Question

8.
The denominator of a rational number is greater than its numerator by 7. #3
is subtracted from the numerator and 2 is added to its denominator, the new
1
number becomes . Find the rational number.
5​

Answer 1

$\bf \underline{ \underline{\maltese\:Given} }$

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.

$\bf \underline{ \underline{\maltese \: To \: find } }$

$\sf \implies \: Original \: fraction = \: ?$

$\bf \underline{ \underline{\maltese \: Solution } }$

$\sf Let \: us \: assume \: that,$

$\sf\Rrightarrow The \: numerator \: of \: the \: rational \: number \: be \: x$

$\sf\Rrightarrow\bf{ \underline{Then}}, \sf the \: denominator \: will \: be \: x+7$

$\sf According \: to \: question,$

$\sf Equation \: = \: \bf \red{ \dfrac{x - 3}{(x + 7) + 2} = \dfrac{1}{5} }$

$\sf \implies\dfrac{x - 3}{x + 9} = \dfrac{1}{5}$

$\sf Doing \: cross \: multiplication,$

$\sf \implies (x - 3) 5 = (x + 9) 1$

$\sf \implies 5x - 15= x + 9$

$\sf \implies 5x - 15 - x = 9$

$\sf \implies 4x - 15 = 9$

$\sf \implies 4x = 9 + 15$

$\sf \implies 4x = 24$

$\sf \implies x = \cancel \dfrac{24}{4}$

$\sf \implies x = 6$

$\bf \underline{ Now},$

$\sf\Rrightarrow Numerator \: ( x) = \bf \: 6$

$\sf\Rrightarrow Denominator \: (x+7) = \bf 6+7 = 13$

$\bf \underline{ Therefore},$

$\underline{ \boxed{\red{\bf \: Original \: fraction= \: \dfrac{6}{13}} }}$

Answer 2

Answer:

• The original fraction is 6/13

Step-by-step explanation:

Given:

• 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 fraction obtained is 1/5

To Find:

• The original rational number

Assumptions:

• Let the numerator of the fraction be x
• Let the denominator of the fraction be x + 7

Solution:

Now the original fraction will be,

$\rightarrow \qquad \tt \dfrac{x}{x + 7}$

According to the question,

$\rightarrow \qquad \tt \dfrac{x - 3}{x + 7 + 2} = \dfrac{1}{5}$

$\rightarrow \qquad \tt 1(x + 7 + 2 ) = 5 ( x - 3 )$

$\rightarrow \qquad \tt x + 9 = 5x - 15$

$\rightarrow \qquad \tt x - 5x = - 15 - 9$

$\rightarrow \qquad \tt -4x = -24$

$\rightarrow \qquad \tt x = \cancel\dfrac{-24}{-4}$

$\rightarrow \qquad \tt {\purple{\boxed{\frak{x = 6}}}\pink\bigstar}$

Now let's find the original fraction,

$:\implies \tt \dfrac{x}{x + 7}$

$:\implies \tt \dfrac{6}{6 + 7}$

$:\implies \tt \dfrac{6}{13}$

Therefore:

• ${\pink{\underline{\boxed{\tt{Original \; Fraction = \dfrac{6}{13} }}}\bigstar}}$