Question

If 1 is added to the denominator of a fraction, the fraction becomes 1/2. If 1 is added to the numerator, the fraction becomes 1. The fraction is:

a. 4/7
b. 5/9
c. 2/3
d. 10/11
​

Answer 1

$\rm{ The \: fraction \: is \: \dfrac{2}{3} }$

Step-by-step explanation:

★Given -

• If 1 is added to the denominator of a fraction, the fraction becomes 1/2.
• If 1 is added to the numerator, the fraction becomes 1.

★To find -

• The fraction.

★Concept -

• Here, we'll use the concept of linear equations in two variables to solve the question.

★Solution -

$\rm{Let \: the \: fraction \: be \: \bf \red{\dfrac{x}{y} }}$

According to the first statement,

$\longmapsto \rm{ \bf{ \dfrac{x}{y + 1} = \dfrac{1}{2} }}$

$\longmapsto \rm{ \bf{2x = y + 1}}$

$\longmapsto \rm{ \bf{ \pink{2x - 1= y} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: (1)}$

Now, according to the second statement,

$\longmapsto \rm{ \bf{ \dfrac{x + 1}{y} = 1}}$

$\longmapsto \rm{ \bf{ {x + 1} = y}}$

${ \longmapsto \rm{ \bf{ {x + 1} = 2x - 1} \: \: \: \: \: \: \: \: \: \: \: \: \: \{from \: (1) \}}}$

${ \longmapsto \rm{ \bf{ {1+ 1} = 2x - x} }}$

${ \longmapsto \rm{ \bf \green{{ {2} = x} }}} \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \:\:\bf \ {\rm \{putting \: in \: (1) \}}$

${ \longmapsto \rm{ \bf{ {2(2) - 1} = y} }}$

${ \longmapsto \rm{ \bf{ {4- 1} = y} }}$

${ \longmapsto \rm{ \bf{ \orange{{3} = y} }}}$

$\bf \rm{ \purple{\therefore the \: fraction \: is \: \dfrac{2}{3} }}$