strange sounding problem

[Bryn]

When 1 is added to the numerator and the denominator of the fraction m/n the new fraction is 3/2. When one is subtracted from the numerator and denominator of the fraction m^2/n^2 the new fraction is 21/8. Find the possible values of m and n.

The values i get for n and m are surds, but the answer is apparently m=8 and n=5, which do work. I wont post my working cos i'm pretty sure i'm totally off track. Any help plz?

[dave]

From the definition of the problem, we have a pair of simultaneous equations:

\frac{m+1}{n+1}=\frac{3}{2} and \frac{m^2-1}{n^2-1}=\frac{21}{8}.

We also have to assume that n\neq-1 otherwise it doesn't work.

Now by simply cross multiplying and rearranging the equations generally we get:

2m-3n=1 which implies m=\frac{1}{2}(1+3n) from the first equation and 21n^2-8m^2=13 from the second.

By substution, therefore,

3n^2-12n-15=0. The solutions for this are n = 5 or n = -1. But we stated in the question that n can't be -1, so disregard this answer. So n=5, which means m = 8.

[Bryn]

ah ok, i was on the right track afterall. I just forgot to square what i subsiuted in for m, i just multiplied by 8 and then used the quadratic formula on the resulting equation, which gave me the surd.

[dave]

That would cause problems :-)