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 53. Find the original rational number.

Answer 1

Correct 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:

The rational number $\sf{ \dfrac{22}{5}}$

Step-by-step explanation:

Let denominator be y

Numerator be 5y - 3

The fraction = $\sf{ \dfrac{(5y - 3)}{y}}$

According to the Question,

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

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

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

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

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

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

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

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

Denominator = 5

★ Numerator =

$\sf{\longrightarrow} \: 5(5) - 3$

$\sf{\longrightarrow} \: 25 - 3$

$\sf{\longrightarrow} \: 22$

The original rational number is :

$\sf{ \dfrac{22}{5} }$

Therefore, the rational number $\sf{ \dfrac{22}{5}}$

Answer 2

Answer:

Correct Question :-

• The numerator of a rational number is 3 less than five times it's 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 number.

Given :-

• The numerator of a rational number is 3 less than five times it's 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 :-

• What is the original number.

Solution :-

Let, the denominator be x

And, the numerator will be 5x - 3

Then, the original number is $\sf \dfrac{5x - 3}{x}$

Now, it is given in the question that, 2 is subtracted from its numerator and 7 is added to its denominator then,

$\longmapsto$ $\sf \dfrac{5x - 3 - 2}{x + 7}$

According to the question,

↦ $\sf \dfrac{5x - 3 - 2}{x + 7} =\: \dfrac{5}{3}$

↦ $\sf \dfrac{5x - 5}{x + 7} =\: \dfrac{5}{3}$

By doing cross multiplication we get,

↦ $\sf 3(5x - 5) =\: 5(x + 7)$

↦ $\sf 15x - 15 =\: 5x + 35$

↦ $\sf 15x - 5x =\: 35 + 15$

↦ $\sf 10x =\: 50$

↦ $\sf x =\: \dfrac{5\cancel{0}}{1\cancel{0}}$

↦ $\sf x =\: \dfrac{5}{1}$

➠ $\sf\bold{\pink{x =\: 5}}$

Hence, the required original number is :

➲ Original Number :

$\leadsto \sf \dfrac{5x - 3}{x}$

$\leadsto \sf \dfrac{5(5) - 3}{5}$

$\leadsto \sf \dfrac{5 \times 5 - 3}{5}$

$\leadsto \sf \dfrac{25 - 3}{5}$

$\leadsto \sf\bold{\red{\dfrac{22}{5}}}$

$\therefore$ $\sf\boxed{\bold{\purple{The\: original\: number\: is\: \dfrac{22}{5}.}}}$

$\rule{150}{2}$

$\sf\boxed{\bold{\green{VERIFICATION\: :-}}}$

⇒ $\sf \dfrac{5x - 3 - 2}{x + 7} =\: \dfrac{5}{3}$

⇒ $\sf \dfrac{5x - 5}{x + 7} =\: \dfrac{5}{3}$

By putting x = 5 we get,

⇒ $\sf \dfrac{5(5) - 5}{5 + 7} =\: \dfrac{5}{3}$

⇒ $\sf \dfrac{25 - 5}{12} =\: \dfrac{5}{3}$

⇒ $\sf \dfrac{\cancel{20}}{\cancel{12}} =\: \dfrac{5}{3}$

➦ $\sf\bold{\dfrac{5}{3} =\: \dfrac{5}{3}}$

Hence, Verified.