Question

14. 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 = . Find the original fraction.​

Answer 1

Correct 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.

Solution :-

Let the numerator be N and denominator

be D.

In a fraction twice the numerator is 2 more than the denominator.

According to question,

=> $\sf{2N \:=\: D\: +\: 2}$

=> $\sf{2N\: - \:2 \:=\: D}$ ___ (eq 1)

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

Now,

• New numerator = N + 3
• New denominator = D + 3

According to question,

=> $\sf{\dfrac{N\:+\:3}{D\:+\:3}\:=\:\dfrac{2}{3}}$

=> $\sf{3(N\:+\:3)\:=\:2(D\:+\:3)}$

=> $\sf{3N\:+\:9\:=\:2D\:+\:6}$

=> $\sf{3N\:+\:9\:-\:6\:=\:2(2N\:-\:2)}$

[From (eq 1)]

=> $\sf{3N\:+\:3\:=\:4N\:-\:4}$

=> $\sf{3N\:-\:4N\:=\:-\:4\:-\:3}$

=> $\sf{-N\:=\:-7}$

=> $\sf{N\:=\:7}$

Substitute value of N in (eq 1)

=> $\sf{2(7)\:-\:2\:=\:D}$

=> $\sf{14\:-\:2\:=\:D}$

=> $\sf{D\:=\:12}$

•°• Fraction = Numerator/Denominator

= 7/12.

Answer 2

$\bold{\underline{\underline{Answer:}}}$

Original fraction = $\bold{\dfrac{7}{12}}$

$\bold{\underline{\underline{Step\:-\:by\:-\:step\:explanation:}}}$

Given :

• 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 $\bold{\dfrac{2}{3}}$

To find :

• The original fraction.

Solution :

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Fraction = $\bold{\dfrac{x}{y}}$

$\bold{\underline{\underline{As\:per\:the\:first\:condition:}}}$

• Twice the numerator is 2 more than the denominator.

Constituting it mathematically,

$\rightarrow$ $\bold{2x=y+2}$

$\rightarrow$ $\bold{2x-y =2}$ ----> (1)

$\bold{\underline{\underline{As\:per\:the\:second\:condition:}}}$

• If 3 is added to the
• numerator and to the denominator, the new fraction is $\bold{\dfrac{2}{3}}$

•°• Numerator = x + 3

Denominator = y + 3

Fraction = $\bold{\dfrac{2}{3}}$

Constituting it mathematically,

$\rightarrow$ $\bold{\dfrac{x+3}{y+3}}$ = $\bold{\dfrac{2}{3}}$

Cross multiplying,

$\rightarrow$ $\bold{3(x+3)= 2(y+3)}$

$\rightarrow$ $\bold{3x+9= 2y+6}$

$\rightarrow$ $\bold{3x-2y= 6-9}$

$\rightarrow$ $\bold{3x-2y= -3}$ ----> (2)

Multiplying equation 1 by 2,

$\bold{4x-2y=4}$ ---> (3)

Solve equation 2 and equation 3 simultaneously by elimination method.

Subtract equation 2 from equation 3,

4x - 2y = 4 ----> (3)

3x - 2y = - 3 ----> (2)

-----------------

x = 7

Substitute x = 7 in equation 3,

4x - 2y = 4

$\rightarrow$ $\bold{4(7)-2y=4}$

$\rightarrow$ $\bold{28-2y=4}$

$\rightarrow$ $\bold{-2y=4-28}$

$\rightarrow$ $\bold{-2y=-24}$

$\rightarrow$ $\bold{y=\:{\dfrac{-24}{-2}}}$

$\rightarrow$ $\bold{y=12}$

$\bold{\boxed{\blue{\rm{Numerator=x=7}}}}$

$\bold{\boxed{\blue{\rm{Denominator=y=12}}}}$

$\bold{\sf{\blue{\boxed{Fraction\:=\:{\sf{\frac{x}{y}}\:=\:{\sf{\dfrac{7}{12}}}}}}}}$