Question

the numerator and denominator of rational number are in the ratio 3: 4. If the denominator is increased by 3 the ratio becomes 3: 5 .Find the rational number.​

Answer 1

Solution :-

Let :-

Numerator = x

Denominator = y

According to the first condition :-

$\longrightarrow \sf \dfrac{x}{y} = \dfrac{3}{4} \\\\\longrightarrow x = \dfrac{3}{4} y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(1)$

According to the second condition :-

$\longrightarrow \sf \dfrac{x}{y+3} = \dfrac{3}{5} \\\\\longrightarrow 5x = 3(y+3) \\\\\longrightarrow 5x= 3y+9 \\\\\longrightarrow 5x-3y = 9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(2)$

Substitute the value of x in eq (2) :-

$\longrightarrow \sf 5 \times \dfrac{3y}{4} -3y = 9 \\\\\longrightarrow \dfrac{15y}{4} - 3y = 9 \\\\\longrightarrow \dfrac{15y-12y }{4} = 9 \\\\\longrightarrow 15y - 12y = 36 \\\\\longrightarrow 3y = 36 \\\\\longrightarrow \bf y = 12$

Substitute y = 12 in eq (1) :-

$\longrightarrow \sf x = \dfrac{3}{4} \times 12 \\\\\longrightarrow x = 3 \times 3 \\\\\longrightarrow \bf x = 9$

$\bf FRACTION = \dfrac{9}{12}$

Answer 2

Given

• The numerator and denominator of rational number are in the ratio 3:4.

To find

• Rational number.

Solution

• Let the ratio be x.

⌬ Fraction = $\dfrac{3x}{4x}$

$\sf\pink{⟶}$ If the denominator is increased by 3 the ratio becomes 3: 5.

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {\dfrac{3x}{4x + 3} = \dfrac{3}{5}}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {5 × 3x = 3(4x + 3)}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {15x = 12x + 9}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {15x - 12x = 9}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {3x = 9}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {x = \dfrac{9}{3}}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {x = 3}$

⠀

★ Therefore

• Required fraction

$\sf\pink{⟶}$ $\dfrac{3 × 3}{4 × 3}$

$\sf\pink{⟶}$ $\dfrac{9}{12}$

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