Question

In a rational number,twice the numerator is 2 more than the denominator. If 3 is added to each , the numerator and the denominator,the new fraction is 2by 3, Find the original number.​

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Answer 1

Step-by-step explanation:

Let a number of notes are 2x, 3x and 5x.

According to question, 100 x 2x + 50 x 3x + 10 x 5x =4,00,000

\Rightarrow\ 200x+150x+50x=4,00,000⇒ 200x+150x+50x=4,00,000

\Rightarrow\ 400x=4,00,000⇒ 400x=4,00,000

\Rightarrow\ \frac{400x}{400}=\frac{400000}{400}⇒

400

400x

=

400

400000

[Dividing both sides by 400]

x = 1000

Hence, number of denominations of ` 100 notes = 2 x 1000 = 2000

Number of denominations of ` 50 notes = 3 x 1000 = 3000

Number of denominations of ` 10 notes = 5 x 1000 = 5000

Therefore, required denominations of notes of 100, 50 and ` 10 are 2000, 3000 and 5000 respectively.

Answer 2

Given that , In a rational number , twice the numerator is 2 more than the denominator & If 3 is added to each , both the numerator and the denominator , the new fraction is ⅔ .

Need To Find : The Rational number ?

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━⠀

❍ Let's say that , the Numerator of Rational number be x and Denominator be ( 2x - 2 ) i.e. In a rational number , twice the numerator is 2 more than the denominator .

$\qquad \bigstar \:\pmb {\underline {\boxed {\sf{ \:Rational \:Number \:=\: \dfrac{x}{2x - 2}\:\:}}}}\:$

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

⠀⠀⠀▪︎⠀If 3 is added to each , both the numerator and the denominator , the new fraction is ⅔ .

$\\:\implies \sf \bigg\{ \dfrac{ \:Numerator \:+\:3\:}{\:Denominator \:+\:3\:}\:\bigg\}\:=\:\dfrac{2}{3}\:$

$:\implies \sf \bigg\{ \dfrac{ \:x \:+\:3\:}{\:(\:2x\:-\:2\:) \:+\:3\:}\:\bigg\}\:=\:\dfrac{2}{3}\:$

$:\implies \sf \bigg\{ \dfrac{ \:x \:+\:2\:}{\:2x\:-\:2 \:+\:3\:}\:\bigg\}\:=\:\dfrac{2}{3} \:$

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

$:\implies \sf \:3\:\big\{ x \:+\:2\:\big\}\:=\:2\:\big\{ 2x + 1 \:\big\}$

$:\implies \sf \:\: 3x \:+\:9\:\:=\:\: 4x + 2 \:$

$:\implies \sf \:\: 4x \:-\:3x\:\:=\:\: 9\:-\: 2 \:$

$:\implies \pmb {\underline {\boxed {\purple {\:\frak{ \:x\:\:=\:7\:}}}}}\:\bigstar \:$

Therefore,

• Numerator of Rational number , x = 7 &
• Denominator of Rational number , ( 2x - 2 ) = [ 2(7) - 2 ] = 12 .

Now ,

$\qquad \twoheadrightarrow \:\sf \:Rational \:Number \:=\: \dfrac{x}{2x - 2}\:\: \\\\\\\qquad \twoheadrightarrow \:\sf \:Rational \:Number \:=\: \dfrac{7}{2(7) - 2}\:\: \\\\\\ \qquad \twoheadrightarrow \:\sf \:Rational \:Number \:=\: \dfrac{7}{14 - 2}\:\: \\\\\\\qquad \twoheadrightarrow \:\sf \:Rational \:Number \:=\: \dfrac{7}{12}\:\: \\\\\\ \qquad \twoheadrightarrow \pmb {\underline {\boxed {\purple {\:\frak{ \:\:Rational \:Number \:=\: \dfrac{7}{12}\:\:\:}}}}}\:\bigstar \:$

∴ Hence , Rational Number is 7/12 .