Question

The denominator of a rational number is greater than its numerator by 7. If the numerator is increased by 6 and the denominator is decreased by 5, the new rational number obtained is 3/2. Find the original rational number.​

Answer 1

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

Given that,

The denominator of a rational number is greater than its numerator by 7.

Let assume that

Numerator of a fraction be x

So,

Denominator of a fraction is x + 7

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf \: Hence - \begin{cases} &\sf{fraction \: = \: \dfrac{x}{x + 7} }\end{cases}\end{gathered}\end{gathered}$

According to second condition,

If the numerator is increased by 6 and the denominator is decreased by 5, the new rational number obtained is 3/2

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

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf \: Hence - \begin{cases} &\sf{fraction \: = \: \dfrac{x + 6}{x + 2} }\end{cases}\end{gathered}\end{gathered}$

So,

$\rm :\longmapsto\:\dfrac{x + 6}{x + 2} = \dfrac{3}{2}$

$\rm :\longmapsto\:3(x + 2) = 2(x + 6)$

$\rm :\longmapsto\:3x +6 = 2x + 12$

$\rm :\longmapsto\:3x - 2x=12 - 6$

$\bf\implies \:x = 6$

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: So-\begin{cases} &\sf{numerator = 6} \\ &\sf{denominator = 7 + 6 = 13} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf \: Hence - \begin{cases} &\sf{fraction \: = \: \dfrac{6}{13} }\end{cases}\end{gathered}\end{gathered}$

Answer 2

$\underline{\underline{ \Large{ \mathfrak{Information \: given \: to \: us:-}}}}$

• The denominator of a rational number is greater than its numerator by 7
• Numerator is increased by 6 whereas denominator is increased by 5
• New rational number obtained is 3/2

$\underline{\underline{ \Large{ \mathfrak{What \: we \: have \: to \: calculate:-}}}}$

• We have to calculate the original rational number.

$\underline{\underline{ \Large{ \mathfrak{Our \: calculations \: started \: here:-}}}}$

___________

$\large{\bf\underbrace{Understanding \: question \: and \: solving \: way}}$

»» First of all we would be assuming the increase in denominator and numerator as any variable (eg:-y , x , z). After assuming we would place it according to the question where, assumed variable would be added with 7 in numerator as well as in denominator.

»» After that we would be increasing 6 in numerator and decrease 5 in denominator. Here assumed variable in the fraction would be added with 6 whereas in denominator it would be subtracted with 5, remember that as their was already an increase of 7 in denominator so 5 would be subtracted with 7.

»» Therefore an equation would be formed which would be kept equal to the new rational number obtained (3/2). And solve it.

________

Let us solve it now!!

Assuming variable be y for denominator and numerator.

• We have been assuming the variable because as we don't know about the actual numerator and denominator.

Calculating the numerator,

$: \ \implies \: \bf{Numerator \: would \: be \: } \bf{y}$

Calculating the denominator,

• As denominator is greater than the numerator by 7. So,

$: \ \implies \: \bf{Denominator \: would \: be \: \: } \bf{y + 7}$

Therefore fraction formed:-

$: \ \leadsto \: \red{\boxed{ \sf{Fraction \: would \: be \: \: } \bf{\dfrac{y}{y + 7}}}}$

________

Increasing numerator by 6,

• Here adding the numerator of the formed fraction with 6.

$: \implies \: \bf{Numerator \: would \: be \: \bf{y + 6}}$

Decreasing denominator by 5,

$: \rightarrow \: \bf{Denominator \: = \: y \: + \: 7 \: - \: 5}$

$: \rightarrow \: \bf{Denominator \: = \: y \: + \: 7 \: - \: 5}$

$: \implies \: \bf{Denominator\: would \: be \: \bf{y + 2}}$

Therefore new fraction formed,

$: \leadsto \: \boxed{\red{\bf{\dfrac{y \: + \: 6}{y \: + \: 2} }}}$

Thus equation formed,

$: \leadsto \: \boxed{\red{\bf{\dfrac{y \: + \: 6}{y \: + \: 2} \: = \: \frac{ 3}{2} }}}$

Solving now!

★ Cross multiplying them,

$: \implies \: \bf{2(y \: + \: 6) \: = \: 3(y \: + \: 2)}$

$: \implies \: \bf{2 \times (y \: + \: 6) \: = \: 3 \times (y \: + \: 2)}$

$: \implies \: \bf{2y \: + \:12 \: = \: 3y \: + \: 6}$

★ Reversing the sides,

$: \implies \: \bf{2y \: - \:3y \: = \: 6 \: - \: 12}$

$: \implies \: \bf{ - y \: = \: - 6}$

★ Negative signs would be cancelled,

$: \implies \: \bf{ \cancel- y \: = \: \cancel - 6}$

$: \implies \: \boxed{ \pink{ \bf{y \: = \: 6}}}$

_________

At last we gets value of y as 6.

• Therefore, numerator would be 6.

Finding out denominator,

$: \implies \: \bf{Denominator \: would \: be \: = \: y \: + \: 7}$

$: \implies \: \bf{Denominator \: would \: be \: = \: 6 \: + \: 7}$

$: \implies \: \bf{Denominator \: would \: be \: = \: 13}$

$\underline{\underline{ \Large{ \mathfrak{Conclusion:-}}}}$

As we got,

• Numerator is 6
• Denominator is 13

Original rational number would be,

• $\boxed{\dfrac{6}{13} }$