Question

$\large\boxed{\bf{ \red{Que} \pink{st} \orange{ion}}}$
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.
$\huge{\boxed{\sf{It's \: Divya}}}$
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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 :-

The original rational number.

Solution :-

Let the denominator of a rational number be X

The numerator of the rational number

= 3 less than five times its denominator.

= 5 times the denominator - 3

= 5X-3

The rational number = Numerator/Denominator

= (5X-3)/X

On subtracting 2 from the Numerator then it will be 5X-3-2 = 5X-5

On adding 7 to the Denominator then it will be X+7

The new rational number = (5X-5)/(X+7)

According to the given problem

The new rational number = 5/3

=> (5X-5)/(X+7) = 5/3

On applying cross multiplication then

=> 3(5X-5) = 5(X+7)

=> 15X-15 = 5X+35

=> 15X-5X = 35+15

=> 10X = 50

=> X = 50/10

=> X = 5

If X = 5 then the numerator = 5X-3

= 5(5)-3

= 25-3

= 22

If X = 5 then the denominator = X = 5

The rational number = 22/5

Answer:-

♦ The Original rational number = 22/5

Check :-

The rational number = 22/5

Denominator = 5

Numerator = 22

= 25-3

= 5(5)-3

= The numerator of a rational number is 3 less than five times its denominator.

and

On subtracting 2 from the Numerator then it will be 22-2 = 20

On adding 7 to the Denominator then it will be 5+7 = 12

The new rational number = 20/12

= (5×4)/(3×4)

= 5/3

Verified the given relations in the given problem.

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,

$\mapsto \bf Denominator =\: x$

$\bigstar$ The numerator of a rational number is 3 less than five times its denominator.

So,

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

Hence, the required original rational number will be :

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

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

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 obtained is 5/3.

So,

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

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

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

By doing cross multiplication we get,

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

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

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

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

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

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

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

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

Hence,

$\dag$ Required Original Rational Number :

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

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

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

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

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