Question

The denominator of a rational number is greater than its numerator by 7. If 3
is subtracted from the numerator and 2 is added to its denominator, the new
number becomes 1/5. Find the rational number.

Answer 1

Let the denominator be x.

Then the numerator will be x + 7.

∴ Original fraction = (x/x + 7)

According to the given condition,

⇒ (x - 3)/(x + 9) = 1/5

⇒ 5(x - 3) = x + 9

⇒ 5x - 15 = x + 9

⇒ 5x - x = 9 + 15

⇒ 4x = 24

⇒ x = 6.

Therefore, rational number = (6/13).

Hope it helps!

Answer 2

$\sf\small\underline\purple{Let:-}$

$\sf{\implies Numerator\:_{(fraction)}=N}$

$\sf{\implies Denominator\:_{(fraction)}=D}$

$\sf{\implies Rational\: number=\dfrac{N}{D}}$

$\sf\small\underline\purple{Given:-}$

$\sf{\implies Denominator=Numerator+7}$

$\sf\small\underline\purple{To\: Find:-}$

$\sf{\implies The\:ratio\: number=?}$

$\sf\small\underline\purple{Solution:-}$

$\bf\small\underline{Calculation\:for\:1st\: equation:-}$

$\sf{\implies Rational\: number\:_{(denominator)}=Rational\: number\:_{(numerator)}+7}$

$\tt{\implies D=N+7----(i)}$

$\bf\small\underline{Calculation\:for\:2nd\: equation:-}$

$\sf{\implies \dfrac{Numerator-3}{Denominator+2}=Rational\: number\:_{(become\:1/5)}}$

$\tt{\implies \dfrac{N-3}{D+2}=\dfrac{1}{5}}$

$\tt{\implies 5N-15=D+2}$

$\tt{\implies 5N-D=2+15}$

$\tt{\implies 5N-D=17-----(ii)}$

In eq (ii) putting the value of D=N+7:-]

$\tt{\implies 5N-(N+7)=17}$

$\tt{\implies 5N-N-7=17}$

$\tt{\implies 5N-N=17+7}$

$\tt{\implies 4N=17+7}$

$\tt{\implies 4N=24}$

$\tt{\implies N=\frac{24}{4}=6}$

Putting the value of N=6 in eq (i):-]

$\tt{\implies D=N+7}$

$\tt{\implies D=6+7}$

$\tt{\implies D=13}$

$\sf\large{Hence,}$

$\sf{\implies Rational\: number=\frac{N}{D}}$

$\bf{\implies Rational\: number=\frac{6}{13}}$