Question

please tell solution .The denominator of a rational number is greater than its numerator by 7. if 3 is subtracted from the numerator and 2 is added to its denominator, the number becomes 1/5. find the rational number​

Answer 1

Given:

• Denominator of rational number is greater than its numerator by 7.
• if 3 is subtracted from the numerator and 2 is added to its denominator, the number becomes $\dfrac{1}{4}$

To find :

• Rational number?

Solution:

Given that denominator is greater then its numerator by 7.

Let suppose that numerator is x.

So, Denominator will be x + 7

Rational number will be as :

• $\dfrac{x}{x + 7}$

Now, given that if 3 is subtracted from the numerator and 2 is added to its denominator, the number becomes $\dfrac{1}{4}$

Let subtract 3 from numerator and add 2 in denominator:

$\dfrac{x}{x + 7}$

$\implies\dfrac{x-3}{x+7 +2}\:=\: \dfrac{1}{5}$

$\implies\dfrac{x-3}{x+9}\:=\:\dfrac{1}{5}$

[On cross multiplication]

$\implies{5(x-3)\:=\: x + 9 }$

$\implies{5x - 15 \:=\: x + 9 }$

[Take like terms one side]

$\implies{5x - x \:=\: 9 + 15}$

$\implies{4x\:=\: 24}$

[Take 4 in RHS]

$\implies{x\:=\:\dfrac{24}{4}}$

$\implies{x\:=\:{\cancel{\dfrac{24}{4}}}}$

$\implies{x\:=\: 6}$

Therefore,

• Numerator of rational number is 6

Now, given that denominator is greater than its numerator by 7.

• So, Denominator of rational number = 6 + 7 = 13

Rational number will be :

• $\large{\boxed{\sf{\pink{Rational\: number\:=\: \dfrac{6}{13}}}}}$

Answer 2

$\sf\small\underline\purple{Let:-}$

$\sf{\implies Numerator\:_{(fraction)}=N}$

$\sf{\implies Denominator\:_{(fraction)}=D}$

$\sf{\implies Rational\: number=\dfrac{N}{D}}$

$\sf\small\underline\purple{Given:-}$

$\sf{\implies Denominator=Numerator+7}$

$\sf\small\underline\purple{To\: Find:-}$

$\sf{\implies The\:ratio\: number=?}$

$\sf\small\underline\purple{Solution:-}$

$\bf\small\underline{Calculation\:for\:1st\: equation:-}$

$\sf{\implies Rational\: number\:_{(denominator)}=Rational\: number\:_{(numerator)}+7}$

$\tt{\implies D=N+7----(i)}$

$\bf\small\underline{Calculation\:for\:2nd\: equation:-}$

$\sf{\implies \dfrac{Numerator-3}{Denominator+2}=Rational\: number\:_{(become\:1/5)}}$

$\tt{\implies \dfrac{N-3}{D+2}=\dfrac{1}{5}}$

$\tt{\implies 5N-15=D+2}$

$\tt{\implies 5N-D=2+15}$

$\tt{\implies 5N-D=17-----(ii)}$

• In eq (ii) putting the value of D=N+7:-]

$\tt{\implies 5N-(N+7)=17}$

$\tt{\implies 5N-N-7=17}$

$\tt{\implies 5N-N=17+7}$

$\tt{\implies 4N=17+7}$

$\tt{\implies 4N=24}$

$\tt{\implies N=\frac{24}{4}=6}$

• Putting the value of N=6 in eq (i):-]

$\tt{\implies D=N+7}$

$\tt{\implies D=6+7}$

$\tt{\implies D=13}$

$\sf\large{Hence,}$

$\sf{\implies Rational\: number=\frac{N}{D}}$

$\bf{\implies Rational\: number=\frac{6}{13}}$