How do I get to the answer (1/(r-1))from (r/(r-1))-((r^2)/(r^2))?

Question

I've been attempting to solve this problem for a while, and I can't seem to be able to simplify (r/(r-1))-((r^2)/(r^2)) into (1/(r-1))*.

 

So far I've gotten the GCF (r^3-r^2) in the denominator, but then when I solve that way, I end up with (1/(r^2))

 

Is there something I've missed in the solving, or am I just going about this completely wrong?

 

*Note: I was given this exact answer by multiple math calculators

Answer 1

r / (r-1) - 1

 

r / (r-1) - (r-1) / (r-1)

 

r - r - 1 / (r-1)

 

-1 / (r-1)

 

1 / 1-r

 

Answer 2

Finn,

 

If I'm reading the problem correctly, is it:

 

1/(r-1) - r²/r² ?

 

So, listen, I'm totally cool with the common denominator approach (and oh, I think you mean the LCD, not the GCF...not that I'm a stickler for vocabulary, but I'd at least like to make sure I'm following you!).  But, I'm wondering if there's any rule that says that you can't simplify the r²/r² part first?  Isn't that, like, 1?

 

Now, even when you do that first, you still need to find the LCD, but it should be a bit more simple, and maybe your error will stand out to you as you go through the simplification.  Go ahead and try it.

 

I'm guessing, when you go your route, it's when you subtract the fractions that some sort of sign is getting goofed up along the way.  My first step, to make the denominators the same, looks like this:

 

1*(r²) / (r²)(r-1) - (r²)(r-1)/(r²)(r-1)

 

Sorry, I'm being a bit sloppy with the parentheses, but I think you get the idea of what is grouped with what.

 

I left the denominator factored, because hopefully, in the end, I'd be able to cancel one of the factors out (and in fact, because you know the answer, you might even have a guess as to which factor is going to go away).

 

Does any of that give you enough of a nudge?

 

Let us know,

 

-- Michael

Answer 3

Hi Finn,

 

To make this a little easier, note that r²/r²=1. So your problem becomes:

 

r/(r-1)-r²/r² = r/(r-1) - 1

 

Now, the common denominator would be r-1, so we can rewrite the above as:

 

r/(r-1) - 1 = r/(r-1) - 1*(r-1)/(r-1)

 

               = r/(r-1) - (r-1)/(r-1)

 

               = (r-(r-1))/(r-1)

 

This should get you to the answer