Question

The denominator of a rational number is greater than its numerator by 10. If the numerator is increased
6.
by 19 and the denominator is decreased by 1, the number obtained is 3/2 find rational no​

Answer 1

Given that , The denominator of a rational number is greater than its numerator by 10 & the numerator is increased by 19 and the denominator is decreased by 1 , the number obtained is 3/2 .

Need To Find : The Rational number ?

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❍ Let's say that , Numerator of Rational number be x & Denominator will be ( x + 10 ) i.e, denominator of a rational number is greater than its numerator by 10 .

$\qquad \:\:\bigstar \:\:\pmb {\underline {\boxed {\sf \:Rational \:number \:=\: \dfrac{x}{x + 10 }\:}}}\:$

$\qquad \:\:\bigstar \:\:\underline {\pmb{\purple {\sf \: According \:To\:The \:Question \:\::\:}}}\:$

⠀⠀⠀▪︎⠀The numerator is increased by 19 and the denominator is decreased by 1 , the number obtained is 3/2 .

$\qquad :\implies \:\sf \bigg\{ \dfrac{\: Numerator \:+\:19\:}{\:Denominator \:-\:1\:}\:\bigg\}\:=\:\dfrac{3}{2}\: \\\\\\ \qquad :\implies \:\sf \bigg\{ \dfrac{\: x \:+\:19\:}{\:\big( x + 10 \big) \:-\:1\:}\:\bigg\}\:=\:\dfrac{3}{2}\: \\\\\\ \qquad :\implies \:\sf \bigg\{ \dfrac{\: x \:+\:19\:}{\: x + 10 \:-\:1\:}\:\bigg\}\:=\:\dfrac{3}{2}\: \\\\\\ \qquad :\implies \:\sf \bigg\{ \dfrac{\: x \:+\:19\:}{\: x + 9\:}\:\bigg\}\:=\:\dfrac{3}{2}\: \\\\\\ \qquad :\implies \:\sf 2\:\big\{\: x \:+\:19\:\big\}\:=\:3\:\big\{\: x + 9\:\big\}\: \\\\\\\qquad :\implies \:\sf \:\big\{\: 2x \:+\:38\:\big\}\:=\:\:\big\{\: 3x + 27\:\big\}\: \\\\\\ \qquad :\implies \:\sf \:\: 2x \:+\:38\:\:=\:\:\: 3x + 27\:\: \\\\\\ \qquad :\implies \:\sf \:\: 3x \:-\:2x\:\:=\:\:38\: -\: 27\:\: \\\\\\ \qquad :\implies \pmb {\underline {\boxed {\purple {\:\frak{ \:x\:\:=\:11\:}}}}}\:\bigstar \:$

Therefore,

• Numerator of Rational number , x = 11 &
• Denominator of Rational number , ( x + 10 ) = ( 11 + 10 ) = 21 .

Now ,

$\qquad \dashrightarrow \sf \: Rational \:number \:=\: \dfrac{x}{x + 10 }\:\:\\\\\\\qquad \dashrightarrow \sf \: Rational \:number \:=\: \dfrac{11}{11 + 10 }\:\:\\\\\\ \qquad \dashrightarrow \sf \: Rational \:number \:=\: \dfrac{11}{21 }\:\:\\\\\\ \qquad \dashrightarrow \pmb {\underline {\boxed {\purple {\:\frak{ \:Rational \:number \:\:=\:\dfrac{11}{21}\:}}}}}\:\bigstar \:$

∴ Hence, Rational number is 11/21 .