Question

The numerator of a fraction is 6 less than the denominator if 3 is added to the numerator the fraction is equal to 2/3, find the original fraction.​

Answer 1

Given that , The numerator of a fraction is 6 less than the denominator & 3 is added to the numerator the fraction is equal to ⅔ .

Exigency To Find : The Original Fraction ?

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❍ Let's say that , the Denominator be x , and Numerator be ( x - 6 ) i.e., the numerator of a fraction is 6 less than the denominator .

$\qquad \star\:\:\underline {\boxed {\pmb{\sf \:\:\: Original \:Fraction \:=\: \dfrac{\:(\:x \:-\:6\:\:)}{\:\:x\:\:}\:\:}}}$

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

⠀⠀⠀━⠀If 3 is added ⠀to the numerator the fraction is equal to 2/3 .

$\\\\ \qquad\dashrightarrow \sf \:\bigg\{ \dfrac{\:Numerator \:+\:3\:}{\:Denominator \:}\:\bigg\}\:=\:\dfrac{2}{3}\:\\\\\\\qquad\dashrightarrow \sf \:\bigg\{ \dfrac{\:(\:x\:-\:6\:) \:+\:3\:}{\:x \:}\:\bigg\}\:=\:\dfrac{2}{3}\:\\\\\\ \qquad\dashrightarrow \:\sf \dfrac{\:\:x\:-\:3\:}{\:x \:}\:\:=\:\dfrac{2}{3}\:\\\\\\ \qquad\dashrightarrow \sf \: 3\:(\:x\:-\:3\:)\:=\:2\:(\:x\:)\:\\\\\\ \qquad\dashrightarrow \:\sf \:\:3x\:-\:9\:\:=\:2x\:\\\\\\ \qquad\dashrightarrow \: \sf\:\:3x\:-\:2x\:\:=\:9\:\\\\\\ \qquad\dashrightarrow \underline {\boxed {\pmb{\frak{ \: \:\:x\:\:=\:9\:}}}}\:\:\bigstar \:$

Therefore,

• Numerator of Original Fraction : ( x - 6 ) = ( 9 - 6) = 3
• Denominator of Original Fraction : x = 9

$\\\\ \qquad:\implies \sf \:\:\: Original \:Fraction \:=\: \dfrac{\:\:Numerator \:\:}{\:\:Denominator \:\:}\:\:\\\\ \qquad:\implies \sf \:\:\: Original \:Fraction \:=\: \dfrac{\:\:3 \:\:}{\:\:9 \:\:}\:\:\\\\\::\implies \:\underline {\boxed {\pmb{\sf \:\:\: Original \:Fraction \:=\: \dfrac{\:3\:\:}{\:\:9\:\:}\:\:}}}\:\:\bigstar$

⠀⠀⠀⠀∴ Hence, Original Fraction is 3/9 or ⅓ .

Answer 2

Let us consider the denominator as x and numerator as (x-6)

By using the formula,

Fraction = numerator/denominator = (x-6)/x

(x – 6 + 3)/x = 2/3

(x – 3)/x = 2/3

By cross-multiplying

3(x-3) = 2x 3x – 9 = 2x 3x – 2x = 9 x = 9

∴ The denominator is x = 9, numerator is (x-6)

(9-6)  = 3

And the fraction = numerator/denominator = (x-6)/x 

3/9

THE ORIGINAL FACTOR IS 1/3