# Simple algebra question

## Main Question or Discussion Point

I am trying to figure out an overall reaction constant, but I am getting stuck on the math behind the problem. I am at this point in the problem:

[Co(H2O)6] = [Co] x (k[H+]/(1+k[H+]))
And I am trying to mathematically get to
[Co(H2O)6] = [Co] x (1-(1/(1+[H+]k)))

When you ignore the chemistry, the equation is the same as:

a = b x (cd/(1+cd))

going to

a = b x (1-(1/(1+cd)))

How do you factor out the cd from the numerator?

Thanks!

Hurkyl
Staff Emeritus
Gold Member
It wasn't factored out. (If it was factored out, it would still be there, but outside. :tongue2:)

Try going the other direction: what sort of things can you do to the latter equation?

So if I work backwards, the step before 1-(1/(1+cd)) could have possibly been
[(1+cd)/(1+cd)] - (1/(1+cd)). What would have led me to that step or is that not the correct step backwards?

Wait a minute! I think a lightbulb just went off...

[(1+cd)/(1+cd)] - (1/(1+cd)) This actually reduces back to cd/(1+cd), right? Because the 1-1=0 and cd is the only thing left in the numerator. So what would prompt me to know that I should change cd/(1+cd) into [(1+cd)/(1+cd)] - (1/(1+cd))?

Hurkyl
Staff Emeritus
Gold Member
Well, whether you should change it is another question entirely.

But to figure out how to change it, you can simply use the division algorithm.

You're dividing cd by (cd + 1)...

How many times does (cd + 1) go into cd? 1 time, with remainder -1.

So, cd/(cd+1) = 1 + (-1)/(cd+1)

(when figuring out how many times, you just look at the "biggest" terms -- in this case, we consider any term involving "cd" to be bigger than any term that is just a number)

Another method is to look at it and think "Hrm, I can probably write that as A + B/(cd+1)", and then try to solve the equation "A + B/(cd+1) = cd/(cd+1)"

A third way is to learn how to do what you just did in the opposite direction!

Last edited:
Thanks so much!