This has been bugging me for some time now, so I ask you to try to help me realize what is going on here. I just can't get my brain around this. I have a proper fraction and a negative integer. The fraction is added to the integer.
$$-100+\frac{1}{2}$$
This admittedly looks like a very simple problem, something very trivial. My first reaction when I see a problem like this is to rewrite the integer as a fraction and then multiply the numerator and the denominator so that I have the same denominator in both fractions. Adding fractions requires the denominators to be the same.
$$-100+\frac{1}{2}=\frac{-200}{2}+\frac{1}{2}=\frac{-200+1}{2}=\frac{-199}{2}=-99,5$$
This is a legitimate statement, and the equality still holds. But I cannot reduce the improper fraction -199/2 any more, it is already in its simplest form. But I should be able to write this as a mixed number.
$$-99\frac{1}{2}$$
This is where my brain work stops. How do I get this mixed number? Does it matter that I can't reduce the fraction -199/2 any more? I think this is what might be bugging me all this time, since it won't go into it evenly and I get a remainder. I am not used to that when writing results as mixed numbers.
I can rewrite this mixed number like this.
$$-99\frac{1}{2}=-(99+\frac{1}{2})=-99-\frac{1}{2}=-99,5$$
As you can see, the value of it, expressed in decimal form, is still the same. But I just don't see the connection between $-100+1/2$ and the mixed number $-99\frac{1}{2}$.
So how do you deal with this? Please lead me through it step by step.
Update
I thank you all for the help you have given me. It is very much appreciated! I can finally rest my mind knowing that I have cracked this nut.
Just to let you know, the original problem was dealing with inequalities. "Check that the given number is a solution to the corresponding inequality." Problem number 4 c was to check that $y=40$ satisfies the inequality $-\frac{5}{2}y + \frac{1}{2} < -18$.
$$-\frac{5}{2}y + \frac{1}{2} < -18 | y=40$$
$$-\frac{5}{2}40 + \frac{1}{2} < -18$$
$$-\frac{5}{1}20 + \frac{1}{2} < -18$$
$$-100 + \frac{1}{2} < -18$$
$$-99 \frac{1}{2} < -18$$
This problem was presented by Sal over at Khan Academy. He is usually very thorough when explaining how to solve some equation, or inequality in this case. He will usually explain every step. But in this video he was trying to squeeze in as many problems as possible in the least amount of time. So I guess he skipped a step or two. But I didn't understand how he got that mixed number on the left side. Now I do! Thank you!
Now, the way I understand it, there are two ways to get from $-100+\frac{1}{2}$ to $-99\frac{1}{2}$. One way is to get the improper fraction and do the long division. Another way, or the "trick" that I think Sal uses here, is to split the numerator into two numbers.
Long division
$$-100+\frac{1}{2}=\frac{-100}{1}+\frac{1}{2}=\frac{-100 \cdot 2}{1 \cdot 2}+\frac{1}{2}=\frac{-200}{2}+\frac{1}{2}=\frac{-200+1}{2}=\frac{-199}{2}=-99\frac{1}{2}$$
When you arrive at $\frac{-199}{2}$ you need to do the long division. You divide 2 into 199. And you get that 2 goes into 199 evenly 99 times with the remainder 1. So 99 is the whole number part and the fraction is 1/2 where remainder 1 is the numerator and 2 is the denominator. The whole thing is of course negative so you put a negative sign in front of it.
Splitting the numerator
$$-100+\frac{1}{2}=\frac{-200}{2}+\frac{1}{2}=\frac{-199}{2}=\frac{-198-1}{2}=\frac{-198}{2}-\frac{1}{2}=-99-\frac{1}{2}=-(99+\frac{1}{2})=-99\frac{1}{2}$$
So the trick here is to rewrite or to "split" -199 in the numerator into two different numbers, so that one of them becomes even (divisible by 2) and when you sum them up you get -199.
You can add -1 to -198 and it becomes -199. Adding a negative is the same thing as subtracting the positive of that number, so this really becomes -198-1 which is indeed -199. So you are not changing the value of the numerator, so this is safe to do.
$$-199=-198+(-1)=-198-1=-199$$
What this allows you to do is to write the two numbers as two separate fractions.
$$\frac{-198-1}{2}=\frac{-198}{2}+\frac{-1}{2}=\frac{-198}{2}-\frac{1}{2}$$
Since one of them is an improper fraction and is divisible by the denominator you have, which is 2 in this case, you can simplify the expression by dividing the improper fraction, and since you made sure that one of the numbers is divisible by 2 it will go into it evenly. And you are left with a whole number and a proper fraction.
$$\frac{-198}{2}-\frac{1}{2}=-99-\frac{1}{2}$$
There are also other numbers you could add together and get the sum -199. But remember, one of them must be even so you can divide it by 2. You could add -3 to -196. Adding -3 to -196 is the same thing as subtracting 3 from -196 which is also -199.
$$-199=-196+(-3)=-196-3=-199$$
So it's the same thing here.
$$\frac{-196-3}{2}=\frac{-196}{2}+\frac{-3}{2}=\frac{-196}{2}-\frac{3}{2}$$
Sure, -196 is divisible by 2, but the problem here is that you also have the improper fraction $\frac{-3}{2}$.
$$\frac{-196}{2}-\frac{3}{2}=-98-\frac{3}{2}$$
So what this tells you is that you can divide 2 into -196 one more time. So notice that $-99\frac{1}{2}$ is the same thing as $-98\frac{3}{2}$. And if you keep on doing this you will see that this is the same thing $-97\frac{5}{2}$ and this in turn is the same thing as $-96\frac{7}{2}$ and so on.
$$-99\frac{1}{2}=-98\frac{3}{2}=-97\frac{5}{2}=-96\frac{7}{2}=-95\frac{9}{2}=-94\frac{11}{2}$$
As you can see a pattern is emerging here. But the simplest of these numbers is the $-99\frac{1}{2}$ and this is the preferred answer for a mixed number.
Wow! That was much more than I wanted to write. I just thought I would share my findings, but I got carried away. The bottom line is that you can handle -100+1/2 like a regular fractions problem. And when you arrive at -199/2 there are two paths you can take. You can either do the long division and write the remainder as the fractional part of the mixed number, or you can rewrite the non-even numerator as a sum of one even number and one non-even number (the remainder). That was the trick I guess, that Sal pulled on me.
It's worth noting how the numerator of the fractional part of the mixed numbers in my last example are all odd (non-even) numbers like 1, 3, 5, etc.
It's problems like these where I have to think hard that really make my math brain grow. Hopefully someone will find this "essay" helpful.
Best Answer
Please lead me through it step by step. \begin{align*} -100+\frac{1}{2} &= \frac{-200}{2}+\frac{1}{2}\\ &= \frac{-200+1}{2} \\ &=\frac{-199}{2}\\ &=-\frac{199}{2}\\ &= -\frac{198+1}{2}\\ &= -\left(\frac{198}{2}+\frac 12\right)\\ &= -\left(99+\frac{1}{2}\right)\\ &= -99\frac12 \end{align*}