Question

the denominator of a rational number is greater than itd numinator by 5 . if d numerator is increased by 11 and denominator decreased by 14 and the new number is 5 .find the original rational number​

Answer 1

Solution :

It is given that in a fraction, the denominator is greater than the numerator by 5.

So, the fraction becomes :

> ( x/x+5)

The numerator is increased by 11 and the denominator is decreased by 14.

The new fraction becomes :

> ( x + 11)/(x - 9)

This is equal to 5.

Hence

> (x+11)/(x-9) = 5

> x + 11 = 5x - 45

> 4x = 56

> x = 14

The fraction thus becomes 14/19.

This is the required answer.

______________________________________________________

Answer 2

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

• The denominator of a rational number is greater than its numerator by 5 . if the numerator is increased by 11 and denominator decreased by 14, the new number is 5. Find the original rational number.

─━─━─━─━─━─━─━─━─━─━─━─━─

$\huge{AηsωeR} ✍$

─━─━─━─━─━─━─━─━─━─━─━─━─

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

• The denominator of a rational number is greater than its numerator by 5.
• If numerator is increased by 11 and denominator decreased by 14, the new number is 5.

─━─━─━─━─━─━─━─━─━─━─━─━─

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

• The original rational number

─━─━─━─━─━─━─━─━─━─━─━─━─

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

─━─━─━─━─━─━─━─━─━─━─━─━─

$\begin{gathered}\begin{gathered}\bf Let = \begin{cases} &\sf{numerator \: be \: x} \\ &\sf{denominator \: be \: x + 5} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf So \: rational \: number = \begin{cases} &\sf{ \dfrac{x}{x + 5} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\bf\red{★ \: According \: to \: statement \: ★}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: Now = \begin{cases} &\sf{numerator \: = \: x + 11} \\ &\sf{denominator \: = \: x + 5 - 14 = x - 9} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf So \: rational \: number = \begin{cases} &\sf{\dfrac{x + 11}{x - 9} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\bf\green{★According \: to \: statement★}\end{gathered}$

$\sf \:  ⟼\dfrac{x + 11}{x - 9} = 5$

$\sf \:  ⟼x + 11 = 5 \times (x - 9)$

$\sf \:  ⟼x + 11 = 5x - 45$

On transposition, we get

$\sf \:  ⟼x - 5x = - 45 - 11$

$\sf \:  ⟼ - 4x = - 56$

$\sf \:  ⟼x = \dfrac{ \cancel{ - 56}}{ \cancel{ - 4}} 14$

$\sf \:  ⟼x = 14$

─━─━─━─━─━─━─━─━─━─━─━─━─

$\begin{gathered}\begin{gathered}\bf \: So, = \begin{cases} &\sf{numerator \: = \: 14} \\ &\sf{denominator \: = \: 14 + 5 = 19} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf So \: rational \: number = \begin{cases} &\sf{\dfrac{14}{19} } \end{cases}\end{gathered}\end{gathered}$

─━─━─━─━─━─━─━─━─━─━─━─━─

Additional Information

─━─━─━─━─━─━─━─━─━─━─━─━─

What is a Rational Number?

• A rational number, in Mathematics, can be defined as any number which can be represented in the form of p/q where q ≠ 0. Also, we can say that any fraction fits under the category of rational numbers, where the denominator and numerator are integers and the denominator is not equal to zero. When the rational number (i.e., fraction) is divided, the result will be in decimal form, which may be either terminating decimal or the repeating decimal. 

Rational Numbers Properties

Since a rational number is a subset of the real number, the rational number will obey all the properties of the real number system. Some of the important properties of the rational numbers are as follows:

• The results are always a rational number if we multiply, add, or subtract any two rational numbers.

• A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.

• If we add zero to a rational number then we will get the same number itself.

• Rational numbers are closed under addition, subtraction, and multiplication.