Question

the numerator of the fraction is less than the denominator. If both the numerator & denominator are increased by 2. the new fraction becomes 6/7. find the original fraction​

Answer 1

Case (1)

• Numerator is 3 less than the denominator

Case (2)

• If both numerator & denominator are increased by 2 then fraction become 6/7.

Let the numerator & denominator be x & y .

$\sf{ \: \: \: \: \: \implies{x-3 = y ~~~~~~~—————(1)}}$

$\\ \sf{ \: \: \: \: \: \implies{ \dfrac{x + 2}{y + 2} = \dfrac{6}{7} ~~~~~~~ \bf—————(2)}}$

On Solving Equ.(1)

$\\ \sf{ \: \: \: \: \: \implies{x = y + 3 }}$

Substituting the value of x in Equ.2

$\sf{ \: \: \: \: \: \implies{ \dfrac{y + 3 + 2}{y + 2} = \dfrac{6}{7} }}$

$\sf{ \: \: \: \: \: \implies{ \dfrac{y + 5}{y + 2} = \dfrac{6}{7} }}$

$\sf{ \: \: \: \: \: \implies{7( y + 5) = 6(y + 2) }}$

$\sf{ \: \: \: \: \: \implies{7y+ 35 = 6y + 12 }}$

$\sf{ \: \: \: \: \: \implies{7y - 6y = 12 - 35 }}$

$\sf{ \: \: \: \: \: \implies{y = - 23 }}$

Putting y in Equ.(1)

$\sf{ \: \: \: \: \: \implies{x-3 = - 23}}$

$\sf{ \: \: \: \: \: \implies{x = - 23 + 3}}$

$\sf{ \: \: \: \: \: \implies{x = - 20}}$

$\boxed{ \mathbb{The ~~Original ~~fraction~~\bf is =\dfrac{x}{y}={-20}{-3} =\dfrac{20}{{23}} }}$

Answer 2

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Cᴀsᴇ::-1

• Numerator is 3 less than the denominator.

Cᴀsᴇ::-2

• If both numerator & denominator are increased by 2 then fraction become 6/7.

⚘Let the numerator & denominator be x & y .

$\tt{ \: \: \: \: \: \implies{x-3 = y ~~~~~~~—————(1)}}$

$\begin{gathered}\\ \tt{ \: \: \: \: \: \implies{ \dfrac{x + 2}{y + 2} = \dfrac{6}{7} ~~~~~~~ \tt—————(2)}}\\\end{gathered}$

• ⚘On Solving Equ.(1)

$\begin{gathered}\\ \tt{ \: \: \: \: \: \implies{x = y + 3 }}\\\end{gathered}$

• ⚘Substituting the value of x in Equ.2

$\tt{ \: \: \: \: \: \implies{ \dfrac{y + 3 + 2}{y + 2} = \dfrac{6}{7} }}$

$\tt{ \: \: \: \: \: \implies{ \dfrac{y + 5}{y + 2} = \dfrac{6}{7} }}$

$\tt{ \: \: \: \: \: \implies{7( y + 5) = 6(y + 2) }}$

$\tt{ \: \: \: \: \: \implies{7y+ 35 = 6y + 12 }}$

$\tt{ \: \: \: \: \: \implies{7y - 6y = 12 - 35 }}$

$\begin{gathered}\tt{ \: \: \: \: \: \implies{y = - 23 }} \\ \end{gathered}$

• ⚘Putting y in Equ.(1)

$\tt{ \: \: \: \: \: \implies{x-3 = - 23}}$

$\tt{ \: \: \: \: \: \implies{x = - 23 + 3}}$

$\begin{gathered}\tt{ \: \: \: \: \: \implies{x = - 20}} \\ \\ \end{gathered}$

$\boxed{ \mathscr{The ~~Original ~~fraction~~\bf is =\dfrac{x}{y}={-20}{-3} =\dfrac{20}{{23}} }}$