Question

The denominator of a rational number is greater than its numerator by 5. If the numerator is
increased by 11 and the denominator is decreased by 14, the new number becomes 5. Find the
original rational number.
​

Answer 1

Let the numerator of the fraction be x respectively.

• The denominator of a rational number is greater than its numerator by 5.

Therefore, Denominator = (x + 5)

$\qquad\quad\rule{150px}{.3ex}$

A/Q,

• According to the given condition, If the Numerator is increased by 11 and the Denominator is decreased by 14, then the number becomes 5.

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$\twoheadrightarrow\:\sf Fraction = \dfrac{Numerator + 11}{Denominator - 14}$

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Therefore,

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$:\implies\sf \dfrac{(x + 11)}{(x + 5 - 14)} = 5 \\\\\\:\implies\sf \dfrac{(x + 11)}{(x - 9)} = 5 \\\\\\:\implies\sf (x + 11 = 5(x - 9)\\\\\\:\implies\sf x + 11 = 5x - 45\\\\\\:\implies\sf x - 5x = - 45 - 11\\\\\\:\implies\sf -4x =-56\\\\\\:\implies\sf x = \cancel\dfrac{-56}{-4}\\\\\\:\implies{\underline{\boxed{\frak{x = 14}}}}\;\bigstar$

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Hence,

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• Numerator of the fraction, x = 14
• Denominator of the fraction, (x + 5) = (14 + 5) = 19

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$\therefore{\underline{\sf{Hence,\;the\; Original\; number\;is\; \bf{\dfrac{14}{19} }.}}}$

Answer 2

${\bf{Given\::}}$

• The denominator of a rational number is greater than its numerator by 5.

• If the numerator is increased by 11 and the denominator is decreased by 14, then the new number becomes 5.

$\\ {\bf{To\: Find\::}}$

• Original rational number (fraction).

$\\ {\bf{Solution\::}}$

Let,

• Numerator of a fraction is x.

• Denominator be y.

Thus,

$:\implies\:\bf{Original\: fraction\:=\:\dfrac{x}{y}\:}$

Given that,

$:\implies\:\bf{y\:=\:x\:+\:5\:}---(1)$

✯ If the numerator is increased by 11 then the new numerator is,

$:\implies\:\bf{(Numerator)'\:=\:x\:+\:11\:}$

✯ If the denominator is decreased by 14 then the new denominator is,

$:\implies\:\bf{(Denominator)'\:=\:y\:-\:14\:}$

Thus,

The new fraction is,

$:\implies\:\bf{New\: fraction\:=\:\dfrac{x\:+\:11}{y\:-\:14}\:}$

It is given that,

✯ The new fraction is equal to 5.

$\dashrightarrow\:\bf{\dfrac{x\:+\:11}{y\:-\:14}\:=\:5}$

$\dashrightarrow\:\tt{\dfrac{x\:+\:11}{y\:-\:14}\:=\:5}$

$\dashrightarrow\:\tt{x\:+\:11\:=\:5y\:-\:70}$

➣ Putting 'y = x + 5' in the above equation, we get

$\dashrightarrow\:\tt{x\:+\:11\:=\:5\times{(x\:+\:5)}\:-\:70}$

$\dashrightarrow\:\tt{x\:+\:11\:=\:5x\:+\:25\:-\:70}$

$\dashrightarrow\:\tt{x\:+\:11\:=\:5x\:-\:45}$

$\dashrightarrow\:\tt{5x\:-\:x\:=\:45\:+\:11}$

$\dashrightarrow\:\tt{4x\:=\:56}$

$\dashrightarrow\:\tt{x\:=\:\dfrac{56}{4}\:}$

$\dashrightarrow\:\bf{x\:=\:14\:}$

Now,

➣ Putting the value of x in equation (1), we get

$:\implies\:\bf{y\:=\:x\:+\:5\:}$

$:\implies\:\tt{y\:=\:14\:+\:5\:}$

$:\implies\:\tt{y\:=\:19\:}$

Hence,

Original fraction is,

➠ $\bf{Original\: fraction\:=\:\dfrac{x}{y}\:}$

➠ $\bf{Original\: fraction\:=\:\dfrac{14}{19}\:}$

$~~~~~~~~~{\boxed{\bf{\therefore\: {\underline{\pink{Original\: fraction\:is\:\dfrac{14}{19}\:}}}}.}}$