Question

The numerator of a rational number is 3 less than five times its denominator. When 2 is subtracted from its numerator, and 7 is added to its denominator, the simplest form of the rational number obtained is 5/3. Find the original rational number.

Answer 1

Given :

• The numerator of a rational number is 3 less than five times its denominator.
• When 2 is subtracted from its numerator, and 7 is added to its denominator, the simplest form of the rational number obtained is 5/3.

To Find :

• Original Rational Number .

Solution :

$\longmapsto\tt{Let\:Denominator\:be=x}$

As Given that the Numerator of a rational number is 3 less than five times its Denominator. So ,

$\longmapsto\tt{Numerator=5x-3}$

Now ,

• If 2 is subtracted from its numerator, and 7 is added to its denominator, the simplest form of the rational number obtained is 5/3.

$\longmapsto\tt{Numerator=5x-3-2=5x-5}$

$\longmapsto\tt{Denominator=x+7}$

A.T.Q :

$\longmapsto\tt{\dfrac{5x-5}{x+7}=\dfrac{5}{3}}$

$\longmapsto\tt{3(5x-5)=5(x+7)}$

$\longmapsto\tt{15x-15=5x+35}$

$\longmapsto\tt{15x-5x=35+15}$

$\longmapsto\tt{10x=50}$

$\longmapsto\tt{x=\cancel\dfrac{50}{10}}$

$\purple\longmapsto\:\large\underline{\boxed{\bf\red{x}\green{=}\blue{5}}}$

Therefore :

$\longmapsto\tt{Numerator=5(5)-3}$

$\longmapsto\tt\bf{22}$

$\longmapsto\tt{Denominator=x}$

$\longmapsto\tt\bf{5}$

So , The Original Fraction is 22/5 .

Answer 2

Answer:

Question :-

✯ The numerator of a rational number is 3 less than five times its denominator. When 2 is subtracted from its numerator, and 7 is added to its denominator, the simplest form of the rational number obtained is 5/3. Find the original rational number.

Given :-

✯ The numerator of a rational number is 3 less than five times its denominator. When 2 is subtracted from its numerator, and 7 is added to its denominator, the simplest form of the rational number obtained is 5/3.

Find Out :-

✯ Find the original rational number.

Solution :-

⦿ Let denominator be y .

⦿ Numerator be 5y - 3

➙ Hence, The fraction = $\bf{ \dfrac{(5y - 3)}{y}}$

So, According to the Question or ATQ :-

$\longrightarrow \sf \bigg[\dfrac{(5y - 3) - 2}{y + 7}\bigg] = \bigg[\dfrac{5}{3}\bigg]$

$\longrightarrow{\sf{ \dfrac{5y - 5}{y + 7}} = \dfrac{5}{3}}$

$\longrightarrow{\sf{3(5y - 5)= 5(y + 7)}}$

$\longrightarrow{\sf{15y - 15= 5y + 35}}$

$\longrightarrow{\sf{15y - 5y = 35 + 15}}$

$\longrightarrow{\sf{10y = 50}}$

$\longrightarrow{\sf{y = \dfrac{50}{10}}}$

$\longrightarrow{\bf{y = 5}}$

Hence, the original rational number will be :

$\small \implies \bf Original\: Rational\: Number =\: \bigg(\dfrac{Numerator}{Denominator}\bigg)$

$\small \implies \sf Original\: Rational\: Number =\: \dfrac{5(5) - 3}{5}$

$\small \implies \sf Original\: Rational\: Number =\: \dfrac{25 - 3}{5}$

$\small \implies \sf Original\: Rational\: Number =\: \dfrac{22}{5}$

Henceforth,, the rational number is $\sf{ \dfrac{22}{5}}$.