Question

The denominator of a rational number is greater than its numerator by 6.If the numerator is increased by 5 and denominator is decreased by 3 the number obtained is 5/4.Find the rational number.​

Answer 1

$\sf\red{What\:we\:have\:to\:find:-}$

• The rational number.

$\sf\blue{Solution:-}$

Let us take the numerator as x

Now ,the denominator will be x+6

It is also given that ,numerator is increased by 5 and denominator is decreased by 3.

Therefore,The numerator of new rational number= x+5

Denominator of new rational number

=(x+6)-3

=x+3

The new rational number=$\dfrac{x+5}{x+3}$

The given rational number=$\dfrac{5}{4}$

$\dfrac{x+5}{x+3}$=$\dfrac{5}{4}$

The next step will solved by Cross Multiplication.

$→\sf{4(x+5)=5(x+3)}$

$→\sf{4x+20=5x+15}$

$→\sf{4x-5x=15-20}$

$→\sf{-x=-5}$

$→\sf{x=5}$

Therefore,

Numerator of rational number=5

Denominator of rational number=5+6=11

Hence ,the rational number=$\dfrac{5}{11}$

$\sf\green{Verification:-}$

$\sf\dfrac{Original\: Numerator\:+5}{Original\: denominator\:-3}$

$=\dfrac{5+5}{11-3}$

$=\dfrac{10}{8}$

$=\dfrac{5}{4}$

Thus,the solution is Verified.

Answer 2

What we have to find:−

The rational number.

Solution:−

Let us take the numerator as x

Now ,the denominator will be x+6

It is also given that ,numerator is increased by 5 and denominator is decreased by 3.

Therefore,The numerator of new rational number= x+5

Denominator of new rational number

=(x+6)-3

=x+3

The new rational number=$\dfrac{x+5}{x+3}$

The given rational number=$\dfrac{5}{4}$

$\dfrac{x+5}{x+3}$

=$\dfrac{5}{4}$

The next step will solved by Cross Multiplication.

→\sf{4(x+5)=5(x+3)}→4(x+5)=5(x+3)

→\sf{4x+20=5x+15}→4x+20=5x+15

→\sf{4x-5x=15-20}→4x−5x=15−20

→\sf{-x=-5}→−x=−5

→\sf{x=5}→x=5

Therefore,

Numerator of rational number=5

Denominator of rational number=5+6=11

Hence ,the rational number=$\dfrac{5}{11}$

Verification:−

\sf\dfrac{Original\: Numerator\:+5}{Original\: denominator\:-3}

Originaldenominator−3

OriginalNumerator+5

=$\dfrac{5+5}{11-3}$=

=$\dfrac{10}{8}$=

=$\dfrac{5}{4}$=

Thus,the solution is Verified.