Question

(i)
The denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator
and denominator respectively, the ratio of numerator and denominator is 3: 5. Find the
fraction.
​

Answer 1

Required Answer:-

Let us consider that the numerator of the fraction is x and the denominator be y. Then the fraction will be $\frac{x}{y}$.

According to question,

The denominator of a fraction is 1 less than twice its numerator. That means,

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

We can keep this equation in this way. Later, we can try solving by substitution.

Also,

If 1 is added to numerator and denominator respectively, the ratio of numerator and denominator is 3 : 5. That means,

$\dfrac{x + 1}{y + 1} = \dfrac{3}{5}$

Cross multiply to get an equation,

$5(x + 1) = 3(y + 1)$

Opening the parentheses,

$5x + 5 = 3y + 3$

$5x - 3y + 2 = 0$

Now from equation (1), substitute y = 2x - 1,

$5x - 3(2x - 1) + 2 = 0$

Simplifying the above equation,

$5x - 6x + 3 + 2 = 0$

$- x + 5 = 0$

$- x = - 5$

And,

$x = 5$

Then,

$y = 2(5) - 1 = 9$

And the required fraction will be $\boxed{\frac{5}{9}}$.

How to check?

• Now you can check both the situations provided in the question.

• Adding 1 to both numerator and denominator gives 6/10 i.e. 3/5. Hence, verified too.

Answer 2

Given :-

Denominator of a fraction is 1 less than twice its numerator. If 1 is added to numerator  and denominator respectively, the ratio of numerator and denominator is 3: 5.

To Find :-

The fraction

Solution :-

Let the numerator be x

And denominator will be 2x - 1

$\sf \dfrac{x + 1}{2x - 1 + 1} = \dfrac{3}{5}$

$\sf\dfrac{x+1}{2x} = \dfrac{3}5$

By cross multiplication

3(2x) = 5(x + 1)

6x = 5x + 5

6x - 5x = 5

x = 5

Finding the fraction

$\sf Fraction =\dfrac{5}{2(5)-1}$

$\sf Fraction = \dfrac{5}{10-1}$

$\sf Fraction = \dfrac{5}{9}$