Question

15. The numerator of a rational number is 5 less than its
denominator. If 2 is subtracted from the numerator and 2
is added to the denominator, the number becomes 2/5.
Find the rational number​

Answer 1

☯ Let Numerator and denominator of a rational number be x and y respectively.

⠀

$\underline{\sf{\bigstar\; According\;to\;the\; question\;:}}$

Numerator of the rational number is 5 less than its denominator.

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf \pink{x = y - 5}\qquad\qquad\bigg\lgroup\bf eq\;(1)\bigg\rgroup$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

☯ If 2 is subtracted from the numerator and 2 is added to the denominator, the number becomes 2/5.

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf \dfrac{(x - 2)}{(y + 2)} = \dfrac{2}{5}$

$:\implies\sf 5(x - 2) = 2(y + 2)$

$:\implies\sf 5x - 10 = 2y + 4$

Now, Putting value of x from eq (1),

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf 5(y - 5) - 10 = 2y + 4$

$:\implies\sf 5y - 25 - 10 = 2y + 4$

$:\implies\sf 5y - 35 = 2y + 4$

$:\implies\sf - 35 - 4 = 2y - 5y$

$:\implies\sf - 39 = - 3y$

$:\implies\sf y = \dfrac{39}{3}$

$:\implies\bf \purple{y = 13}$

Now, Putting value of y in eq (1),

⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf x = 13 - 5$

$:\implies\bf \purple{x = 8}$

Therefore,

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Required rational number is,

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$:\implies{\boxed{\sf{\pink{ \dfrac{8}{13}}}}}\;\bigstar$

$\therefore$ Hence, Required rational number is 8/13.

Answer 2

Answer:

Let Numerator and denominator of a rational number be x and y respectively.

⠀

\begin{gathered}\underline{\sf{\bigstar\; According\;to\;the\; question\;:}}\\ \\\end{gathered}

★Accordingtothequestion:

Numerator of the rational number is 5 less than its denominator.

⠀⠀⠀⠀⠀⠀⠀

\begin{gathered}:\implies\sf \pink{x = y - 5}\qquad\qquad\bigg\lgroup\bf eq\;(1)\bigg\rgroup\\ \\\end{gathered}

:⟹x=y−5

⎩

⎪

⎪

⎪

⎧

eq(1)

⎭

⎪

⎪

⎪

⎫

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

☯ If 2 is subtracted from the numerator and 2 is added to the denominator, the number becomes 2/5.

⠀⠀⠀⠀⠀⠀⠀

\begin{gathered}:\implies\sf \dfrac{(x - 2)}{(y + 2)} = \dfrac{2}{5}\\ \\\end{gathered}

:⟹

(y+2)

(x−2)

=

5

2

\begin{gathered}:\implies\sf 5(x - 2) = 2(y + 2)\\ \\\end{gathered}

:⟹5(x−2)=2(y+2)

\begin{gathered}:\implies\sf 5x - 10 = 2y + 4\\ \\\end{gathered}

:⟹5x−10=2y+4

Now, Putting value of x from eq (1),

⠀⠀⠀⠀⠀⠀⠀

\begin{gathered}:\implies\sf 5(y - 5) - 10 = 2y + 4\\ \\\end{gathered}

:⟹5(y−5)−10=2y+4

\begin{gathered}:\implies\sf 5y - 25 - 10 = 2y + 4\\ \\\end{gathered}

:⟹5y−25−10=2y+4

\begin{gathered}:\implies\sf 5y - 35 = 2y + 4\\ \\\end{gathered}

:⟹5y−35=2y+4

\begin{gathered}:\implies\sf - 35 - 4 = 2y - 5y\\ \\\end{gathered}

:⟹−35−4=2y−5y

\begin{gathered}:\implies\sf - 39 = - 3y\\ \\\end{gathered}

:⟹−39=−3y

\begin{gathered}:\implies\sf y = \dfrac{39}{3}\\ \\\end{gathered}

:⟹y=

3

39

\begin{gathered}:\implies\bf \purple{y = 13}\\ \\\end{gathered}

:⟹y=13

Now, Putting value of y in eq (1),

⠀⠀⠀⠀⠀⠀⠀

\begin{gathered}:\implies\sf x = 13 - 5\\ \\\end{gathered}

:⟹x=13−5

\begin{gathered}:\implies\bf \purple{x = 8}\\ \\\end{gathered}

:⟹x=8

Therefore,

⠀⠀⠀⠀⠀⠀⠀

Required rational number is,

⠀⠀⠀⠀⠀⠀⠀

\begin{gathered}:\implies{\boxed{\sf{\pink{ \dfrac{8}{13}}}}}\;\bigstar\\ \\\end{gathered}

:⟹

13

8

★

\therefore∴ Hence, Required rational number is 8/13.