Question

the sum of the numerator and denominator of a fraction is 3 less than twice the denominator if the numerator and denominator are decreased by 1 the numerator becomes half the denominator. determine tha fraction
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Answer 1

${\mathcal{\red{ANSWER}}}$

Fraction is $\tt{\dfrac{4}{7}}$

${\mathcal{\red{EXPLANATION}}}$

Let fraction be $\tt{\dfrac{x}{y}.}$

According to the question,

⇒ x + y + 3 = 2y

x = y - 3     ......................(1)

and,

$\tt{\implies \dfrac{x-1}{y-1} = \dfrac{y}{2y} = \dfrac{1}{2}}$

$\tt{\implies x - 1 = \dfrac{y-1}{2}}$

$\tt{\implies x = \dfrac{y-1}{2}+1}$

$\tt{\implies x = \dfrac{y+1}{2}}$

$\tt{\implies 2x = y+1}$

$\tt{2x-y=1\;\;\;\;\;\;...........(2)}$

Substitute the value of (1) and (2) we get,

$\tt{\implies 2(y-3)-y=1\;\;\;\;\;\;\;\;[Substitute\;value\;of\;x\;from\;eq(1)]}$

$\tt{\implies 2y-6-y=1}$

$\tt{\implies y-6=1}$

$\tt{\implies y=7}$

Put the value of y in equation(1), we get

$\tt{\implies x=y-3}$

$\tt{\implies x=7-3}$

$\tt{\implies x=4}$

${\boxed{\boxed{\bf{Therefore\;fraction\;is\;\dfrac{x}{y} = \dfrac{4}{7}}}}}$