Question

When 1 is added to both the numerator and the denominator of a certain fraction it
becomes 1/2 and when 1 is subtracted from the numerator and the denominator it
becomes 1/3. Find the fraction. (Ans.: 3/7)​

Answer 1

SOLUTION :-

Let,

Numerator = x

Denominator = y

According to the first condition,

$\longrightarrow\sf \dfrac{x+1}{y+1}=\dfrac{1}{2} \\\\\longrightarrow 2(x+1)=y+1\\\\\longrightarrow 2x+2 = y+1 \\\\\longrightarrow 2 x-y = -1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ........................(1)$

According to the second condition,

$\longrightarrow\sf \dfrac{x-1}{y-1}=\dfrac{1}{3}\\\\\longrightarrow 3(x-1)=y-1\\\\\longrightarrow 3x-3 = y - 1 \\\\\longrightarrow 3x-y = 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ............................(2)$

Equation (2) - Equation (1),

$\longrightarrow \bf x = 3$

Substitute x = 3 in equation (2),

$\longrightarrow\sf 3\times 3-y = 2 \\\\\longrightarrow 9 - y = 2\\\\\longrightarrow -y = -7 \\\\\longrightarrow\bf y = 7$

$\bf FRACTION = \dfrac{3}{7}$

Answer 2

Answer:

Let the Numerator be x and Denominator be y of a fraction.

Thus, the given fraction will be x/y

❍ According to first condition :

➸ Numerator - 1/ Denominator - 1 = 1/3

➸ (x - 1)/(y - 1) = 1/3

➸ (x - 1)3 = 1(y - 1)

➸ 3x - 3 = y - 1

➸ 3x = y - 1 + 3

➸ 3x - y = 2

➸ y = 3x - 2 .......[Equation (i)]

______________________

❍ According to Second condition :

➸ Numerator + 1/ Denominator + 1 = 1/2

➸ x + 1/y + 1 = 1/2

➸ (x + 1)2 = 1(y + 1)

➸ 2x + 2 = y + 1

➸ y - 2x = 1 ........[Equation (ii)]

Now, Putting the value of y in equation (ii) we get,

➸ 3x - 2 - 2x = 1

➸ x - 2 = 1

➸ x = 1 + 2

➸ x = 3

Now, Substituting the value of x = 3 in equation (i) :

➸ y = 3x - 2

➸ y = 3(3) - 2

➸ y = 9 - 2

➸ y = 7

Therefore,

• Required fraction = x/y = 3/7