Question

The numerator of a rational number is two greater than its denominator. If two is added to the denominator and two is subtracted from the numerator, it becomes 7/9. Find the rational number

Answer 1

$\underline{\underline{\huge{\gray{\tt{\textbf Answer :-}}}}}$

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$\sf\large\bold{\orange{\underline{\blue{ Given :-}}}}$

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• The numerator of a rational number is two greater than its denominator

• If two is added to the denominator and two is subtracted from the numerator, it becomes 7/9

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$\sf\large\bold{\orange{\underline{\blue{ To \: Find :-}}}}$

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• The rational number

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$\sf\large\bold{\orange{\underline{\blue{ Solution :-}}}}$

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• Let the numerator be 'n'

• Let the denominator be 'd'

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$\underline{ \underline{\bold{\texttt{Original Rational number :}}}}$

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➠ $\bf \dfrac { n } { d }$ ⚊⚊⚊⚊ ⓵

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Given that , The numerator of a rational number is two greater than its denominator

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So,

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: ➜ n = d + 2 ⚊⚊⚊⚊ ⓶

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$\underline{ \underline{\bold{\texttt{Adding 2 to denominator :}}}}$

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➠ d + 2

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$\underline{ \underline{\bold{\texttt{Subtracting 2 from numerator :}}}}$

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➠ n - 2

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Also given that , If two is added to the denominator and two is subtracted from the numerator, it becomes 7/9

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Thus ,

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: ➜ $\sf \dfrac { n - 2 } { d + 2 } = \dfrac { 7 } { 9 }$

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: ➜ 9(n - 2) = 7(d + 2)

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: ➜ 9n - 18 = 7d + 14

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: ➜ 9n - 7d = 14 + 18

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: ➜ 9n - 7d = 32 ⚊⚊⚊⚊ ⓷

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⟮ Putting n = d + 2 from ⓶ to ⓷ ⟯

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: ➜ 9n - 7d = 32

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: ➜ 9(d + 2) - 7d = 32

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: ➜ 9d + 18 - 7d = 32

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: ➜ 2d = 32 - 18

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: ➜ 2d = 14

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: ➜ $\sf d = \dfrac { 14 } { 2 }$

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: ➜ d = 7 ⚊⚊⚊⚊ ⓸

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• Hence the denominator is 7

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⟮ Putting d = 7 from ⓸ to ⓶ ⟯

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: ➜ n = d + 2

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: ➜ n = 7 + 2

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: ➜ n = 9 ⚊⚊⚊⚊ ⓹

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• Hence the numerator is 9

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⟮ Putting d = 7 from ⓸ & n = 9 from ⓹ to ⓵ ⟯

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: : ➨ $\sf \dfrac { 9} { 7}$

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• $\sf Hence \: the \: rational \: number \: is \: \dfrac { 9 } { 7 }$

Answer 2

$\sf\large\bold{\orange{\underline{\blue{ Solution :-}}}}$

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Let the numerator be 'n'

Let the denominator be 'd'

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$\underline{ \underline{\bold{\texttt{Original Rational number :}}}}$

$\:\:$

➠ $\bf \dfrac { n } { d }$ ⚊⚊⚊⚊ ⓵

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Given that , The numerator of a rational number is two greater than its denominator

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So,

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: ➜ n = d + 2 ⚊⚊⚊⚊ ⓶

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$\underline{ \underline{\bold{\texttt{Adding 2 to denominator :}}}}$

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➠ d + 2

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$\underline{ \underline{\bold{\texttt{Subtracting 2 from numerator :}}}}$

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➠ n - 2

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Also given that , If two is added to the denominator and two is subtracted from the numerator, it becomes 7/9

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Thus ,

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: ➜ $\sf \dfrac { n - 2 } { d + 2 } = \dfrac { 7 } { 9 }$

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: ➜ 9(n - 2) = 7(d + 2)

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: ➜ 9n - 18 = 7d + 14

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: ➜ 9n - 7d = 14 + 18

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: ➜ 9n - 7d = 32 ⚊⚊⚊⚊ ⓷

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⟮ Putting n = d + 2 from ⓶ to ⓷ ⟯

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: ➜ 9n - 7d = 32

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: ➜ 9(d + 2) - 7d = 32

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: ➜ 9d + 18 - 7d = 32

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: ➜ 2d = 32 - 18

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: ➜ 2d = 14

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: ➜ $\sf d = \dfrac { 14 } { 2 }$

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: ➜ d = 7 ⚊⚊⚊⚊ ⓸

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Hence the denominator is 7

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⟮ Putting d = 7 from ⓸ to ⓶ ⟯

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: ➜ n = d + 2

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: ➜ n = 7 + 2

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: ➜ n = 9 ⚊⚊⚊⚊ ⓹

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Hence the numerator is 9

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⟮ Putting d = 7 from ⓸ & n = 9 from ⓹ to ⓵ ⟯

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: : ➨ $\sf \dfrac { 9} { 7}$

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$\sf Hence \: the \: rational \: number \: is \: \dfrac { 9 } { 7 }$