Question

In a fraction twice the numerator is 2 more the denominator, if three is added to a numerator and to the denominator the new fraction is $$\dfrac{2}{3}$$, find the original fraction.

Answer 1

Hint: So to find the solution we first need to assume the numerator and after that by the given condition we will get the value of the denominator and since the new fraction is $$\dfrac{2}{3}$$, then by equating the ratio of the numerator(assumed) and denominator with $$\dfrac{2}{3}$$ we will get the solution.

Complete step by step answer:
Let the numerator of the fraction be x.
So here it is given that twice the numerator is 2 more the denominator, i,e we can say that the denominator is 2 less than that of twice the numerator.
Therefore, denominator = 2x - 2.

Now if we add 3 to the numerator and to the denominator the new fraction will be $$\dfrac{2}{3}$$.
Therefore we can write,
$$\dfrac{x+3}{2x-2+3} =\dfrac{2}{3}$$
$$\Rightarrow \dfrac{x+3}{2x+1} =\dfrac{2}{3}$$
$$\Rightarrow 3\left( x+3\right) =2\left( 2x+1\right) $$ [by cross multiplication]
$$\Rightarrow 3x+3\times 3=2\times 2x+2\times 1$$
$$\Rightarrow 3x+9=4x+2$$
$$\Rightarrow 4x+2=3x+9$$
$$\Rightarrow 4x+2-3x=9$$
$$\Rightarrow 4x-3x+2=9$$
$$\Rightarrow x+2=9$$
$$\Rightarrow x=9-2$$
$$\Rightarrow x=7$$
Therefore the numerator is 7 and denominator (2x - 2) = $$(2\times 7-2)$$ = 14-2 = 12
Therefore the number $$\dfrac{7}{12}$$.

Note:
While solving any equation you need to follow some rules:
Simplify each side of the equation by removing parentheses and combining like terms.
Use addition or subtraction to isolate the variable term on one side of the equation.
Use multiplication or division to solve for the variable.