Question

The denominator of a rational number is greater than its numerator by 5. If the numerator and denominator decreased by 15 and 1 respectively,the number obtained is 3/2. Find the rational number​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

$\displaystyle{\boxed{\red{\sf\:The\:rational\:number\:=\:\dfrac{42}{37}\:}}}$

Step-by-step-explanation:

Let the numerator of the rational number be p.

And the denominator of the rational number be q.

$\displaystyle{\therefore\:\sf\:The\:rational\:number\:=\:\dfrac{p}{q}}$

From the first condition,

Denominator = Numerator + 5

⇒ q = p + 5 $\displaystyle{\;\;\;\sf\qquad\cdots(\:1\:)}$

From the second condition,

$\displaystyle{\sf\:\dfrac{p\:-\:15}{q\:-\:1}\:=\:\dfrac{3}{2}}$

$\displaystyle{\implies\sf\:2\:(\:p\:-\:15\:)\:=\:3\:(\:q\:-\:1\:)}$

$\displaystyle{\implies\sf\:2p\:-\:30\:=\:3q\:-\:3}$

$\displaystyle{\implies\sf\:2p\:-\:3q\:=\:-\:3\:+\:30}$

$\displaystyle{\implies\sf\:2p\:-\:3q\:=\:27}$

$\displaystyle{\implies\sf\:2p\:-\:3\:(\:p\:+\:5\:)\:=\:27\:\qquad\cdots[\:From\:(\:1\:)\:]}$

$\displaystyle{\implies\sf\:2p\:-\:3p\:-\:15\:=\:27}$

$\displaystyle{\implies\sf\:-\:p\:=\:27\:+\:15}$

$\displaystyle{\implies\sf\:-\:p\:=\:42}$

$\displaystyle{\implies\:\pmb{\sf\:p\:=\:-\:42}}$

$\displaystyle{\therefore\:\boxed{\pink{\sf\:Numerator\:=\:-\:42\:}}}$

By substituting p = - 42 in equation ( 1 ), we get,

q = p + 5

⇒ q = - 42 + 5

⇒ q = - 37

$\displaystyle{\therefore\:\boxed{\blue{\sf\:Denominator\:=\:-\:37\:}}}$

Now,

$\displaystyle{\sf\:The\:rational\:number\:=\:\dfrac{Numerator}{Denominator}}$

$\displaystyle{\implies\sf\:The\:rational\:number\:=\:\dfrac{\cancel{-}\:42}{\cancel{-}\:37}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:The\:rational\:number\:=\:\dfrac{42}{37}\:}}}}$