Question

19. The numerator of a fraction is 1 less than the denominator.

When both numerator and denominator are increased by 2,
the fraction is increased by 1/12Find the original fraction.

Answer 1

Let the original fraction be

`(n-1)/(n)`

Now let's increase both numerator and denominator by 2:

`(n+1)/(n+2)`

This produces a 1/12 increase:

`(n+1)/(n+2)-(n-1)/(n)=(1)/(12)`

The left hand side can be rearranged as

`(2)/(n^2+2n)=(1)/(12)`

Invert both sides:

`(n^2+2n)/(2)=12 \iff n^2+2n=24`

Solve the quadratic equation:

`n^2+2n-24=0 \iff n=-6 \lor n=4`

So, in the first case, the original fraction is

`(n-1)/(n)=(-6-1)/(-6)=(7)/(6)`

In the second case, we have

`(n-1)/(n)=(4-1)/(4)=(3)/(4)`