Question

the denominator of a rational number is greater than its numerator by 6 if the numerator is decreased by 1 and the denominator is increased by 1 and the number obtained is 1/3rd find the rational number answer is 5 /11 ​

Answer 1

Answer :

The required rational number is 5/11

Given :

• The denominator of a rational number is greater than its numerator by 6
• If the numerator is decreased by 1 and the denominator is increased by 1 then the number obtained is 1/3

To Find :

• The rational number

Solution :

Let us consider the numerator be x and the denominator by y

So that the rational number becomes :

$\rm \dashrightarrow \dfrac{x}{y}$

According to question

$\rm \implies y = x + 6 .........(1)$

Again by question

$\rm \implies \dfrac{x-1}{y+ 1}=\dfrac{1}{3} \\\\ \rm\implies 3(x -1) = y + 1 \\\\ \rm \implies 3x - 3 = y + 1 \\\\ \rm \dagger \: \: Putting \: the \: value \: of \: y \: from \: (1) \\\\ \rm \implies 3x - 3 = x + 6 + 1 \\\\ \rm \implies 3x - x = 7 + 3 \\\\ \rm \implies 2x = 10 \\\\ \rm \implies x = 5$

Using the value of x in (1) :

$\rm \implies y = 5 + 6 \\\\ \rm \implies y = 11$

Thus the rational number is

$\rm \dashrightarrow \dfrac{5}{11}$

Answer 2

✰$\huge{Answer}$✰

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Let the numerator be $\red{x}$

So, the denominator of rational number= $\red{x+6}$

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Rational number= $\red{\frac{x}{x+6}}$

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When numerator is decreased by 1, its value = $\purple{x-1}$

When denominator is increased by 1, it's value= $x+6+1$ =$\purple{x+7}$

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It's given that:-

${\boxed{\green{\frac{x - 1}{x + 7} = \frac{1}{3}}}}$

So by method of cross multiplication we get:-

$⇝3(x - 1) = 1(x + 7) \\⇝ 3x - 3 = x + 7 \\ ⇝3x - x = 7 + 3 \\ ⇝2x = 10 \\ ⇝x = \frac{10}{2} \\⇝ x = 5$

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∴ $\red{x = 5}$

∴ $\red{ x + 6 =5 + 6 = 11}$

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∴ The rational number = ${\boxed{\green{\frac{x}{x+6}= \frac{5}{11}}}}$