manasan

New member
$$\displaystyle if, \frac{a}{b} = \frac{c}{d} = \frac{e}{f} \\ Show \ that, \frac{ 2a^4b^2 + 3a^2c^2-5e^4f }{ 2b^6 + 3b^2d^2-5f^5 } = \frac{a^2ce}{b^2df}$$
I tried making a,b,c as subject but it resulted in two variable equation which was unable to be plugged into the 2nd equation and arrive in single variable solution.
So I don't even know where to start actually?

Dr.Peterson

Elite Member
I would set $$\displaystyle \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k$$, and replace a, c, and e with expressions in terms of b, d, and f. See what happens.

manasan

New member
Thank You I solved, but is there any other short way to solve.
$$\displaystyle By\ substituting\ {\text{'k'}}\ I\ got\ \frac{2 k^{2} a^2 b^{4}+3 k^{4} b^{2} d^{2}-5 k^{3} f^{4} e}{2 a^{6} k^{-6}+3 k^{4} a^{2} c^{2}-5 e^{5} k^{-5}}=k^4\\ \text{and I divided by}\ k^4\ \text{and solving lead me to same denominator and numerator on one side and 1 on the other side.}$$
Is this the shortest route or are there any better methods?
What should I do to tackle these kinds of problems easily and quickly next time?

Dr.Peterson

Elite Member
This is basically a standard technique when you are given a proportion, especially one like this one.

It doesn't look like you did quite what I suggested, so maybe your work was harder than mine. I replaced a with bk, c with ck, and e with fk everywhere on both sides, and both sides easily become $$\displaystyle k^4$$. I'm also not sure how you proceeded from what you show.

Try doing what I suggested, and show me your work if it isn't really easy, so I can make suggestions.