maths- fractions

[masonbm]

<XMP>When I was at school, we learned</XMP><XMP>that a quick way of finding a fraction</XMP><XMP>between the value of two other </XMP><XMP>fractions</XMP><XMP>was simply to add together the</XMP><XMP>numerators of the two other fractions,</XMP><XMP>do the same for the two denominators</XMP><XMP>and put one figure over the other</XMP><XMP>e.g between 1/2 and 1/3 you have 2/5.</XMP><XMP>Does anyone know if this always works</XMP><XMP>- and why?</XMP>

[Gnu]

It is true, as long as the fractions are positive, but it gets a bit algebraic and tricky to set out clearly!

The fractions could be a/b and c/d, with a/b being less than c/d. A fraction formed in the way you've said is (a+c)/(b+d).

We need to show that (a+c)/(b+d) is larger than a/b i.e.

(a+c)/(b+d) > a/b

Assuming b and (b+d) are positive, we can cross multiply to get.

b(a+c)>a(b+d), or ba + bc > ab + ad

the ba and ab cancel, to give

bc > ad

dividing (as long as d and b are positive), this gives

c/d>a/b, which is true from our first assumption.

The same logic can be used to show that the new fraction is less than c/d.

Hope this helps!