Question

$\huge\sf{Question}$
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.

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Answer 1

$\large\underline{\sf{Solution-}}$

Given that, The numerator of a rational number is 3 less than five times its denominator.

So, Let assume that

Denominator of a rational number = x

Numerator of a rational number = 5x - 3

Therefore,

$\rm \: Rational\:number = \dfrac{5x - 3}{x}$

Further given that, 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.

Now,

Denominator = x + 7

Numerator = 5x - 3 - 2 = 5x - 5

So,

$\rm \: Rational\:number = \dfrac{5x - 5}{x + 7}$

Now, According to statement

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

$\rm \: 3(5x - 5) = 5(x + 7)$

$\rm \: 15x - 15 = 5x + 35$

$\rm \: 15x - 5x = 15 + 35$

$\rm \: 10x = 50$

$\rm\implies \:x = 5$

Hence,

$\rm \: Rational\:number = \dfrac{5 \times 5 - 3}{5} = \frac{25 - 3}{5} = \frac{22}{5}$

Thus,

$\rm\implies \:\boxed{ \rm{ \:Rational\:number \: = \: \frac{22}{5} \: \: }}$

Answer 2

Answer:

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 :-

• What is the original rational number.

Solution :-

Let,

$\mapsto \bf Denominator =\: x$

$\mapsto \bf Numerator =\: 5x - 3$

Hence, the required original rational number is :

$\mapsto \sf Original\: Rational\: Number =\: \dfrac{Numerator}{Denominator}$

$\mapsto \sf\bold{\blue{Original\: Rational\: Number =\: \dfrac{5x - 3}{x}}}$

✯ According to the question :

$\bigstar$ When 2 is subtracted from its numerator, and 7 is added to its denominator, the simplest form of the rational number is 5/3.

So,

$\implies \bf \bigg\{\dfrac{Numerator - 2}{Denominator + 7}\bigg\} =\: \bigg\{\dfrac{5}{3}\bigg\}$

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

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

By doing cross multiplication we get :

$\implies \sf 5(x + 7) =\: 3(5x - 5)$

$\implies \sf 5x + 35 =\: 15x - 15$

$\implies \sf 5x - 15x =\: - 15 - 35$

$\implies \sf {\cancel{-}} 10x =\: {\cancel{-}} 50$

$\implies \sf 10x =\: 50$

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

$\implies \sf x =\: \dfrac{5}{1}$

$\implies \sf\bold{\green{x =\: 5}}$

Hence, the required original rational number is :

$\dashrightarrow \mathtt{Original\: Rational\: Number =\: \dfrac{5x - 3}{x}}$

$\dashrightarrow \mathtt{Original\: Rational\: Number =\: \dfrac{5(5) - 3}{5}}$

$\dashrightarrow \mathtt{Original\: Rational\: Number =\: \dfrac{25 - 3}{5}}$

$\dashrightarrow \mathtt{\bold{\red{Original\: Rational\: Number =\: \dfrac{22}{5}}}}$

$\small \sf\bold{\underline{\purple{\therefore\: The\: required\: original\: rational\: number\: is\: \dfrac{22}{5}\: .}}}$