Question

The denominator of a rational number is greater than its numerator by 3. If
numerator and denominator are increased by 1 and 4, respectively, the number
obtained is Find the rational number.
(4
1
2
1​

Answer 1

Corrected Question:

• The denominator of a rational number is greater than its numerator by 3 If numerator and denominator are increased by 1 and 4 respectively the number obtained is ½. Find the rational number.

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❍ Let the numerator of the fraction be x respectively. Then, it's denominator becomes (x + 3).

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$\underline{\bigstar\:\boldsymbol{According \;to\; the \;given\; Question :}}$

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• If the numerator and denominator of the fraction are increased by 1 and 4 respectively. The number obtained is ½.

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Therefore,

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$\dashrightarrow\sf \dfrac{Numerator + 1}{Denominator + 4} = \dfrac{1}{2} \\\\\\\dashrightarrow\sf \dfrac{(x + 1)}{(x + 3) + 4} = \dfrac{1}{2} \\\\\\\dashrightarrow\sf \dfrac{(x + 1)}{x + 7} = \dfrac{1}{2}\\\\\\\dashrightarrow\sf 2(x + 1) = 1(x + 7) \\\\\\\dashrightarrow\sf 2x + 2 = x + 7\\\\\\\dashrightarrow\sf 2x - x = 7-2\\\\\\\dashrightarrow\underline{\boxed{\frak{\pink{x = 5}}}}\;\bigstar$

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Hence,

• Numerator of the fraction is x = 5.

• Denominator of the fraction is (x + 3) = (5 + 3) = 8.

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$\therefore{\underline{\sf{Hence,\;the\; required\; rational\; number\;is\; \textbf{ {}^{\text5}\!/{}_{\text{8}} }.}}}$

$\rule{250px}{.3ex}$

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

• It is given that, when numerator and denominator of the fraction are increased by 1 and 4 respectively. The number obtained is ½. So, let's verify :

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$\twoheadrightarrow\sf\dfrac{Numerator + 1}{Denominator + 4} = \dfrac{1}{2} \\\\\\\twoheadrightarrow\sf \dfrac{5 + 1}{8 + 4} = \dfrac{1}{2} \\\\\\\twoheadrightarrow\sf \cancel\dfrac{6}{12} = \dfrac{1}{2} \\\\\\\twoheadrightarrow\underline{\boxed{\frak{\dfrac{1}{2} = \dfrac{1}{2}}}}$

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

Answer 2

Answer:

Correct Question :-

• The denominator of a rational number is greater than its numerator by 3. If numerator and denominator are increased by 1 and 4 respectively. The number obtained is ½. Find the rational number.

Given :-

• The denominator of a rational number is greater than its numerator by 3. If numerator and denominator are increased by 1 and 4 respectively. The number obtained is ½.

To Find :-

• What is the rational number.

Solution :-

Let, the numerator be x

And, the denominator will be x + 3

Then, the rational number is $\sf \dfrac{x}{x + 3}$

According to the question,

↦ $\sf \dfrac{x + 1}{x + 3 + 4} =\: \dfrac{1}{2}$

↦ $\sf \dfrac{x + 1}{x + 7} =\: \dfrac{1}{2}$

By doing cross multiplication we get,

↦ $\sf 2(x + 1) =\: 1(x + 7)$

↦ $\sf 2x + 2 =\: x + 7$

↦ $\sf 2x - x =\: 7 - 2$

➠ $\sf\bold{\green{x =\: 5}}$

Hence, the required rational number is :-

➲ $\sf \dfrac{x}{x + 3}$

⇒ $\sf \dfrac{5}{5 + 3}$

➦ $\sf\bold{\purple{\dfrac{5}{8}}}$

$\therefore$ The rational number is $\sf\boxed{\bold{\red{\dfrac{5}{8}}}}$.