Question

Select the correct alternative
from the given choices. the sum of the numerator and the denominator of a positive proper fraction is 9. if 4 is added to twice
the numrator and 5 is subracted from 4 times
the denominator then the fraction remains the same. the fraction is​

Answer 1

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

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

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• The sum of the numerator and the denominator of a positive proper fraction is 9

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• If 4 is added to twice the numerator and 5 is subtracted from 4 times the denominator then the fraction remains the same

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

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• The original fraction

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

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The sum of the numerator and the denominator of a positive proper fraction is 9

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➜ n + d = 9

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➜ n = 9 - d ⚊⚊⚊⚊ ⓵

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

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➠ $\sf \dfrac { n } { d }$

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$\underline{\bold{\texttt{New numerator :}}}$

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If 4 is added to twice the numerator

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➠ 2n + 4

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$\underline{\bold{\texttt{New denominator :}}}$

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5 is subtracted from 4 times the denominator

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➠ 4d - 5

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Given that if 4 is added to twice the numerator and 5 is subtracted from 4 times the denominator then the fraction remains the same

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➜ $\sf \dfrac {2n + 4 } { 4d - 5 } = \dfrac { n } { d }$ ⚊⚊⚊⚊ ⓶

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⟮ Putting n = 9 - d from ⓵ to ⓶ ⟯

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➜ $\sf \dfrac {2(9 - d) + 4 } { 4d - 5 } = \dfrac { 9 - d} { d }$

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➜ $\sf \dfrac { 18 - 2d + 4 } { 4d - 5 } = \dfrac { 9 - d } { d }$

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➜ $\sf \dfrac { 22 - 2d } { 4d - 5 } = \dfrac { 9 - d } { d }$

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➜ (9 - d)(4d - 5) = (22 - 2d)(d)

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➜ 36d - 45 - 4d² + 5d = 22d - 2d²

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➜ 4d² - 2d² + 22d - 36d - 5d + 45 = 0

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➜ 2d² - 19d + 45 = 0

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Solving quadratic equation

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For a quadratic equation in general form :

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➠ ax² + bx + c = 0

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Roots can be form by :

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➠ $\sf \dfrac { - b \pm \sqrt { b ^2 - 4ac } } { 2a }$ ⚊⚊⚊⚊ ⓷

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For our equation 2d² - 19d + 45 = 0

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We have ,

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• a = 2

• b = - 19

• c = 45

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⟮ Putting these values in ⓷ ⟯

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➜ $\sf \dfrac { - b \pm \sqrt { b ^2 - 4ac } } { 2a }$

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➜ $\sf \dfrac { - (-19) \pm \sqrt { 19^2 - 4(2)(45)} } { 2(2)}$

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➜ $\sf \dfrac { 19\pm \sqrt { 361 - 360} } { 4}$

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➜ $\sf \dfrac { 19\pm \sqrt { 1} } { 4}$

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➜ $\sf \dfrac { 19\pm 1} { 4}$

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Roots are ;

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➜ $\sf \dfrac { 19 + 1} { 4}$

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➜ $\sf \dfrac { 20} { 4}$

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➜ d = 5 ⚊⚊⚊⚊ ⓸

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Or

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➜ $\sf \dfrac { 19 - 1} { 4}$

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➜ $\sf \dfrac { 18} { 4}$

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➜ d = 4.5 ⚊⚊⚊⚊ ⓹

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From ⓸ & ⓹ we get the value of d is either 5 or 4.5

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We will take the value as 5 because given that the fraction is a proper fraction

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• Hence the denominator is 5

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⟮ Putting d = 5 from ⓸ to ⓵ ⟯

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➜ n = 9 - d

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➜ n = 9 - 5

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➨ n = 4

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• Hence the numerator is 4

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

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➨ $\sf \dfrac { 4 } { 5 }$

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