Question

A fraction is such that, if the numerator is multitiplied by 3 and denominator reduced by 3 we get 18/11 but, if numerator is increased by 8 and denominator is double we get 2/5. Find the fraction ​

Answer 1

S O L U T I O N :

Let the numerator place digit be x & let the denominator place digit be y respectively.

$\boxed{\bf{The\:fraction = \frac{x}{y} }}$

A/q

$\underbrace{\sf{1^{st}\;Case\::}}$

$\mapsto\tt{\dfrac{3 \times x}{y - 3} =\dfrac{18}{11} }$

$\mapsto\tt{\dfrac{3 x}{y - 3} =\dfrac{18}{11} }$

$\mapsto\tt{18(y-3) = 11(3x)\:\:\:\underbrace{\sf{Cross-multiplication}}}$

$\mapsto\tt{18y - 54 = 33x}$

$\mapsto\tt{18y =33x + 54}$

$\mapsto\tt{y =33x + 54/18.................(1)}$

$\underbrace{\sf{2^{nd}\;Case\::}}$

$\mapsto\tt{\dfrac{x+8}{2y} =\dfrac{2}{5} }$

$\mapsto\tt{2(2y) = 5(x+8)\:\:\:\underbrace{\sf{Cross-multiplication}}}$

$\mapsto\tt{4y = 5x +40}$

$\mapsto\tt{4y-5x = 40}$

$\mapsto\tt{4\bigg(\dfrac{33x+54}{18} \bigg) - 5x = 40\:\:[from(1)]}$

$\mapsto\tt{\dfrac{132x+216}{18} - 5x = 40}$

$\mapsto\tt{132x +216 - 90x = 720}$

$\mapsto\tt{132x - 90x = 720-216}$

$\mapsto\tt{42x = 504}$

$\mapsto\tt{x = \cancel{504/42}}$

$\mapsto\bf{x=12}$

∴ Putting the value of x in equation (1),we get;

$\mapsto\tt{y = \dfrac{33(12) + 54}{18} }$

$\mapsto\tt{y = \cancel{\dfrac{450}{18} }}$

$\mapsto\bf{y = 25}$

Thus,

$\boxed{\bf{The\:fraction = \frac{x}{y} = \frac{12}{25} }}$