Question

In a rational number, twice the numerator is 2 more than the denominator. If 3 is added to each, the numerator and the denominator, the new fraction is $\dfrac{2}{3}$. Find the original number.

Answer 1

Hint: Assume that the rational number is of the form $\dfrac{x}{y}$, where ‘x’ is the numerator and ‘y’ is the denominator. Write equations based on data given in the question. Solve the linear equations in two variables by the elimination method to calculate the value of ‘x’ and ‘y’ and thus, the number.

Complete step-by-step answer:
We have data relating the numerator and denominator of a rational number. We have to calculate the exact number.

We will assume that the rational number is of the form $\dfrac{x}{y}$, where ‘x’ represents the numerator of the number and ‘y’ represents the denominator of the number.

We know that twice the numerator is 2 more than the denominator.

Multiplying the numerator by 2, we get $2x$.

Thus, we have $2x=y+2$.

Rearranging the terms of the above equation, we have $y=2x-2.....\left( 1 \right)$.

We know that if 3 is added to both numerator and denominator, the new fraction is

$\dfrac{2}{3}$.

Adding 3 to both numerator and denominator, the new numerator and denominator are

$x+3$ and $y+3$ respectively.

Thus, we have $\dfrac{x+3}{y+3}=\dfrac{2}{3}$.

Multiplying the above equation, we have $3\left( x+3 \right)=2\left( y+3 \right)$.

Simplifying the above equation, we have $3x+9=2y+6$.

Thus, we have $3x+3=2y.....\left( 2 \right)$.

We will now solve equation (1) and (2) by the elimination method.

Substituting equation (1) in equation (2), we have $3x+3=2\left( 2x-2 \right)$.

Simplifying the above equation, we have $3x+3=4x-4$.

Thus, we have $4x-3x=3+4\Rightarrow x=7.....\left( 3 \right)$.

Substituting equation (3) in equation (1), we have $y=2\left( 7 \right)-2=14-2=12$.

Hence, the rational number is $\dfrac{x}{y}=\dfrac{7}{12}$.



Note: We can check if the calculated values are correct or not by substituting the values in the equation and checking if they satisfy the data given in the question or not. We can also solve this question by writing a linear equation in one variable. Assume that the numerator is ‘x’ and write the denominator in terms of ‘x’ based on the data given in the question.