Question

The numerator of a fraction is 1 less than denominator. If the denominator is added to the
numerator and the mumerater is subtracted from the denominator, we get 11. Find the fraction ?

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Answer 1

Given : The numerator of a fraction is 1 less than denominator. If the denominator is added to the numerator and the numerator is substracted from the denominator, We get 11.

To Find : The Fraction ?

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❍ Let's consider the denominator be x

Given that,

• The numerator of a fraction is 1 less than denominator.

$~~~~\therefore$ The Numerator is x - 1.

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Then,

$~~~~~~~~~~:\implies{\sf{Original~Fraction~:~\dfrac{x~-~1}{x}}}$

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Given that,

• If the denominator is added to the numerator and the numerator is substracted from the denominator, We get 11.

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$\underline{\bf{According~ to ~the ~Given ~Question~:}}$

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$~~~~~~~~~~:\implies{\sf{\Bigg(\dfrac{x~-~1~+~(x)}{x~-~(x~+~1)}~=~11\Bigg)}}$

$~~~~~~~~~~:\implies{\sf{\dfrac{x~-~1~+~(x)}{x~+~(x~-~1)}~=~11}}$

$~~~~~~~~~~:\implies{\sf{\dfrac{x~-~1~+~x}{x~+~x~-~1}~=~11}}$

$~~~~~~~~~~:\implies{\sf{\dfrac{x~+~x~-~1}{1}~=~11}}$

$~~~~~~~~~~:\implies{\sf{\dfrac{2x~-~1}{1}~=~11}}$

By Cross Multiplication :

$~~~~~~~~~~:\implies{\sf{2x~-~1~=~1}}$

$~~~~~~~~~~:\implies{\sf{2x~=~11~+~1}}$

$~~~~~~~~~~:\implies{\sf{2x~=~12}}$

$~~~~~~~~~~:\implies{\sf{x~=~\cancel\dfrac{12}{2}}}$

$~~~~~~~~~~:\implies{\underline{\boxed{\frak{\pink{x~=~6}}}}}$★

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Therefore,

• The Numerator of the Fraction is (x - 1) = 6 - 1 = 5.

• The Denominator of the Fraction is x = 6.

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Hence,

$\therefore\underline{\sf{The~ Fraction~ is ~\dfrac{5}{6}}}$

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V E R I F I C A T I O N :

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$\underline{\bf{Now~By~Substituting~the~Found~Values~in~Eqⁿ.~1~:}}$

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$~~~~~~~~~~:\implies{\rm{\Bigg(\dfrac{x~-~1~+~(x)}{x~+~(x~-~1)}~=~11\Bigg)}}$

$~~~~~~~~~~:\implies{\rm{x~=~6}}$$~~~\dashrightarrow{\sf{Putting ~in ~Eqⁿ.~1}}$

$~~~~~~~~~~:\implies{\rm{\dfrac{6~-~1~+~(6)}{6~-~(6~+~1)}~=~11}}$

$~~~~~~~~~~:\implies{\rm{\dfrac{12~-~1}{6~-~6~+~1}~=~11}}$

$~~~~~~~~~~:\implies{\rm{\dfrac{11}{1}~=~11}}$

$~~~~~~~~~~:\implies{\underline{\boxed{\frak{\pink{11~=~11}}}}}$★

$~~~~$$\qquad\quad\therefore{\underline{\textsf{\textbf{Hence Verified!}}}}$