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. 6 points (a) 21/13 (b) 13/21 (c) 14/13

Answer 1

⇝ 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,

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

Therefore,

$\large \pmb{ \red\bigstar \: \: \orange{ \underbrace{ \underline{  \frak{Option \: (b) \: is \: Correct }}}}}$

Answer 2

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.

★ Options ::

• (a) 21/13

• (b) 13/21

• (c) 14/13

Answer:

• Option (b) 13/21 is correct.

Explanation:

Given that:

• 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:

• Rational number?

Solution:

• Let the numerator of rational number be m. As it is stated in question that its denominator is 8 more than its numerator. So, its denominator is m + 8.

• Hence, rational number = numerator/denominator = m/(m + 8).

According to the question,

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

Therefore,

↣ $\sf \dfrac{m + 17}{(m + 8) - 1} = \dfrac{3}{2}$

↣ $\sf \dfrac{m + 17}{m + 8 - 1} = \dfrac{3}{2}$

↣ $\sf \dfrac{m + 17}{m + 7} = \dfrac{3}{2}$

By doing cross multiplication we get,

↣ $\sf 2(m + 17) = 3(m + 7)$

↣ $\sf 2m + 34 = 3m + 21$

↣ $\sf 34 - 21 = 3m - 2m$

↣ $\sf 13 = 1m$

➼ $\bf\red{m = 13}$

$\boxed{\sf{\therefore\:Numerator\:of\:rational\:number = {\textsf{\textbf{13}}}}}$

Now,

We know that,

• Denominator is 8 more than numerator.

↣ $\sf Denominator = m + 8$

Put m = 13 in above equation we get,

↣ $\sf Denominator = 13 + 8$

➼ $\bf\purple{Denominator = 21}$

$\boxed{\sf{\therefore\:Denominator\:of\:rational\:number = {\textsf{\textbf{21}}}}}$

Again we know that,

• Rational number = $\bf\dfrac{Numerator}{Denominator}$

$\large{\boxed{\sf{\therefore\:Required\:rational\:number = \dfrac{{\textsf{\textbf{13}}}}{{\textsf{\textbf{21}}}}}}}$

Let's Verify:

We know that,

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

Therefore,

⇒ $\sf \dfrac{m + 17}{(m + 8) - 1} = \dfrac{3}{2}$

Put m = 13 in above equation we get,

⇒ $\sf \dfrac{13 + 17}{(13 + 8) - 1} = \dfrac{3}{2}$

⇒ $\sf \dfrac{30}{21 - 1} = \dfrac{3}{2}$

⇒ $\sf \dfrac{3\cancel{0}}{2\cancel{0}} = \dfrac{3}{2}$

⇒ $\sf \dfrac{3}{2} = \dfrac{3}{2}$

➠ $\bf\pink{LHS = RHS}$

$\large{\boxed{\sf{Hence,\:Verified\:\checkmark}}}$

∴ Correct option is (b) 13/21.

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