Question

The denominator of a rational numbers is greater than its numerator by 3. If 3 is subtracted from the numerator and 2 is added to the denominator, the new number becomes 1 by 3. Find the original number

Answer 1

$\huge{\blue{\bold{\underline{\underline{Answer :}}}}}$

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$\large\underline{\green{\bf Given :-}}$

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• The denominator of a rational numbers is greater than its numerator by 3

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• If 3 is subtracted from the numerator and 2 is added to the denominator, the new number becomes 1/3

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$\large\underline{\red{\bf To \: Find :-}}$

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• The original number

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$\large\underline{\orange{\bf Solution :-}}$

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• Let the numerator be "n"

• Let the denominator be "d"

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$\underline{\bold{\texttt{Original fraction :}}}$

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➠ $\sf \dfrac { n } { d }$ ⚊⚊⚊⚊ (a)

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$\purple{\underline \bold{According \: to \: the \ question :}}$

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The denominator of a rational numbers is greater than its numerator by 3

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➜ d = n + 3 ⚊⚊⚊⚊ ⓵

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$\underline{\bold{\texttt{3 is subtracted from the numerator :}}}$

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➠ n - 3

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$\underline{\bold{\texttt{2 is added to the denominator :}}}$

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➠ d + 2

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Also given that , If 3 is subtracted from the numerator and 2 is added to the denominator, the new number becomes 1/3

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So ,

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➜ $\sf \dfrac { n - 3 } { d + 2 } = \dfrac { 1 } { 3 }$

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➜ 3n - 9 = d + 2 ⚊⚊⚊⚊ ⓶

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⟮ Putting d = n + 3 from ⓵ to ⓶ ⟯

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➜ 3n - 9 = n + 3 + 2

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➜ 3n - n = 3 + 2 + 9

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➜ 2n = 14

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➜ $\sf n = \dfrac { 14 } { 2 }$

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➨ n = 7 ⚊⚊⚊⚊ ⓷

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• Hence the numerator is 7

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⟮ Putting n = 7 from ⓷ to ⓵ ⟯

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➜ d = 7 + 3

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➨ d = 10 ⚊⚊⚊⚊ ⓸

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• Hence the denominator is 10

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⟮ Putting equation ⓷ & ⓸ in equation (a) ⟯

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➨ $\sf \dfrac {7 } { 10}$

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• Hence the original fraction $\rm\dfrac {7 } { 10}$

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