Question

The denominator of a rational number is greater than its numerator by 2. if the numerator is decreased by 3 and the denominator is increased by 4, the new rational number obtained is 2/5 . find the rational number .​

Answer 1

Given :-

• The denominator of a rational number is greater than its numerator by 2.

• If the numerator is decreased by 3 and the denominator is increased by 4, then the new rational number obtained is 2/5 .

To find :-

• The rational number .

Solution :-

Given that

Let the numerator of the rational number be X

Then, The denominator of the rational number

= Numerator + 2

= X+2

The rational number = X/(X+2)

If the numerator is decreased by 3 then it will be

(X-3)

If the denominator is increased by 4 then it will be (X+2)+4 = (X+6)

The new rational number = (X-3)/(X+6)

According to the given problem

The new rational number = 2/5

Therefore, (X-3)/(X+6) = 2/5

On applying cross multiplication then

=> 5(X-3) = 2(X+6)

=> 5X-15 = 2X+12

=> 5X-2X = 12+15

=> 3X = 27

=> X = 27/3

=> X = 9

The Numerator = (X) = 9

The denominator = (X+2)= 9+2 = 11

The rational number = X/(X+2) = 9/11

Answer :-

The required rational number = 9/11

Check :-

The rational number = 9/11

Denominator = 11 = 9+2

=> Denominator = Numerator +2

and

If the numerator is decreased by 3 then it will be

= 9-3 = 6

If the denominator is increased by 4 then it will be

= 11+4 = 15

The new rational number = 6/15

= (2×3)/(5×3)

= 2/5

Verified the given relations in the given problem.

Answer 2

Given : The Denominator of a rational number is greater than its numerator by 2 . If the Numerator is decreased by 3 and denominator is increased by 4 ,the new rational number is 2/5 .

To Find : Find the Original Rational Number

$\\ \qquad{\rule{200pt}{2pt}}$

SolutioN : Going to Solve this question directly So ,first let the number be y .After that , by using the Equation we'll derive the Value of y . Than, we can Calculate the Rational Number . Let's Solve :

$\maltese$ Solving the Equation :

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { \bigg\{ \dfrac{Numerator}{Denominator} \bigg\} = \bigg\{ \dfrac{2}{5} \bigg\} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { \bigg\{ \dfrac{y - 3}{ ( y + 2 ) + 4 } \bigg\} = \bigg\{ \dfrac{2}{5} \bigg\} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { 2 \bigg\{ \bigg( y + 2 \bigg) + 4 \bigg\} = 5 \bigg(y - 3 \bigg) } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { 2 \bigg( y + 6 \bigg) = 5 \bigg(y - 3 \bigg) } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { 2y + 12 = 5y - 15 } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { 5y - 2y = 15 + 12 } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { 3y = 27 } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { y = \dfrac{27}{3} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; \sf { y = \cancel\dfrac{27}{3} } \\ \\ \\ \end{gathered}$

$\begin{gathered} \qquad \; \longrightarrow \; \; {\underline{\boxed{\pmb{\frak{ y = 9 }}}}} \; \purple\bigstar \\ \\ \\ \end{gathered}$

$\maltese$ Calculating the Rational Number :

• Numerator = y = 9
• Denominator = y + 2 = 9 + 2 = 11

$\therefore \;$ Original Rational Number is 9/11 .

$\\ \qquad{\rule{200pt}{2pt}}$