Question

The mumerator of a fraction is 3 less than the denominator. If 2 is added to both
the numerator and the denominator, then the sum of the new fraction and the
original fraction is
Find the original fraction.​

Answer 1

Correct Question:

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is 29 / 20 Find the original fraction.

Answer:

The required fraction is 7 / 10.

Step-by-step-explanation:

Let the numerator of fraction be x.

And denominator of fraction be y.

∴ The fraction = x / y

From the first condition,

The numerator of a fraction is 3 less than the denominator.

∴ y = x + 3 - - - ( 1 )

From the second condition,

If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is 29 / 20.

∴ ( x / y ) + ( x + 2 ) / ( y + 2 ) = 29 / 20

⇒ ( xy + 2x + xy + 2y ) / ( y² + 2y ) = 29 / 20

⇒ ( 2xy + 2x + 2y ) / ( y² + 2y ) = 29 / 20

⇒ 20 ( 2xy + 2x + 2y ) = 29 ( y² + 2y )

⇒ 40xy + 40x + 40y = 29y² + 58y

⇒ 40 * x * ( x + 3 ) + 40x + 40 ( x + 3 ) = 29 ( x + 3 )² + 58 ( x + 3 ) - - - [ From ( 1 ) ]

⇒ 40 ( x² + 3x ) + 40x + 40x + 120 = 29 ( x² + 2 * x * 3 + 3² ) + 58x + 174

⇒ 40x² + 120x + 80x + 120 = 29 ( x² + 6x + 9 ) + 58x + 174

⇒ 40x² + 200x + 120 = 29x² + 174x + 261 + 58x + 174

⇒ 40x² + 200x + 120 = 29x² + 174x + 58x + 261 + 174

⇒ 40x² + 200x + 120 = 29x² + 232x + 435

⇒ 40x² + 200x + 120 - 29x² - 232x - 435 = 0

⇒ 40x² - 29x² + 200x - 232x + 120 - 435 = 0

⇒ 11x² - 32x - 315 = 0

⇒ 11x² - 77x + 45x - 315 = 0

⇒ 11x ( x - 7 ) + 45 ( x - 7 ) = 0

⇒ ( x - 7 ) ( 11x + 45 ) = 0

⇒ ( x - 7 ) = 0 OR ( 11x + 45 ) = 0

⇒ x - 7 = 0 OR 11x + 45 = 0

⇒ x = 7 OR 11x = - 45

⇒ x = 7 OR x = - 45 / 11

If x = - 45 / 11,

y = x + 3

⇒ y = ( - 45 / 11 ) + 3

⇒ y = ( - 45 + 33 ) / 11

⇒ y = - 12 / 11

And,

Fraction = x / y

⇒ Fraction = ( - 45 / 11 ) / ( - 12 / 11 )

⇒ Fraction = - 45 / 11 * 11 / - 12

⇒ Fraction = 45 / 12

As the numerator of this fraction is not 3 less than the denominator, x = - 45 / 11 is unacceptable.

∴ x = 7

∴ Numerator = 7

By substituting x = 7 in equation ( 1 ),

y = x + 3 - - - ( 1 )

⇒ y = 7 + 3

⇒ y = 10

∴ Denominator = 10

Now,

The fraction = Numerator / Denominator

⇒ The fraction = 7 / 10

∴ The required fraction is 7 / 10.