Question

The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes 12. Find the fraction

Answer 1

Solution :

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

The sum of numerator & denominator of a fraction is 3 less than twice the denominator. If each of the numerator & denominator is decreased by 1, the fraction becomes 1/2.

$\underline{\bf{Explanation\::}}}}$

Let the numerator digit be r & denominator digit be m

$\boxed{\bf{The\:fraction\:becomes=\frac{r}{m} }}}$

A/q

$\longrightarrow\sf{r+m+3=2m}\\\\\longrightarrow\sf{r+m-2m=-3}\\\\\longrightarrow\sf{r-m=-3}\\\\\longrightarrow\bf{r=-3+m......................(1)}$

&

$\longrightarrow\sf{\dfrac{r-1}{m-1} =\dfrac{1}{2} }\\\\\longrightarrow\sf{2(r-1)=1(m-1)}\\\\\longrightarrow\sf{2r-2=m-1}\\\\\longrightarrow\sf{2(-3+m)-2=m-1\:\:[from(1)]}\\\\\longrightarrow\sf{-6+2m-2=m-1}\\\\\longrightarrow\sf{-8+2m=m-1}\\\\\longrightarrow\sf{2m-m=-1+8}\\\\\longrightarrow\bf{m=7}$

∴ Putting the value of m in equation (1),we get;

$\longrightarrow\sf{r=-3+7}\\\\\longrightarrow\bf{r=4}$

Thus;

$\boxed{\sf{The\:fraction\:becomes=\frac{r}{m}=\boxed{\bf{\frac{4}{7} }}}}$

Answer 2

Solution :

The sum of numerator & denominator of a fraction is 3 less than twice the denominator. If each of the numerator & denominator is decreased by 1, the fraction becomes 1/2.

Let the numerator digit be r & denominator digit be m

A/q

\begin{lgathered}\longrightarrow\sf{r+m+3=2m}\\\\\longrightarrow\sf{r+m-2m=-3}\\\\\longrightarrow\sf{r-m=-3}\\\\\longrightarrow\bf{r=-3+m......................(1)}\end{lgathered}

⟶r+m+3=2m

⟶r+m−2m=−3

⟶r−m=−3

⟶r=−3+m......................(1)

&

\begin{lgathered}\longrightarrow\sf{\dfrac{r-1}{m-1} =\dfrac{1}{2} }\\\\\longrightarrow\sf{2(r-1)=1(m-1)}\\\\\longrightarrow\sf{2r-2=m-1}\\\\\longrightarrow\sf{2(-3+m)-2=m-1\:\:[from(1)]}\\\\\longrightarrow\sf{-6+2m-2=m-1}\\\\\longrightarrow\sf{-8+2m=m-1}\\\\\longrightarrow\sf{2m-m=-1+8}\\\\\longrightarrow\bf{m=7}\end{lgathered}

⟶

m−1

r−1

=

2

1

⟶2(r−1)=1(m−1)

⟶2r−2=m−1

⟶2(−3+m)−2=m−1[from(1)]

⟶−6+2m−2=m−1

⟶−8+2m=m−1

⟶2m−m=−1+8

⟶m=7

∴ Putting the value of m in equation (1),we get;

\begin{lgathered}\longrightarrow\sf{r=-3+7}\\\\\longrightarrow\bf{r=4}\end{lgathered}

⟶r=−3+7

⟶r=4

Thus;

\boxed{\sf{The\:fraction\:becomes=\frac{r}{m}=\boxed{\bf{\frac{4}{7} }}}}

Thefractionbecomes=

m

r

=

7

4

Hope this helps you