Question

in a rational number twice the numerator is 2 more than the denominator . if 3 is added to each the numerator and denominator the new fraction is 2/3 . find the original number.​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

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

$\large\underline{\bold{Given- }}$

• A rational number such that twice the numerator is 2 more than the denominator.

• When 3 is added to each the numerator and denominator the new fraction is 2/3 .

$\large\underline{\bold{To\:Find - }}$

• The original number.

$\large\underline{\bold{Solution-}}$

• Let numerator be 'x'.

Then

According to statement,

• Twice the numerator is 2 more than the denominator.

$\bf\implies \:2(numerator) = 2 + denominator$

$\rm :\implies\:denominator = 2x - 2$

$\begin{gathered}\begin{gathered}\bf \:Hence - \begin{cases} &\sf{numerator = x} \\ &\sf{denominator = 2x - 2} \end{cases}\end{gathered}\end{gathered}$

So,

$\begin{gathered}\begin{gathered}\bf \:fraction - \begin{cases} &\sf{\dfrac{x}{2x - 2} }  \end{cases}\end{gathered}\end{gathered}$

According to statement again,

• When 3 is added to each the numerator and denominator the new fraction is 2/3 .

$\begin{gathered}\begin{gathered}\bf \:So - \begin{cases} &\sf{numerator \: = x + 3} \\ &\sf{denominator = 2x - 2 + 3 = 2x + 1} \end{cases}\end{gathered}\end{gathered}$

Now,

$\begin{gathered}\begin{gathered}\bf \:fraction - \begin{cases} &\sf{\dfrac{x + 3}{2x + 1} }  \end{cases}\end{gathered}\end{gathered}$

Therefore,

$\rm :\longmapsto\:\dfrac{x + 3}{2x + 1} = \dfrac{2}{3}$

$\rm :\longmapsto\:3x + 9 = 4x + 2$

$\rm :\longmapsto\:4x - 3x = 9 - 2$

$\bf :\implies\:x \: = \: 7$

$\begin{gathered}\begin{gathered}\bf \therefore \: fraction \: is- \begin{cases} &\sf{\dfrac{x}{2x - 2} = \dfrac{7}{7 \times 2 - 2} = \dfrac{7}{12} }  \end{cases}\end{gathered}\end{gathered}$

Basic Concept :-

• Writing System of Linear Equations from Word Problems

Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find.

Translate the problem to an equation.

• Assign a variable (or variables) to represent the unknown.

• Clearly state what the variable represents.

Carry out the plan and solve the problem.

Answer 2

Step-by-step explanation: