0

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

Lino
  • 23
  • The $(3 + \frac 14) - (2 \color{red}{-} \frac 78)$ part. In particular, $-2-\frac 78$ equals $-(2\color{red}+\frac 78)$, not $-(2\color{red}-\frac 78)$. This is due to the distributive property. –  Nov 22 '16 at 21:49
  • Thank you. Can you tell me why the first time I added brackets I didn't use the distributive property and everything is working fine? – Lino Nov 22 '16 at 22:00
  • If by "the first time" you mean going from $3 + \frac 14 - 2 + \frac 78$ to $(3 + \frac 14) - (2 + \frac 78)$, then it didn't work. The first expression is incorrect -- it's the second that is correct. The first is equal to $3\frac 14 -1\frac18 = 2\frac 18$ while the second is equal to $3\frac 14-2\frac 78=\frac 38$. –  Nov 22 '16 at 22:04
  • Rule of thumb: Whenever you have $a\frac bc$ it always means $(a+\frac bc)$. If you have $-a\frac{b}{c}$ then it means $-(a+\frac bc) = -a-\frac bc$. –  Nov 22 '16 at 22:11
  • Thank you, I think I understand now. And, I guess, $ -a \frac bc $ always means $ -(a + \frac bc)$ ? – Lino Nov 22 '16 at 22:17
  • That's correct. –  Nov 22 '16 at 22:17

2 Answers2

0

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.

Arthur
  • 199,419
  • Thank you for your answer. Why it is not correct? I have done the same in the line before CASE1. (added brackets and didn't change any sign). I can see the difference, I don't understand what is the error, I have done the same thing as before – Lino Nov 22 '16 at 21:48
  • Well, note that $3 \frac 14 - 2 \frac 78$ is actually $(3 + \frac 14) - (2 + \frac 78)$. When you wrote 3 + \frac 14 - 2 + \frac 78$, that was not really correct either. – Arthur Nov 22 '16 at 21:52
  • Thank you, I think I have now understood! – Lino Nov 22 '16 at 22:19
0

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, $-(2-1) = -2 + 1$.

fuglede
  • 6,716
  • Thank you for your answer. I understand, but when I added brackets the first time everything was fine. Then when I have added them the 2nd time it's an error, why? I cannot see any difference in these two steps. – Lino Nov 22 '16 at 21:53
  • As @Arthur noted, that only worked because there's also an error in the second line, which happens to exactly cancel the error in the third line. – fuglede Nov 22 '16 at 21:57
  • I have finally understood everything. Thank you so much for the help – Lino Nov 22 '16 at 22:33