Question

In a fraction, twice the numerator is 2 more than the denominator. If 3 is added to the
numerator and to the denominator, the new fraction is 2/3. Find the original fraction.

Answer 1

Answer:

The original fraction is

$\huge\frac{7}{12}$

Step-by-step explanation:

Given that

• In a fraction, twice the numerator is 2 more than the denominator

let the numerator be x

$\therefore$ the denominator is 2x-2

Also

• If 3 is added to the numerator and to the denominator, the new fraction is 2/3

$\implies$ $\large\frac{x + 3}{(2x - 2) + 3} = \frac{2}{3}$

By cross multiplying we have,

(x+3)3 = 2(2x+1)

3x+9 = 4x+2

4x-3x = 9-2

$\therefore$ x = 7

And

denominator is (2×7)-2 i.e. 12.

Answer 2

Given :

• Twice the numerator is 2 more than the Denominator .
• If 3 is added to both numerator and denominator the fraction becomes 2/3 .

To Find :

• Original Fraction = ?

Solution :

$\bigstar$ According to the Question :

$\longrightarrow$ Twice the numerator is 2 more than the Denominator .So,

$\qquad \; \large{\underline{\overline{\boxed{\pink{\pmb{\frak{ \dfrac{Numerator}{Denominator} = \dfrac{y}{2y - 2} }}}}}}}$

$\longrightarrow$ If 3 is added to both numerator and denominator the fraction becomes 2/3 .So,

$\qquad \; \large{\dag{\underline{\overline{\boxed{\pink{\pmb{\frak{ \dfrac{y + 3}{(2y - 2) + 3} = \dfrac{2}{3} }}}}}}}}{\dag}$

$\\ \qquad{\rule{150pt}{1pt}}$

$\bigstar$ Let's Cross Multiply :

${\dashrightarrow{\qquad{\frak{ 3(y + 3) = 2 \bigg\{ (2y - 2) + 3 \bigg\} }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\frak{ 3y + 9 = 4y - 4 + 6 }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\frak{ 4y - 3y = 9 + 4 - 6 }}}} \\ \\ \\ \ {\dashrightarrow{\qquad{\frak{ 4y - 3y = 13 - 6 }}}} \\ \\ \\ \ {\qquad \; \; {\therefore \; {\underline{\boxed{\purple{\pmb{\frak{ y = 7 }}}}}}}}$

$\\ \qquad{\rule{150pt}{1pt}}$

$\bigstar$ Calculating the Numerator and Denominator :

$\large{\color{cyan}{\dashrightarrow \; {\underline{\underline{\red{\sf{ Numerator = 7 }}}}}}}$

$\large{\color{cyan}{\dashrightarrow \; {\underline{\underline{\red{\sf{ Denominator = 2y - 2 = 2(7) - 2 = 12 }}}}}}}$

$\\ \qquad{\rule{150pt}{1pt}}$

$\bigstar$ Therefore :

❛❛ Original fraction is ${\sf{ \dfrac{7}{12} }}$ . ❜❜

$\\ {\underline{\rule{300pt}{9pt}}}$