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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?

basic division

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    $\begingroup$ it should give you $5$ instead. Check your long division. $\endgroup$ – Kenny Lau May 3 '17 at 16:41
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    $\begingroup$ Clear the denominators before doing the long division to make it easier, since $\frac{a}{b} = \frac{ac}{bc}$. So in this case $\frac{2.5}{0.5} = \frac{25}{5}$. $\endgroup$ – Nick May 3 '17 at 16:43
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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.

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  • $\begingroup$ Yujie why do you multiply 0.1 x 10? The 5/0.5 is 10 times,so do we multilpy by 10 to give the .1 its proper place value since the answer was 10? $\endgroup$ – jitterbug May 4 '17 at 0:25
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    $\begingroup$ @jitterbug Because that $5/0.5$ is done on the first decimal point, it is of one tenth, and that why if you do long divide mechanically, you get wrong answer. Essentially, you could understand it as $4.\{10\}$, and here the whole $\{10\}$ account for just one decimal point (like $4.9$, but here it's $10$ on that single place), and thus we need to advance it into $5$. $\endgroup$ – Yujie Zha May 4 '17 at 0:29
  • $\begingroup$ I thought so!! :) when I did it yesterday but I didnt know what to do with it, ok respect the 10 in place values!, thankyou Yujie. And so this mechanical long division actually throws curve balls I see to the beginner. no wonder I didnt get it years ago... thanks again for clarifying that. $\endgroup$ – jitterbug May 4 '17 at 0:39
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    $\begingroup$ @jitterbug yes, you were correct in your above comment, I just elaborated a bit more. No problem, yea, dividing non-integer number mechanically like this could be confusing, and hopefully this helps you to understand :) $\endgroup$ – Yujie Zha May 4 '17 at 0:42
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    $\begingroup$ @jitterbug My suggestion is always make it an integers for long divide, because the principal behind long divide is that for each digital position, we will not create "value overflow" (larger than 9), but with non integer, you kinda cannot guarantee that. I'm not sure if there are any other methodology other than long divide, but I'll say for complex division, just leverage calculators. $\endgroup$ – Yujie Zha May 4 '17 at 0:57
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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.

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  • $\begingroup$ Thanks Ethan, exactly thats what I was thinking, I thought I aced long division until I tried this for myself. obviously I dont know all the arithmetic or algebraic rules of it. I would do 25/5 as 5 into 2 goes 0 times. 2 minus 0 = 2. bring down the 5. Then 5 into 25 goes 5 times. so the answer is 5... can you explain step by step how to do the original division above without moving the decimal points? $\endgroup$ – jitterbug May 4 '17 at 0:34
  • $\begingroup$ Ethan its cool, I get it now, the .1 answer was actually 10 so I need to respect that place value. $\endgroup$ – jitterbug May 4 '17 at 0:40
  • $\begingroup$ Ethan thanks for your story I appreciated it, mm I had that notion a long time ago, sometimes I overthink it and very quickly lose track of the steps, i`m trying to dumb down my thinking, and at the same time, as you say the opposite it true, thinking things through ( I guess once I master the basics) will allow me to work with algorithms down the line, which is what I want to do!. .. best, thanks. $\endgroup$ – jitterbug May 4 '17 at 13:15
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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!)

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  • $\begingroup$ Thanks John! appreciated, I did try that method after and it worked, but I was stuck as to how this long division algorithm worked, I guess no one or any books ive read ever explained the steps and workings/conditions of the division, I always just saw it as divide into the next number,multiply the answer by the divisor and minus this from the dividend, bring down the next column and do the same, maintain your decimal point and place values. $\endgroup$ – jitterbug May 4 '17 at 0:30

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