Adding/Removing round brackets question

Question

I have these mixed numbers operation:

$ 3 \frac 14 - 2 \frac 78 $

I can re-write this to:

$ 3 + \frac 14 - 2 + \frac 78 $

Now, I have some issues with the negative sign, that's the part I am interested.

I add round brackets to make things more clear as shown below

$ (3 + \frac 14) - (2 + \frac 78) $

CASE 1 I solve the brackets and this would result in:

$ (\frac {12}{4} + \frac 14) - (\frac {16}{8} + \frac 78) $

$ \frac {13}{4} - \frac {23}{8} $

CASE2 instead of solving the brackets let's assume I just want to remove the brackets,so I change the signs where needed and I will have:

$ 3 + \frac 14 - 2 - \frac 78 $

Assuming everything done is correct (so far), I now want to add brackets again, to make things clear (Sorry, I am silly and I want to put them back again).

$ (3 + \frac 14) - (2 - \frac 78) $

this result in a completely different result:

$ (\frac {12}{4} + \frac 14) - (\frac {16}{8} - \frac 78) $

$ \frac {13}{4} - \frac {9}{8} $

Where I am doing wrong? :-(

Thank you

Answer 1

When you add brackets, you need to be careful about the signs: Consider the difference between $-2 -1$ and $- (2 - 1)$. The first one equals $-3$, but the second one equals $-1$.

Instead, the rule is that you can remove the brackets when there's a minus in front of them by changing the sign of everything in the brackets. That is, $-(2-1) = -2 + 1$.

Answer 2

You went from $$(3 + \frac 14) - (2 + \frac 78)$$ to $$3 + \frac 14 - 2 - \frac 78,$$which is correct, but then you change it to $$(3 + \frac 14) - (2 - \frac 78),$$ which is not correct. See that the only difference between this last one and the first one is that you've changed $+\frac78$ into $-\frac78$. I hope you can see that those two aren't the same.