Question

(4) The denominator of a fraction is 4 more than twice its numerator. Denominator
becomes 12 times the numerator, if both the numerator and the denominator are
reduced by 6. Find the fraction.​

Answer 1

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

• The denominator of a fraction is 4 more than twice its numerator.

• When both the numerator and the denominator are reduced by 6, denominator becomes 12 times the numerator.

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

• The fraction.

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

Basic Concept Used :-

Writing System of Linear Equations from Word Problem

1. Understand the problem.

• Understand all the words used in stating the problem.

• Understand what you are asked to find.

2. Translate the problem to an equation.

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

• Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Let's solve the problem now!!

Given that

• The denominator of a fraction is 4 more than twice its numerator.

So,

$\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{numerator = x} \\ &\sf{denominator = 2x + 4} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \:So - \begin{cases} &\sf{fraction = \dfrac{x}{2x + 4}}  \end{cases}\end{gathered}\end{gathered}$

According to statement,

• When both the numerator and the denominator are reduced by 6,

$\begin{gathered}\begin{gathered}\bf \:Now - \begin{cases} &\sf{numerator = x - 6} \\ &\sf{denominator = 2x + 4 - 6 = 2x - 2} \end{cases}\end{gathered}\end{gathered}$

• Denominator becomes 12 times the numerator

$\rm :\implies\:2x - 2 = 12(x - 6)$

$\rm :\longmapsto\:2x - 2 = 12x - 72$

$\rm :\longmapsto\:2x - 12x = - 72 - 2$

$\rm :\longmapsto\: - 10x = - 70$

$\bf\implies \:x = 7$

$\begin{gathered}\begin{gathered}\bf \:Hence - \begin{cases} &\sf{numerator = 7} \\ &\sf{denominator = 2 \times 7 + 4 = 18} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \:So - \begin{cases} &\sf{fraction = \dfrac{x}{2x + 4} = \dfrac{7}{18}}  \end{cases}\end{gathered}\end{gathered}$

Answer 2

Solution

The denominator of a fraction is 4 more than twice its numerator.

So,

$\begin{gathered}\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{numerator = x} \\ &\sf{denominator = 2x + 4} \end{cases}\end{gathered}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \:So - \begin{cases} &\sf{fraction = \dfrac{x}{2x + 4}} \end{cases}\end{gathered}\end{gathered} \\ So−{ fraction= 2x+4x}$

According to statement,

When both the numerator and the denominator are reduced by 6,

$\begin{gathered}\begin{gathered}\begin{gathered}\bf \:Now - \begin{cases} &\sf{numerator = x - 6} \\ &\sf{denominator = 2x + 4 - 6 = 2x - 2} \end{cases}\end{gathered}\end{gathered}\end{gathered}$

$Now−{numerator=x−6 \: denominator=2x+4−6=2x−2}$

Denominator becomes 12 times the numerator

$\rm :\implies\:2x - 2 = 12(x - 6)$

$\rm :\longmapsto\:2x - 2 = 12x - 72$

$\rm :\longmapsto\:2x - 12x = - 72 - 2:$

$\rm :\longmapsto\: - 10x = - 70$

$\bf\implies \:x = 7⟹x=7$

$\begin{gathered}\begin{gathered}\begin{gathered}\bf \:Hence - \begin{cases} &\sf{numerator = 7} \\ &\sf{denominator = 2 \times 7 + 4 = 18} \end{cases}\end{gathered}\end{gathered}\end{gathered} </p><p>$

\begin{gathered}\begin{gathered}\bf \:So - \begin{cases} &\sf{fraction = \dfrac{x}{2x + 4} = \dfrac{7}{18}} \end{cases}\end{gathered}\end{gathered}So−{fraction= 2x+4x = 187}[/tex]