Maths

Fizban140

Just don't understand the mystery rule here, I got the asnwer in the back of the book I just don't know why they did it like this.

1+1/x

---

1/x

Why doesn't the 1/x just cancel out the top one to make it one? I guess the book wants me to do x/1 (1/x) but I am not sure why you don't cancel it out. Obviously both can't work since the former gives me 1 and the latter 1+x

ceres

The bottom one can't cancel the top one because the top one is being added and is not a factor. You can only cancel factors.

minirhyder

1 + 1/x is a term
If you're talking about canceling something the entire term must possess the element you want to cancel. So in this case you would only be able to cancel out the 1/x if the top term was 1/x(1+1/x) or 1/x + (1/x)*(1/x)

Al_wat

you can't cancel it out because that operation you wrote there is essentially adding fractions, not necessarily just dividing.

Basically -> [1 + (1/x) ] / (1/x) = [1 / (1/x) ] + [(1/x) / (1/x)]

Does that make sense?

from here you can see that [(1/x) / (1/x)] = 1 and you can figure out that [1 / (1/x) ] = x

and thus this = x+1

does this help at all?

Al_wat

like heres an example that might be easier to read.

A + B

---

C


this expression is the same as

A + B
--- ----
C C




To us an example that is more of the form of your question; it would be like this:

A + C

---

C
=

A + C
--- ----
C C

minirhyder

I'll try again Al_wat's terminology:

A + B

---

B

You cannot cancel out because both of the terms in the numerator must have the term you're trying to cancel out; these are just the rules of math.
Think of it as doing the opposite of the distributive property [ie A(B+C) = AB + AC], that is, in order to cancel out an element from the term, every single element of that term must have the element which you are trying to factor out or cancel out. For instance if you had

AB + CB

---

B

You can cancel out the B and end up with A + B
You can also think of this as factoring B out of the expression and turning it into B(A+C)

However in the example you have and in this example

A + B

---

B

You cannot do so because the entire term does not have B in it. If you were to try to factor out B from the term A + B you would be unable to do so, and that is your problem.

dobi

Here's the full solution, if you're interested in figuring out exactly what happens in it. It's got some pretty important algebra rules in it.

soxbox

That's an awfully convoluted solution - the inverting fraction knowledge used in the second last step could have been used at step one (ie, if you know the dividing by a fraction is the same as multiplying by an inverse, you can just do that straight off and get x * (1 + 1/x) as step 2).

The key thing to know here is that if you multiply the numerator and the denominator by the same thing, the fraction remains unchanged (because x/x = 1 and multiplying by 1 changes nothing)

(1 + 1/x) / (1/x)

= (x * (1 + 1/x)) / (x * (1/x))

= (x + 1) / (1)

= x + 1

Fizban140

soxbox wrote: »

That's an awfully convoluted solution - the inverting fraction knowledge used in the second last step could have been used at step one (ie, if you know the dividing by a fraction is the same as multiplying by an inverse, you can just do that straight off and get x * (1 + 1/x) as step 2).

The key thing to know here is that if you multiply the numerator and the denominator by the same thing, the fraction remains unchanged (because x/x = 1 and multiplying by 1 changes nothing)

(1 + 1/x) / (1/x)

= (x * (1 + 1/x)) / (x * (1/x))

= (x + 1) / (1)

= x + 1

That makes a lot of sense, thanks for the responses everyone.