Question

The sum of numerator and denominator of a fraction is greater by 1 than thrice the numerator is decreased by 1 then the fraction reduced to 1/3 find the fraction​

Answer 1

Given:

The sum of the numerator and denominator of a fraction is greater by 1. Than thrice the numerator.

If the numerator is decreased by 1 then the fraction reduces to 1/3.

Find:

The fraction.

Solution:

Let the numerator of the fraction be 'x' and the denominator of the fraction be 'y'.

So, the fraction will become.

Fraction = Numerator/Denominator = x/y

The sum of the numerator and denominator of a fraction is greater by 1. Than the numerator.

=> x + y = 3x + 1

=> -1 = 3x - x - y

=> -1 = 2x - y

=> y = 2x + 1 .......(i).

If the numerator is decreased by 1 then the fraction reduces to 1/3.

=> x - 1/y = 1/3

=> 3(x - 1) = y

=> 3x - 3 = y

Putting the value y from Eq (i). in Eq (ii).

=> 3x - (2x + 1) = 3

=> 3x - 2x - 1 = 3

=> 3x - 2x = 3 + 1

=> x = 4

Putting the value of x in Eq (i).

=> 2x - y = -1

=> 2(4) - y = -1

=> 8 - y = -1

=> -y = -1 - 8

=> -y = -9

=> y = 9

So, fraction = x/y

=> 4/9

Hence, the fraction is 4/9.

I hope it will help you.

Regards

Answer 2

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

⠀⠀⠀⠀______________________________

$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

• The sum of numerator and denominator of a fraction is greater by 1 than thrice the numerator.

⠀⠀⠀⠀

$:\implies\sf x + y = 3x + 1$

$:\implies\sf x - 3x + y = 1$

$:\implies\sf -2x + y = 1$

$:\implies{\boxed{\sf{y = 1 + 2x}}}\qquad\qquad\sf\bigg[ eq\:(1) \bigg]$

Also,

• If numerator is decreased by 1 then the fraction reduced to 1/3.

⠀⠀⠀⠀

$:\implies\sf \dfrac{x - 1}{y} = \dfrac{1}{3}$

$:\implies\sf 3(x - 1) = y$

$:\implies\sf 3x - 3 = y$

$:\implies{\boxed{\sf{y = 3x - 3}}}\qquad\qquad\sf\bigg[ eq\:(2) \bigg]$

⠀⠀⠀⠀______________________________

Now, From eq (1) and eq (2),

⠀⠀⠀⠀

$:\implies\sf 1 + 2x = 3x - 3$

$:\implies\sf 2x - 3x = - 3 - 1$

$:\implies\sf - x = - 4$

$:\implies{\underline{\boxed{\sf{\purple{x = 4}}}}}\;\bigstar$

⠀⠀⠀⠀______________________________

Putting value of x in eq (1),

⠀⠀⠀⠀

$:\implies\sf y = 1 + 2 \times 4$

$:\implies\sf y = 1 + 8$

$:\implies{\underline{\boxed{\sf{\pink{y = 9}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Hence,\:the\:fraction\:is\: \bf{ \dfrac{4}{9}}.}}}$