Question

The denominator of a rational number is greater than it's
numerator by 8. If the numerator is increased
by 17 and denominator is decreased by 1.the number obtained is 3/2.Find the number
​

Answer 1

Given :-

• The denominator of a rational number is greater than it's numerator by 8.

• If the numerator is increased by 17 and denominator is decreased by 1, the number obtained is 3/2.

To Find :-

• The number

Concept Used :-

Formulation of 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 mention what the variable represents.

3. Carry out the plan and solve the problem.

Solution :-

$\begin{gathered}\begin{gathered}\bf\: Let-\begin{cases} &\sf{numerator = x} \\ &\sf{denominator = x + 8} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: Hence \: number \: is-\begin{cases} &\sf{\dfrac{x}{x + 8} }\end{cases}\end{gathered}\end{gathered}$

According to statement,

If the numerator is increased by 17 and denominator is decreased by 1.

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{numerator = x + 17} \\ &\sf{denominator = x + 8 - 1 = x + 7} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: Hence \: number \: is-\begin{cases} &\sf{\dfrac{x + 17}{x + 7}}\end{cases}\end{gathered}\end{gathered}$

According to statement, the number become 3/2.

$\bf\implies \:\dfrac{x + 17}{x + 7} = \dfrac{3}{2}$

$\rm :\longmapsto\:3(x + 7) = 2(x + 17)$

$\rm :\longmapsto\:3x + 21 = 2x + 34$

$\rm :\longmapsto\:3x - 2x = 34 - 21$

$\rm :\implies\:x = 13$

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{numerator = 13} \\ &\sf{denominator = 13 + 8 = 21} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: So \: number \: is-\begin{cases} &\sf{\dfrac{13}{21}}\end{cases}\end{gathered}\end{gathered}$