Question

The denominator of a rational number is three less than the numerator. If the denominator is doubled and the denominator is increased by 15, the new number obtained is ⅘. Find the original rational number.​

Answer 1

Correct Question :-

The denominator of a rational number is three less than the numerator. If the denominator is doubled and the Numerator is increased by 15, the new number obtained is ⅘. Find the original rational number.

Let :-

• The Numerator of fraction = x
• The denominator of fraction = y
• The original rational number = x/y

To Find :-

• The original rational number = ?

Solution :-

To calculate the original rational number at first we have to set up equation as per the given clue in the question. Then solve the equation.

Given in case - (i) :-

• The denominator of a rational number is three less than the numerator.

[Numerator = x. Denominator = y ]

⇢Denominator = Numerator - 3

⇢ y = x - 3--------(i)

Given in case - (ii) :-

• If the denominator is doubled and the Numerator is increased by 15, the new number obtained is ⅘.

[Numerator = x + 15. Denominator = 2 × y ]

⇢ Numerator + 15/Denominator × 2 = New number

⇢ (x + 15)/2 × y = 4/5

⇢ 5(x + 15) = 2y(4)

⇢ 5x + 75 = 8y

⇢ 5x - 8y = - 75---------(ii)

[substituting y = x - 3. in equation (ii) ]

⇢ 5x - 8(x - 3) = - 75

⇢ 5x - 8x + 24 = - 75

⇢ - 3x = - 75 - 24

⇢ - 3x = - 99

⇢ 3x = 99

⇢ x = 33

[Putting x = 33 in equation (ii) ]

⇢ 5x - 8y = - 75

⇢ 5 × 33 - 8y = - 75

⇢ - 8y = -75 - 165

⇢ - 8y = - 240

⇢ 8y = 240

⇢ y = 30

Therefore, the original rational number = x/y

⇢ The orginal rational number = 33/30 or 11/10

Answer 2

Answer:

$\qquad\qquad\underline{\textsf{\textbf{ \color{red}{Question :-} }}}$

• The denominator of a rational number is three less than the numerator. If the denominator is doubled and the Numerator is increased by 15, the new number obtained is 4/5. Find the original rational number.

$\qquad\qquad\underline{\textsf{\textbf{ \color{magenta}{Given :-} }}}$

• The denominator of a rational number is three less than the numerator. If the denominator is doubled and the Numerator is increased by 15, the new number obtained is 4/5.

$\qquad\qquad\underline{\textsf{\textbf{ \color{green}{Find\: Out :-} }}}$

• Find the original rational number.

$\qquad\qquad\underline{\textsf{\textbf{ \color{blue}{Solution :-} }}}$

Let, the Numerator of fraction = x

And, the denominator of fraction = y

Hence, the original rational number = $\sf \dfrac{x}{y}$

☣ At first case :-

➙ y = x - 3 ____________ (Eqn i)

☣ At second case :

➙ $\sf \dfrac{(x + 15)}{2 × y} = \dfrac{4}{5}$

➙ $\sf \dfrac{(x + 15)}{2y} = \dfrac{4}{5}$

Cross multiplication we get

➙ 5(x + 15) = 2y(4)

➙ 5x + 75 = 8y

➙ 5x - 8y = - 75 __________ (Eqn ii)

By solving the equation we get,

➔ 5x - 8(x - 3) = - 75

➔ 5x - 8x + 24 = - 75

➔ - 3x = - 75 - 24

➔ - 3x = - 99

➔ 3x = 99

➔ x = $\sf \cancel{\dfrac{99}{3}}$

$\bigstar \: \boxed{\sf{x =\: 33}}$

By putting x = 33 in equation (ii) we get,

➔ 5x - 8y = - 75

➔ 5 × 33 - 8y = - 75

➔ - 8y = -75 - 165

➔ - 8y = - 240

➔ 8y = 240

➔ y = $\sf \cancel{\dfrac{240}{8}}$

$\bigstar \: \boxed{\sf{y =\: 30}}$

Hence, the rational number is :

➣ $\sf \dfrac{x}{y}$

➣ $\sf \cancel{\dfrac{33}{30}}$

➣ ${\small{\bold{\purple{\dfrac{11}{10}}}}}$

Henceforth, the orginal rational number is 11/10.