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 = . Find the original fraction??​

Answer 1

Appropriate 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.​

$\rule{200}1$

Given,

• In a Fraction , twice the numerator is 2 more than the denominator .
• If 3 is added to the numerator and to the denominator then the new fraction became 2/3 .

To Find,

• The Original Fraction

Solution :

$\implies$ Suppose the Numerator be a

And, Suppose the Denominator be b

$\mapsto \underline{\sf{\pink{According \ to \ the \ First \ Condition :}}}$

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

$\implies \sf{2a = b + 2}$

$\implies \boxed{\sf{b = 2a - 2}}$   1)Equation

$\mapsto \underline{\sf{\pink{According \ to \ the \ Second \ Condition :}}}$

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

$\implies \sf{\dfrac{a + 3}{b + 3} = \dfrac{2}{3}}$

$\implies \sf{3(a + 3) = 2(b + 3)}$

$\implies \sf{3a + 9 = 2b + 6}$

$\implies \sf{3a + 9 = 2( 2a - 2) + 6}$

$\implies \sf{3a + 9 = 4a - 4 + 6}$

$\implies \sf{3a - 4a = 2 - 9}$

$\implies \sf{-a = -7}$

$\implies \boxed{\sf{a = 7}}$

Now Put the Value of a in First Equation :-

$\implies \sf{b = 2a - 2}$

$\implies \sf{b = 2(7)- 2}$

$\implies \sf{b = 14 - 2}$

$\implies \boxed{\sf{b = 12}}$

Therefore,

$\boxed{\sf{\red{The \ Fraction = \dfrac{a}{b} = \dfrac{7}{12}}}}$

$\rule{200}2$

Answer 2

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.

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 2/3.

To find :

• The original fraction.

Solution :

Let the numerator be x and the denominator be y .

According to the 1st condition :-

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

$\implies\sf{2x=y+2}$

$\implies\sf{y=2x-2........eq(1)}$

According to 2nd condition :-

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

$\implies\sf{\frac{x+3}{y+3}=\frac{2}{3}}$

$\implies\sf{3x+9=2y+6}$

✪ Now put the value of y=2x-2 from eq (1)✪

$\implies\sf{3x+9=2(2x-2)+6}$

$\implies\sf{3x+9=4x-4+6}$

$\implies\sf{3x-4x=-9+2}$

$\implies\sf{-x=-7}$

$\implies\sf{x=7}$

✪ Now put x = 7 in eq(1) ✪

$\implies\sf{y=2x-2}$

$\implies\sf{y=2\times\:7-2}$

$\implies\sf{y=14-2}$

$\implies\sf{y=12}$

★ Numerator = 7

★ Denominator = 12

${\boxed{\bold{Fraction=\dfrac{Numerator}{Denominator}}}}$

Therefore,

${\boxed{\purple{\bold{Original\: fraction=\dfrac{x}{y}=\dfrac{7}{12}}}}}$