Question

The denominator of a fraction in the simplest form is greater than the numerator by 1. If 4 is added to the numerator, and 3 is subtracted from the denominator, then the fraction itself is increased by 2 1/6 . Find the original fraction.

Answer 1

The original fraction is equal to
`(5)/(6)`

Explanation:

Let

The fraction in the simplest form equal to

`(x)/(y)`

we know that

The denominator of a fraction in the simplest form is greater than the numerator by 1

so

`y=x+1` ----> equation A

If 4 is added to the numerator, and 3 is subtracted from the denominator, then the fraction itself is increased by 2 1/6

Remember that

`2(1)/(6)=2+(1)/(6)=(13)/(6)`

so

`(x+4)/(y-3)=(13)/(6)+((x)/(y))` ----> equation B

substitute equation A in equation B

`(x+4)/(x+1-3)=(13)/(6)+((x)/(x+1))`

solve for x

`(x+4)/(x-2)=(13)/(6)+((x)/(x+1))`

`(x+4)/(x-2)=(13(x+1)+6x)/(6(x+1))`

`(x+4)/(x-2)=(19x+13)/(6(x+1))`

Multiply in cross

`6(x^2+x+4x+4)=19x^2-38x+13x-26`

`6(x^2+x+4x+4)=19x^2-25x-26`

`6x^2+30x+24=19x^2-25x-26`

`19x^2-6x^2-25x-30x-24-26=0`

`13x^2-55x-50=0`

solve the quadratic equation by graphing

The solution is x=5

therefore

Find the value of y

`y=5+1=6`

The original fraction is equal to
`(5)/(6)`