Question

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

Answer 1

${\purple{\underline{\underline{\large{\mathtt{Answer:}}}}}}$

Given:

It is given that the sum of numerator and denominator of a fraction is 3 less than twice the denominator and if each numerator and denominator is decreased by 1, the fraction becomes 1/2.

To Find:

We need to find the fraction.

Solution:

Let the numerator be x and denominator be y.

So, the fraction is x/y.

Sum of numerator and denominator is

x + y.

But, it is given that x + y is 3 times less than twice the denominator

$=> x + y = 2y - 3$

$= x - y +3 = 0 \: .... .....(1)$

Also when numerator and denominator are decreased by 1 we have numerator=

(x - 1) and denominator= (y - 1)

The numerator becomes half of the denominator. So,

$= > x - 1 = \frac{1}{2} (y - 1)$

$= >2x -2 = y - 1$

$= > 2x - y - 2 = - 1$

$= > 2x - y = - 1 + 2$

$= > 2x - y = 1.........(2)$

Now, subtracting equation 1 from equation 2 we get,

$x - y + 3 - (2x - y - 1) = 0$

$x - y + 3 - 2x + y + 1 = 0$

$- x + 4 = 0$

$- x = - 4$

$or \: x = 4.............(3)$

Now, substituting the value of x from equation 3 in equation 1 we get,

$4 - y + 3 = 0$

$7 - y = 0$

$7 = y$

Therefore the fraction is 4/7.

Answer 2

Given,

• The sum of numerator and denominator of a fraction of a fraction is 3 less than the twice the denominator .
• When numerator and denominator is decreased by 1 then the fraction became 1/2

To Find,

• The Fraction

Solution,

⇒Suppose the numerator be a

And, Suppose the denominator be b

According to the First Condition :-

• The sum of numerator and denominator of a fraction of a fraction is 3 less than the twice the denominator .

$\implies \sf{a + b = 2b - 3}$

$\implies \sf{a = 2b - b - 3}$

$\implies \boxed{\sf{a = b - 3}}$

According to the Second Condition :-

• When numerator and denominator is decreased by 1 then the fraction became 1/2

$\implies \sf{\dfrac{a - 1}{b - 1} = \dfrac{1}{2}}$

$\implies \sf{2(a - 1) = 1(b - 1)}$

$\implies \sf{2a - 2 = b - 1}$

$\implies \sf{2a - b = 1}$

(Now Put the value of a from the First Condition)

$\implies \sf{2(b - 3) - b = 1}$

$\implies \sf{2b - 6 - b = 1}$

$\implies \sf{b = 1 + 6}$

$\implies \boxed{\sf{b = 7}}$

Now Put the value of b in First Condition :-

$\implies \sf{a = b - 3}$

$\implies \sf{a = 7 - 3}$

$\implies \boxed{\sf{a = 4}}$

Therefore,

$\large{\boxed{\bold{Fraction\:=\:\dfrac{a}{b}\:=\:\dfrac{4}{7}}}}$

$\rule{200}2$