Question

the numerator of a rational number is 4 less than its denominator if one is added to both it's numerator and denominator it becomes 2/3. find the rational number.​

Answer 1

Given that , The numerator of a rational number is 4 less than it's denominator & if one is added to both it's numerator and denominator it becomes ⅔ .

Exigency To Find : The Rational Number ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

❍ Let's say that , Denominator of rational number be x & Numerator of rational number be ( x - 4 ) , i.e. Numerator of a rational number is 4 less than it's denominator .

$\qquad \star \:\: \underline {\boxed {\pmb{\sf \:\:Rational \:\:Number \:=\: \dfrac{\:(\:x \:-\:4\:)\:}{x}\:\:}}}$

$\qquad \bigstar \:\:\underline {\purple {\pmb{\sf According \:\:to\:\:the\:Given \:\:Question \:\:\::\:}}}$

⠀⠀⠀⠀━━━ If one is added to both it's numerator and denominator it becomes ⅔ .

$\\ \qquad \dashrightarrow \sf \bigg\{ \dfrac{\:Numerator \:+\:1\:}{\:Denominator \:+\:1\:}\:\bigg\}\:=\:\dfrac{2}{3}\:\\\\\\ \qquad \dashrightarrow \sf \bigg\{ \dfrac{\:(\:x\:-\:4\:) \:+\:1\:}{\:x \:+\:1\:}\:\bigg\}\:=\:\dfrac{2}{3}\:\\\\\\ \qquad \dashrightarrow \sf \bigg\{ \dfrac{\:(\:x\:-\:3\:)\:}{\:x \:+\:1\:}\:\bigg\}\:=\:\dfrac{2}{3}\:\\\\\\ \qquad \dashrightarrow \sf \dfrac{\:(\:x\:-\:3\:)\:}{\:x \:+\:1\:}\:=\:\dfrac{2}{3}\:\\\\\\ \qquad \dashrightarrow \sf 3\:(\:x\:-\:3\:)\:=\:2\:(\:x \:+\:1\:)\:\:\\\\\\\qquad \dashrightarrow \sf 3x\:-\:9\:\:=\:2x \:+\:2\:\:\:\\\\\\ \qquad \dashrightarrow \sf 3x\:-\:2x\:=\:\:2\:+\:9\:\\\\\\\qquad \dashrightarrow \underline {\boxed {\pmb{\frak{\purple { \:x\:\:=\:\:11\:}}}}}\:\:\bigstar \:$

Therefore,

• Numerator of Rational number is : ( x - 4 ) = ( 11 - 4 ) = 7 &,
• Denominator of Rational number is : x = 11 .

$\\ \qquad:\implies \sf \:\:Rational \:\:Number \:=\: \dfrac{\:Numerator \:}{\:Denominator\:}\:\:\\\\\\ \qquad:\implies \sf \:\:Rational \:\:Number \:=\: \dfrac{\:7 \:}{\:11\:}\:\:\\\\\\ \qquad:\implies \underline {\boxed {\pmb{\sf \:\:Rational \:\:Number \:=\: \dfrac{\:7\:}{11}\:\:}}}\:\:\bigstar$

⠀⠀⠀⠀∴ Hence , The Rational Number is 7/11 .