Question

Numerator of a fraction is 2 less than the denominator If 1 is added to the numerator and 3 to the denominator then the fraction becomes 2/3. Find the original fraction.​

Answer 1

$\large \dag$ Question :-

Numerator of a fraction is 2 less than the denominator If 1 is added to the numerator and 3 to the denominator then the fraction becomes 2/3. Find the original fraction.

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf  \green{The   \: Original \:  Fraction  \: is \:  \frac{7}{9} }} }$

$\large \dag$ Step by step Explanation :-

Let Numerator and Denominator of Original Fraction be :

• Numerator = x

As Per the question numerator of the fraction is 2 less than the denominator so,

• Denominator should be = x + 2

So ,

$\text{Original Fraction = } \frac{\text x}{\text x + 2}$

❒ When 1 is added to the numerator and 3 to the denominator :

$\rm \text{Fraction Becomes } :  \frac{x + 1}{x + 5}$

⏩ According To Question :

$\large \blue \bigstar  \:   \red{ \bf  \frac{x + 1}{x + 5} =  \frac{2}{3}  }$

$:\longmapsto \rm 3(x + 1) = 2( + 5)$

$:\longmapsto \rm 3x + 3 = 2x + 10$

$:\longmapsto \rm 3x - 2x = 10 - 3$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf x = 7} }}}$

$\blue\dashrightarrow\underline{\underline{\sf  \orange{Numerator  \: of \:  Original  \: Fraction = 7 }} }$

☆ As numerator of the fraction is 2 less than the denominator :

$\rm\therefore \:  Denominator  = 7   + 2$

$\blue\dashrightarrow\underline{\underline{\sf  \orange{Denominator  \: of \:  Original  \: Fraction = 9 }} }$

Therefore,

$\large\underline{\pink{\underline{\frak{\pmb{ Original  \: Fraction  =  \dfrac{7}{9} }}}}}$

Answer 2

Given : Numerator of a fraction is 2 less than the denominator If 1 is added to the numerator and 3 to the denominator then the fraction becomes ⅔. Find the original fraction.

★ Cᴏɴᴄᴇᴘᴛ :

According to the question, the numerator of a fraction is 2 less than the denominator henceforth 2 will be added to the denominator with the variable. Now, 1 is added to both the numerator and 3 is added to the denominator. New Fraction becomes ⅔. So after which we'll get Original Fraction = New Fraction. Putting the values we'll find variable's value than put the value in the original fraction to get the required answer

★ Sᴏʟᴜᴛɪᴏɴ :

Let us assume the Numerator be a

Hence,

$\star \quad \small \boxed{ \frak{ \pink{Original \: Fraction} = \green{ \frac{Numerator}{Denominator}}}} \quad \star$

$\rightarrow \frak{Original \: Fraction = \frac{a}{a + 2}}$

Now, 1 is added to the numerator and 3 is added to the denominator

$\rightarrow \frak{Original \: Fraction = \frac{a + 1}{a + 2 + 3}}$

$\rightarrow \frak{Original \: Fraction = \blue{\frac{a + 1}{a + 5}}}$

Now after which the new fraction becomes ⅔

$\dashrightarrow \frak{New \: Fraction = \frac{2}{3}}$

So, Original Fraction = New Fraction

$\hookrightarrow \frak{ \frac{a + 1}{a + 5} = \frac{2}{3}}$

$\hookrightarrow \frak{3(a + 1) = 2(a + 5)}$

$\hookrightarrow \frak{3a + 3 = 2a + 10}$

$\hookrightarrow \frak{3a - 2a = 10 - 3}$

$\star \quad \underline{ \boxed{ \red{ \frak{a = 7}}}}$

★ Fɪɴᴅɪɴɢ Oʀɪɢɪɴᴀʟ Fʀᴀᴄᴛɪᴏɴ :

$\hookrightarrow \frak{ Original \: Fraction = \frac{a}{a + 2}}$

$\hookrightarrow \frak{ Original \: Fraction = \frac{7}{7 + 2} }$

$\star\quad \frak{ Original \: Fraction = \frac{7}{9} }$

• Hence, the original fraction is ⁷/₉