Question

the denominator of a rational number is greater than its numerator by 8. if the numeris increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. find the rational number​

Answer 1

⇝ Correct Question :-

The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. find the Rational number

⇝ Given :-

The denominator of a rational number is greater than its numerator by 8.

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

⇝ To Find :-

The Rational Number.

⇝ Solution :-

Let Numerator and Denominator of Original Rational Number be :

Numerator = x

Accordingly :

Denominator should be = x + 8

So ,

$\text{Original Number = } \frac{\text x}{\text x + 8}$

❒ When Numerator is increased by 17 and the denominator is decreased by 1 :

$\text{Number Becomes : } \frac{\text x + 17}{\text x + 7}$

⏩ According To Question :

$\frac{\text x + 17}{\text x + 7} = \frac{3}{2}$

$:\longmapsto2(\text x + 17) = 3(\text x + 7)$

$:\longmapsto2\text x + 34 = 3\text x + 21$

$:\longmapsto2\text x - 3\text x = 21 - 34$

$:\longmapsto \cancel - \text x = \cancel - 13$

$\purple{ \Large :\longmapsto  \underline {\boxed{{\bf x = 13} }}}$

So,

$\text{Original Number = } \frac{13}{13+ 8}$

Hence,

$\large\underline{\pink{\underline{\frak{\pmb{ Original \: Number = \dfrac{13}{21} }}}}}$

Answer 2

☯GIVEN :

• The denominator of a rational number is 8 more than its numerator.

• The new number becomes $\bf{ \dfrac{3}{2}}$ , if the numerator is increased by 17 and the denominator is decreased by 1.

☯TO FIND :

• Required Number.

➲SOLUTION :

Let the numerator be x.

Then , denominator = x+8

• After increasing the numerator by 17 and decreasing the denominator by 1 , the new number becomes $\bf{ \dfrac{3}{2}}$.

$: \implies{\sf{ \dfrac{Numerator + 17}{Denominator - 1} = \dfrac{3}{2} }}$

$: \implies{\sf{ \dfrac{x + 17}{(x + 8) - 1} = \dfrac{3}{2} }}$

$: \implies{\sf{ \dfrac{x + 17}{x + 8- 1} = \dfrac{3}{2} }}$

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

• By Cross multiplication :

$: \implies{\sf{2(x + 17) = 3(x + 7) }}$

$: \implies{\sf{2x + 34 = 3x + 21 }}$

• Transposing 21 to LHS and 2x to RHS :

$: \implies{\sf{34-21= 3x - 2x }}$

$:\implies{\sf{13= x }}$

$:\implies{ \underline{ \huge{ \boxed{\red{ \bf{x= 13 }}}}}}$

★Numerator :

• $\bf{x = 13}$

★ Denominator :

• Substituting the value of x :

$:\leadsto {\sf{x+8}}$

$:\leadsto{\sf{13+8}}$

$:\leadsto\ {\bf{21}}$

$\huge {\pink{\therefore}}$ The Required Number is $\bf{\dfrac{13}{21}}$ .