\ \purple{\boxed{x = 3}} \\ \red{\sf \therefore \: Fraction = \frac{x}{y} = \frac{3}{5} }$