Question

the denominator of a rational number is greater than its numerator by 2. if the numerator is decreased by 3 and the denominator is increased by 4, the new rational number obtained is 2/5 . find the rational number​

Answer 1

Given: The denominator of a rational number is greater than its numerator by 2. If the numerator is decreased by 3 and the denominator is increased by 4. & the new number obtained is ⅖.

Need to find: The rational number?

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Let's Consider that the numerator of the number be x. So, the denominator will be (x + 2).

A/q,

• If the numerator is decreased by 3 and the denominator is increased by 4, the new rational number obtained is ⅖.

⠀

$\longrightarrow\sf\qquad \Bigg\{\dfrac{Numerator - 3}{Denominator + 4}\Bigg\} = \Bigg\{\dfrac{2}{5}\Bigg\}$

$\longrightarrow\sf\qquad \Bigg\{\dfrac{x - 3}{x + 2 + 4}\Bigg\} = \Bigg\{\dfrac{2}{5}\Bigg\}$

$\longrightarrow\sf\qquad \Bigg\{\dfrac{x - 3}{x + 6}\Bigg\}= \Bigg\{\dfrac{2}{5}\Bigg\}$

$\longrightarrow\sf\qquad 5\Big\{x - 3\Big\} = 2\Big\{x + 6\Big\}$

$\longrightarrow\sf\qquad 5x - 15 = 2x + 12$

$\longrightarrow\sf\qquad 5x - 2x = 12 + 15$

$\longrightarrow\sf\qquad 3x = 27$

$\longrightarrow\sf\qquad x = \cancel\dfrac{27}{3}$

$\longrightarrow\qquad\underline{\boxed{\pmb{\frak{x = 9}}}}$

Therefore,

• Numerator of the number, x = 9.
• Denominator of the number, (x + 2) = (9 + 2) = 11.

⠀

$\therefore$ Hence, the required rational number is ⁹⁄₁₁.

Answer 2

Information provided with us:

• The denominator of a rational number is greater than its numerator by 2.

• If the numerator is decreased by 3 and the denominator is increased by 4, the new rational number obtained is 2/5 .

What we have to find :

• ➡ The required rational number

Assumption :

Consider the original numerator be x .

Now :

According to first condition given in question .

• The denominator of a rational number is greater than its numerator by 2 .

Then original denominator ,

• ➡ ( x + 2 )

The original number ,

$\rm \implies \: \dfrac{x}{x + 2}$

According to second condition given in question .

• The numerator is decreased by 3

The new numerator

• ➡ ( x - 3)

According to third condition given in question .

• The denominator is increased by 4

The new denominator

• ➡ (x+2) + 4 = x + 6

The new number

$\rm \implies \: \dfrac{x - 3}{x + 6}$

According to the question ,

• The new rational number obtained is 2/5

Here :

• By Using the given information, we get

$\rm \implies \: \dfrac{x - 3}{x + 6} = \dfrac{2}{5}$

• By cross multiplication ❌

$\rm \implies \: 5(x - 3) = 2(x + 6)$

$\rm \implies \: 5x - 15= 2x + 12$

$\rm \implies \: 5x - 2x= 12 + 15$

$\rm \implies \: 3 \: x=27$

$\rm \implies \: x = \dfrac{27}{3}$

$\bf \implies \: x =9$

Henceforth :

The original numerator

• ➡ x = 9

The original denominator

• ➡ x + 2 = 9 + 2 = 11

Therefore :

➡ The required original rational number

$\rm \implies \: \dfrac{x}{x+2} = \dfrac{9}{9+2}$

$\bf\implies \: \dfrac{9}{11}$

Now :

By placing the value of x = 9 in $\rm \implies \: \dfrac{x - 3}{x + 6}$ we get ,

➡ The new rational number

$\rm \implies \: \dfrac{9- 3}{9+ 6} = \dfrac{6}{15}$

Therefore :

➡ The new rational number is

$\bf\implies \: \dfrac{6}{15}$

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