Question

The denominator of a fraction is two more than its numerator. If the sum of the fraction and its reciprocal is find the
fraction,
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Answer 1

Answer:

please tell what is their sum

Step-by-step explanation:

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Answer 2

The Fraction for the given statements is $\dfrac{5}{3}$  and  $\dfrac{3}{5}$  

Step-by-step explanation:

Given as :

Let The numerator = x

Let The denominator = y

So, The Fraction = $\dfrac{numerator}{denominator}$

Or, The Fraction = $\dfrac{x}{y}$

According to question

Statement I

The denominator of a fraction is two more than its numerator.

i.e    denominator = 2 + numerator

i.e     y = 2 + x              ...........      1

Statement II

If the sum of the fraction and its reciprocal = $\dfrac{34}{15}$

∵     Fraction = $\dfrac{x}{y}$      and   Reciprocal = $\dfrac{y}{x}$

So,  Fraction +  Reciprocal =  $\dfrac{34}{15}$

i.e   $\dfrac{x}{y}$  +  $\dfrac{y}{x}$ = $\dfrac{34}{15}$

Or,  $\dfrac{x^{2}+ y^{2} }{xy}$ = $\dfrac{34}{15}$

Or,  $\dfrac{x^{2}+ (2+x)^{2} }{x(2+x)}$ = $\dfrac{34}{15}$

Or,  $\dfrac{x^{2}+ 4+x^{2}+4x }{2x+x^{2} }$   =  $\dfrac{34}{15}$

By cross multiplication

15 ( 2 x² + 4 x + 4 ) = 34 ( x² + 2 x )

Or,   30 x² + 60 x + 60  = 34 x² + 68 x

Or, 34 x² + 68 x - 30 x² - 60 x - 60  = 0

Or,  4 x² + 8 x- 60 = 0

Or,  x² + 2 x- 15 = 0

Or,  x² + 5 x - 3 x - 15 = 0

Or,  x ( x + 5 ) - 3 ( x + 5 ) = 0

∴    ( x + 5 ) ( x - 3 ) = 0

So,  ( x + 5) = 0         and   ( x - 3 ) = 0

i.e      x = - 5             and    x = 3

Put the value of x into eq 1

y = 2 + x  

For x = - 5      ,   y = 2 - 5 = - 3

For  x = 3        ,   y = 2 + 3 = 5

So, The Fraction = $\dfrac{x}{y}$  = $\dfrac{-5}{-3}$  = $\dfrac{5}{3}$

And                        $\dfrac{x}{y}$ = $\dfrac{3}{5}$

Hence, The Fraction for the given statements is $\dfrac{5}{3}$   ,  $\dfrac{3}{5}$   Answer