cant understand this basic division My long division of $0.5$ into $2.5$ gives the answer $4.1$ , but $0.5 \times 4.1$ does not equal the dividend of $2.5?$ so what gives here? 

 A: Notice that in your long division, you are dividing $.5$ for each position. You did it so for the first position and you get $4$.
However, when you do it for the second position (first decimal point), it should be $5/0.5=10$, not $1$ as you get. Thus it should be $4+0.1 \times 10 =5$ instead.
Suggestion: better to always put the denominator into integer when doing long division (i.e. $25 \div 5$), and it'll be much less error-prone.
A: The correct answers here tell you how to do the problem right, but don't seem to address where you went wrong. I think you were thinking:

The $.5$ goes into the $2$ four times, so write down $4$ over the $2$.
The $.5$ goes into the $.5$ one time, so write down $.1$ over the
  $.5$.

But that's not how the division algorithm works. You can't just "write down the numbers" that way, you have to follow the rules for what you write and when: multiply by the answer digit you've just found, subtract the product from what's left in the thing you're dividing, ...
You can see your mistake clearly if you try dividing $25$ by $5$ with your "rule". It would tell you to write $4$ over the $2$ for the number of times $5$ goes into $20$ and then $1$ over the $5$ for the number of times it goes into $5$, for an answer of $41$.
Edit:
In fact, most of what you were thinking was just right. There are four $0.5$'s in $2$ and one more in $0.5$ for a total of $5$ - the correct answer.
Your error was in "writing down the numbers in the right places".
Here's the moral to the story. There are often two ways to solve a simple math problem. One is to think about it. The other is to pretend you are a computer and follow rules you may or may not understand. (The computer surely does not understand them.) If you choose the second method you have to stop thinking and just follow the rules. Don't mix the two strategies.
Sometimes the algorithm is best. Thinking is hard. But in the long run learning to think is more useful. Someone has to think the problems through to invent the algorithms and program the computers. If you end up a mathematician that kind of thinking will be your job. And you'll solve hard problems just for fun.
A: I was taught to move the decimal points of the dividend (under the bar) and divisor (at the left) in tandem until the divisor was a whole number. Then, I could pull where the decimal point ended up on the dividend directly to the top of the bar, and it would be in the correct place. Then, I'd divide normally.
(What this does is it multiplies the top and bottom each by $10$ every time the decimal points are moved. This doesn't change the answer.)
But for this one, if you move the decimal of both dividend and divisor one place to the right, you get $25 \div 5$, which you should be able to do in your head (or maybe have even memorized!)
