The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is
2 9/10(mixed fraction).find the fraction
Answers
Answer:
Fraction discussed here is 5 / 2.
Step-by-step explanation:
Let the denominator of the required fraction be b and numerator of that same be a.
Now the required fraction is a / b.
As mentioned, denominator of this fraction is one more than twice of the numerator.
It means : Denominator of this fraction = 1 + 2( numerator of this fraction ) = > b = 1 + 2a or 2a + 1.
Now, substituting the value of b in a / b.
= > a / b
= > a / ( 2a + 1 )
Given, sum of the fraction and its reciprocal is 2 9/10 or 29 / 10.
It means : a / ( 2a + 1 ) + ( 2a + 1 ) / a = 29 / 10
= > a / ( 2a + 1 ) + ( 2a + 1 ) / a = 29 / 10
= > { a^2 + ( 2a + 1 )^2 } / a( 2a + 1 ) = 29 ) / 10
= > { a^2 + 4a^2 + 1 + 4a }10 = 29a( 2a + 1 )
= > 50a^2 + 10 + 40a = 58a^2 + 29a
= > 58a^2 - 50a^2 + 29a - 40a - 10 = 0
= > 8a^2 - 11a - 10 = 0
= > 8a^2 - 16a + 5a - 10 = 0
= > 8a( a - 2 ) + 5( a - 2 ) = 0
= > ( 8a + 5 )( a - 2 ) = 0
= > a = - 5 / 8 or 2.
Thus the required fraction is :
Case 1 : Value of a is 2.
= > a / ( 2a + 1 )
= > 2 / { 2( 2 ) + 1 }
= > 2 / ( 4 + 1 )
= > 2 / 5
Case 2 : Value of a is - 5 / 8
= > ( - 5 / 8 ) / { 2( - 5 / 8 ) + 1 }
= > - 5 / ( - 10 + 8 )
= > 5 / 2
Hence the fraction discussed here is 5 / 2.