Question

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

Answer 1

Let the fraction be (x-3) / x

By the given condition, new fraction

= (x−3+2) / x +2

= x−1/ x+2

By the given condition

x -3 + x -1 = 29

x x +2 20

⇒20[(x−3)(x+2)+x(x−1)]=29(x²+2x)

⇒20[x²−x−6+x²−x]=29x²+58x

⇒20[2x²−2x−6]=29x²+58x

⇒40x²−40x−120−29x²−58x=0

⇒11x²−98x−120=0

⇒11x²−110x+12x−120=0

⇒11x(x−10)+12(x−10)=0

⇒(x−10)(11x+12)=0

⇒x=10 11x = -12

x = -12 /11

∴ the fraction is 7/10

Answer 2

$\large\underline{\sf{Solution-}}$

Let assume that denominator be x

So, numerator is x - 3

$\sf \: \sf \: \implies \: Fraction = \dfrac{x - 3}{x} - - - (1)$

When 2 is added to numerator and denominator, we get

Numerator = x - 3 + 2 = x - 1

Denominator = x + 2

$\sf \: \sf \: \implies \: Fraction = \dfrac{x - 1}{x + 2} - - - (2)$

According to statement,

$\sf \: \dfrac{x - 3}{x} + \dfrac{x - 1}{x + 2} = \dfrac{29}{20}$

$\sf \: \dfrac{(x - 3)(x + 2) + x(x - 1)}{x(x + 2)} = \dfrac{29}{20}$

$\sf \: \dfrac{ {x}^{2} - 3x + 2x - 6 + {x}^{2} - x}{ {x}^{2} + 2x} = \dfrac{29}{20}$

$\sf \: \dfrac{2{x}^{2} - 2x - 6}{ {x}^{2} + 2x} = \dfrac{29}{20}$

$\sf \: 20( {2x}^{2} - 2x - 6) = 29( {x}^{2} + 2x)$

$\sf \: {40x}^{2} - 40x - 120 = 29{x}^{2} +58x$

$\sf \: {11x}^{2} - 98x - 120 = 0$

$\sf \: {11x}^{2} - 110x + 12x - 120 = 0$

$\sf \: 11x(x - 10) + 12(x - 10) = 0$

$\sf \: (x - 10)(11x + 12) = 0$

$\bf\implies \:x = 10 \: \: or \: \: - \dfrac{12}{11} \: \{rejected \}$

So,

$\sf \: \bf \: \implies \: Fraction = \dfrac{7}{10}$

$\rule{190pt}{2pt}$

Additional Information

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac