Question

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 to denominator is 3.5. Find the fraction​

Answer 1

Answer:

Fraction = 5/9

Step-by-step explanation:

Given:

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

To Find:

• The fraction

Solution:

Let the numerator of the fraction be x

Hence by given,

Denominator = 2 × (numerator) - 1

Denominator = 2x - 1

Therefore the fraction is given by,

$\sf Fraction = \dfrac{x}{2x-1}$

Also by given adding 1 to numerator and denominator, the fraction becomes 3 : 5 = 3/5

Hence,

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

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

Cross multiplying,

5 (x + 1) = 3 × 2x

5x + 5 = 6x

6x - 5x = 5

x = 5

Hence numerator of the fraction is 5.

Now finding the denominator,

Denominator = 2x - 1

Denominator = 2 × 5 - 1

Denominator = 9

Hence the denominator of the fraction is 9.

Hence the fraction is 5/9.

$\boxed{\bold{Fraction=\dfrac{5}{9}}}$

Answer 2

Answer:

$\huge \rm \: solution$

$\small \sf \blue {let}$

Numerator = x

Denominator = 2x - 1

$\sf \: now$

Ratio of numerator to denominator

Then,

If we add 1 to numerator and denominator.

$\sf \: fraction \: = \frac{x +1}{2x - 1 +1} = \frac{3}{5}$

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

Cross multiplication

$\sf 5(x + 1) = 3 \times 2x$

$\sf \: 5x + 5 = 6x$

$\sf \: 6x - 5 x = 5$

${ \sf \bold {x = 5}}$

Numerator = 5

Now finding Denominator

$\sf \: denominator \: = 2x - 1$

$\sf \: denominator \: = 2 \times 5 - 1$

$\sf \: denominator \: = 10 - 1$

$\sf denominator \: = 9$

Let's verify

Case 1

$\sf \frac{x}{2x - 1} = \frac{5}{9}$

$\sf \: \frac{5}{2 \times 5 - 1} = \frac{5}{9}$

$\sf \: \frac{5}{10 - 1} = \frac{5}{9}$

$\sf \: \frac{5}{9} = \frac{5}{9}$

LHS = RHS

Hence case 1 is verified

Case 2

$\frac{5+1}{9+1} = 3 \ratio \: 5$

$\sf \: \frac{6}{10} = 3 \ratio \: 5$

$6 \ratio \: 10 = 3 \ratio \: 5$

$\sf \: 3 \ratio5 = 3 \ratio \: 5$

LHS = RHS

Hence Case 2 is verified

$\huge \bf \: fraction \: = \frac{5}{9}$