Question

If a÷b=c÷d=e÷f,prove that a^3+c^3+e^3÷b^3+d^3+f^3=ace÷bdf

Answer 1

With this setup (a/b=c/d=e/f) you can only ever express one variable in terms of 3 others, but I just wanted to put in a good word for one of my favourite little rules that sadly does not get much publicity today: The Rule of Threes.

Basically, if you have three numbers a, b, c, such that, a/b = c/d (also represented as ratios or proportions, a:b = c:d, or “a is to b what c is to d”) then you can calculate any one of them using the other 3 known values by multiplying on the diagonal that doesn’t contain the unknown, then dividing by the remaining number.

Here’s how it works:

ab=cd

Then you have the following:

a=b×cd , b=a×dc , c=a×db , etc.

(Easily proved by solving the standard way: bringing fractions to same denominator, then isolating your unknown, etc. — but it saves at least 2 steps in your calculations.)