Question

the denominator of a rational number is greater than its numerator by 5. if the numerator is increased by 8 and the denominator is decreased by 1 ,the number becomes 5/3.find the original number​

Answer 1

$\LARGE{ \underline{\underline{ \sf{Required \: answer:}}}}$

GiveN:

• Denominator is greater than its numerator by 5.
• If the numerator is increased by 8 and the denominator is decreased by 1, the number becomes 5/3.

To FinD:

• The original number?

Step-by-step Explanation:

Let the numerator be x and denominator be y.

According to question,

⇒ Denominator = Numerator + 5

⇒ y = x + 5

⇒ x - y = -5

And, It is also given that:

When,

• Numerator increased by 8 = x + 8
• Denominator decreased by 1 = y - 1

The Fraction becomes 5/3

Then,

$\rm{ \dfrac{x + 8}{y - 1} = \dfrac{5}{3} }$

Cross multiplying,

$\rm{3(x + 8) = 5(y - 1)}$

Expanding the parentheses,

$\rm{3x + 24 = 5y - 5}$

$\rm{3x - 5y = - 5 - 24}$

$\rm{3x - 5y = - 29} \: - - - - - (2)$

Multiplying 3 with eq. (1),

⇒ 3(x - y) = -15

⇒ 3x - 3y = -15

Subtracting eq. (1) from eq. (2),

⇒ 3x - 3y - (3x - 5y) = -15 + 29

⇒ 3x - 3y - 3x + 5y = 14

⇒ 2y = 14

⇒ y = 7

Then,

⇒ x - 7 = -5

⇒ x = 2

Thus,

The original number is:

$\Large{ \boxed{ \sf{ \orange{ \dfrac{2}{7} }}}}$

Note:

• After finding the answer, always recheck or verify by comparing with the conditions given.

Answer 2

$The \: Original \: Rational \: Number \: is \: \dfrac{2}{7}$

Step-by-step explanation:

Given :

Denominator is = greater than its numerator by 5

Numerator when increased by 8 and denominator when decreased by 1,

$new fraction = \dfrac{5}{3}$

To find :

• The Original Rational Number

Solution :

$\textbf{\small{\underline{Let the - }}}$

• Numerator be x
• Denominator be (x + 5)

$\textbf{\small{\underline{According to the Question - }}}$

$Numerator \: when \: increased \: by \: 8 \: and \: denominator \: when \: decreased \: by \: 1, \: new \: fraction \: = \dfrac{5}{3}$

$★ \boxed{\bf{\frac{x + 8}{(x + 5) - 1} = \frac{5}{3}}}$

$\begin{gathered} \sf{\implies} \:\frac{x + 8}{(x + 5) - 1} = \frac{5}{3} \\ \sf{\implies} \: \frac{x + 8}{x + 4} = \frac{5}{3} \\ \sf{\implies} \:3(x + 8) = 5(x + 4) \\ \sf{\implies} \:3x + 24 = 5x + 20 \\ \sf{\implies} \:5x - 3x = 24 - 20 \\ \sf{\implies} \:2x = 4 \\ \sf{\implies} \:x = \frac{4}{2} \\ \sf{\implies} \:x = 2\end{gathered}$

$\rule{300}{1.5}$

★ Value of (x + 5)

$\begin{gathered} \sf{\implies} \:2 + 5 \\ \sf{\implies} \:7\end{gathered}$

• Numerator = 2
• Denominator = 7
• $Original Rational Number = \boxed{\boxed{\sf{\frac{2}{7}}}}$

$\therefore∴ The Original Rational Number is \dfrac{2}{7}$

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