Question

The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by
17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{\orange{Question}}}}}}$

The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.

$\Large{\underline{\underline{\mathfrak{\bf{\pink{Solution}}}}}}$

$\Large{\underline{\mathfrak{\bf{\blue{Given}}}}}$

• The denominator of a rational number is greater than its numerator by 8
• If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2

$\Large{\underline{\mathfrak{\bf{\blue{Find}}}}}$

• Rational number

$\Large{\underline{\underline{\mathfrak{\bf{\red{Explanation}}}}}}$

Let,

• Numerator = x
• Denominator = y

So , A/C to question,

( The denominator of a rational number is greater than its numerator by 8 )

$:\mapsto\sf{\:y\:=\:x+8} \\ \\ :\mapsto\sf{\pink{\:x\:-\:y\:=\:-8.......(1)}}$

Again,

( If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2 )

$:\mapsto\sf{\:\dfrac{x+17}{y-1}\:=\:\dfrac{3}{2}} \\ \\ :\mapsto\sf{\:2.(x+17)\:=\:3.(y-1)} \\ \\ :\mapsto\sf{\pink{\:2x-3y\:=\:-37......(2)}}$

multiply by 2 in equ(1),

• 2x - 2y = -16
• 2x - 3y = -37

___________Sub. it's

$:\mapsto\sf{\:(-2y+3y)\:=\:(-16+37)} \\ \\ :\mapsto\sf{\orange{\:y\:=\:21}}$

keep value of y in equ(2),

$:\mapsto\sf{\:2x-3\times21\:=\:-37} \\ \\ :\mapsto\sf{\:2x\:=\:-37+63} \\ \\ :\mapsto\sf{\:x\:=\:\cancel{\dfrac{26}{2}}} \\ \\ :\mapsto\sf{\pink{\:x\:=\:13}}$

$\large{\underline{\mathfrak{\bf{\blue{Thus}}}}}$

$:\mapsto\sf{\green{\:Value\:of\:x\:=\:13}} \\ \\ :\mapsto\sf{\green{\:Value \:of\:y\:=\:21}}$

$\large{\underline{\mathfrak{\bf{\blue{Hence}}}}}$

$:\mapsto\small\sf{\blue{\:Required\:rational\:number\:will\:be\:=\:\dfrac{13}{21}}}$

Answer 2

$\huge{\underline{\purple{\rm{Solution:}}}}$

$\small{\underline{\red{\rm{Let:}}}}$

$\sf{The\:Numerator\:of\:fraction=N}$

$\sf{The\:Denominator\:of\:fraction=D}$

$\orange{To\: Find}$

• $\bold\green{The\: rational\: Number}$

$\sf{The\:denominator\:of\:a\:rational\: Number}$

$\sf{is\: greater\:than\:its\: Numerator\:by 8}$

• $\bold\green{As\:per\:the\: Question}$

$\sf{\dashrightarrow D=N+8}$

• $\bold\red{D=N+8-----(1)}$

$\sf{If\:the\:numerator\:is\:increased\:by\:17\:and\:the\:denominator}$

$\sf{is\:less\:by\:1\:the\:Number\:obtained\:3\:by\:2}$

$\sf{\dashrightarrow \frac{N+17}{D-1}=\frac{3}{2}}$

$\sf{\dashrightarrow 2(N+17)=3(D-1)}$

$\sf{\dashrightarrow 2N+34=3D-3}$

$\sf{\dashrightarrow 2N-3D=-3-34}$

$\sf{\dashrightarrow 2N-3D=-37}$

• $\bold\red{2N-3D=-37----(2)}$

$\sf{putting\: the\: value\: of\:D=N+8\:in\:eq\:2}$

$\sf{\dashrightarrow 2N-3(N+8)=-37}$

$\sf{\dashrightarrow 2N-3N-24=-37}$

$\sf{\dashrightarrow -N=-37+24}$

$\sf{\dashrightarrow -N=-13}$

$\sf{\dashrightarrow N=13}$

$\sf{putting\: the\: value\: of\:N=13\:in\:eq\:1}$

$\sf{\dashrightarrow D=N+8}$

$\sf{\dashrightarrow D=13+8}$

$\sf{\dashrightarrow D=21}$

$\bold\green{Hence,}$

$\small{\boxed{\purple{\rm{The\: rational\: number=\frac{N}{D}=\frac{13}{21}:}}}}$