Question

if 2 is added to the numerator of a
fraction it reduce to 1/2 and if 1 is
Subtradad
from
the denominater it
subtracted
From
1/3
find the
fraction​

Answer 1

Correct question :

• If 2 is added to the numerator of a fraction it reduces to 1/2 and if 1 is subtracted from the denominator, it reduces to 1/3. Find the fraction.

Solution :

Let the numerator be x and the denominator be y.

★ According to the question,

• Case 1

If two is added to the numerator of the fraction, it reduces to 1/2.

➟ (x + 2)/y = 1/2

• Cross multiply

➟ 2(x + 2) = y

➟ 2x + 4 = y

➟ 2x = y - 4 ⠀⠀⠀⠀⠀— eq (1)

• Case 2

If one is subtracted from the denominator of the fraction, it reduces to 1/3.

➟ x/(y - 1) = 1/3

• Cross multiply

➟ 3x = y - 1 ⠀⠀⠀⠀⠀— eq (2)

• Subtracting eq (1) from eq (2), we get

➟ x = 3

• Putting the value of x in eq (1)

➟ 2x = y - 4

➟ 2(3) = y - 4

➟ 6 = y - 4

➟ y = - 4 - 6

➟ y = 10

Therefore,

• Numerator = x = 3
• Denominator = y = 10

•°• The required fraction is 3/10.

Answer 2

❍ Let the numerator of the fraction be x and denominator of the fraction be y respectively.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\underline{\bigstar\;\boldsymbol{According\;to\;the\;Question:}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\underline{\sf{Case\;}\it{1}:}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• If two is added to the numerator of the fraction, it reduces to 1/2. Therefore,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\begin{gathered}:\implies\sf{\dfrac{x+2}{y}=\dfrac{1}{2}}\\\\\\:\implies\sf{2x+4=y}\\\\\\:\implies\sf{2x=y-4}\qquad\qquad\bigg\lgroup\sf{eq^n\;1}\bigg\rgroup\end{gathered}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\underline{\sf{Case\;}\it{2}:}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• If one is subtracted from the denominator of the fraction, it reduces to 1/3. Therefore,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\begin{gathered}:\implies\sf{\dfrac{x}{y-1}=\dfrac{1}{3}}\\\\\\:\implies\sf{3x=y-1}\qquad\qquad\bigg\lgroup\sf{eq^n\;2}\bigg\rgroup\end{gathered}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\qquad\qquad\underline{\bf{\dag\;}\frak{Subtracting\;eq^n\;2\;from\;eq^n\;1:}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\begin{gathered}:\implies\sf{3x-2x=y-y-1+4}\\\\\\:\implies\sf{x=-1+4}\\\\\\:\implies\underline{\boxed{\frak{\pink{x=3}}}}\;\bigstar\end{gathered}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\qquad\qquad\underline{\bf{\dag\;}\frak{Substituting\;value\;of\;x\;in\;eq^n\;1:}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\begin{gathered}:\implies\sf{2(3)=y-4}\\\\\\:\implies\sf{6=y-4}\\\\\\:\implies\sf{6+4=y}\\\\\\:\implies\underline{\boxed{\frak{\purple{y=10}}}}\;\bigstar\end{gathered}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\therefore\;{\underline{\sf{Hence,\;the\;fraction\;is\;\;\frak{\dfrac{3}{10}}}.}}$⠀⠀