Can someone give me a real life example of division in practice?? How does that work?
In: Mathematics
Long division is just taking the divisor and multiplying it so it is at most as big as the dividend. Then you subtract and repeat the process until you are left with a number smaller than your divisor.
You do it all the time without thinking about it. Think about how you know how many of each coin you need when calculating exact change.
Trying to format a long division in this editor is going to be hellish – I’d suggest searching ‘long division example’ on YouTube, maybe.
If you want to divide 70000 by 6, it helps to break down the 70000 into parts that you can divide by 6 using mental math alone. It’s easy to verify that 70000 = 60000 + 6000 + 3600 + 360 + 36 + 4. And it’s easy to compute that (60000 + 6000 + 3600 + 360 + 36)/6 = 10000 + 1000 + 600 + 60 + 6.
So 70000/6 = 11666 with a remainder of 4 that couldn’t be divided by 6.
Long division is the technique we use to break the number apart for easier division.
The first step of writing out long division breaks out 60000, shows us we still have 10000 to go, and puts a 1 up as the confirmed first digit of the answer:
1
____
6 | 70000
6
__
10000
I can explain more, but you’d be better served, as another commenter said, watching a video or another form of learning this than a reddit comment.
If you have eight oranges and three people and were to divide them evenly amongst those people, how many whole oranges would each person get? Each person would get two oranges and there would be two oranges off to the side.
Now what if you wanted to use up all the oranges and have none left over? Then what you could do is take those two spare oranges and slice them each into thirds. One third of each orange to each person. So now each person has two and two-thirds oranges.
Long division works by utilising all your other arithmetic skills – multiplication, subtraction, addition, and simple division … and knowing how to apply them in solving a long division problem.
Take the example of 1/7 (one divided by seven) … you could also use 1 billion (1,000,000,000) divided by seven since we’re going to be using a lot of zeroes, and as long as we remember where we want a decimal point later it doesn’t really matter if it’s one or one billion.
First, make a list of multiples of your divisor (7) from 0*7 to 9*7 (0, 7, 14, 21, 28, 35, 42, 49, 56, & 63). This might seem simplistic, but when you have a bigger divisor it will be more useful.
Write the dividend (the number to be divided) as 1.0000000000 … or as many trailing zeroes as you like. Then select the largest number from your list of divisors that you can subtract from the dividend. … in this case it will be 0*7, because 1*7 is obviously larger than 1. Because our first subtraction was ‘0’ we place a 0 then a decimal point, as we know our answer will be less than 1
Whenever you cannot subtract a multiple of 7 you ‘borrow’ one of those zeroes, and add it to your remaining dividend … in this case that would give us ’10’ … we can now subtract 1*7 from 10, so we take that 1 and add it to the end of the ‘0.’ from before, giving 0.1
After having subtracted 1*7 from 10 we are left with a ‘3’, which we cannot wholly divide by any of our divisors, so we ‘borrow’ another ‘0’ which gives us ’30’ … and from 30 we can subtract a maximum of 4*7 (28) so we add the ‘4’ to our answer, which is now 0.14 … and we are left with ‘2’ remaining.
Again ‘borrow’ another ‘0’ so that we have ’20’, and we can subtract 2*7 from 20, meaning our answer is now 0.142 … with 6 remaining.
6 –> 60 – (8*7) –> 0.1428 & 4 remaining
4 –> 40 – (5*7) –> 0.14285 & 5 remaining
5 –> 50 – (7*7) –> 0.142857 & 1 remaining
1 –> 10 … now we’re back to our original problem, so repeating the steps above is just going to give the same answers which means our answer is going to have a repeating pattern …
0.142857142857142857…. etc.
*While not quite* ELI5 *level, this was how I learned long division when I was seven … which probably dates me somewhat :)*
It takes advantage on how we write numbers.
Lets do 123/5 with the same steps as in long division but in more rigorous fashion.
123 can be written as 1×100 + 2×10 + 3×1 (or with exponents as 1×10^2 + 2×10^1 + 3×10^(0). We’ll need this later).
So now we have ( 1×100 + 2×10 + 3×1 ) / 5.
We can write this as
( 1×100 ) / 5 + ( 2×10 ) / 5 + ( 3×1 ) / 5
And further.
1/5 × 100 + 2/5 × 10 + 3/5 × 1
Now we can start doing the divisions.
1/5 would result in some fraction. We do not accept any fractions here. So lets go back few steps and combine the 100 term and 10 terms together.
( 1×100 + 2×10 + 3×1 ) / 5 = ( 12×10 + 3×1 ) / 5. = 12/5 × 10 + 3/5 × 1
12/5 is 2 and leftover 2/5. Lets keep the 2 here and move the leftover to the next term. So 12/5 ×10 = 2×10 + 2/5×10.
12/5 × 10 + 3/5 × 1 = 2×10 + 2/5 × 10 + 3/5 ×1 = 2×10 + 23/5 × 1
23/5 is 4 and 3/5 as leftover.
2×10 + 23/5 × 1 = 2×10 + 4×1 + 3/5 × 1
Again lets keep the 4 here and move 3/5 to the next term.
But we do not have any terms left? We can just add more.
At the beginnign we wrote the number using decreasing powers of 10 (10^(2), 10^(1), 10^(0)). So we can just add 0×10^-1 (=0×0.1) in there! (it is just zero, we can add as many zeros to number as we like)
2×10 + 4×1 + 3/5 × 1 + 0×10^-1 = 2×10 + 4×1 + 30/5 ×10^-1
30/5 is nice even 6.
So our result is
2×10 + 4×1 + 6 ×10^-1
Now we just turn this back into normal number.
24.6
Long division does these steps. (lets see how long division works with reddit formatting…)
5|123
-0 5 doesn’t go into 1.
12 carry the 1 to the next term.
-10 5 goes into 12 2 times. 2×10 goes into the result so remove 2×5 from here.
23 And carry the remaining 2 to next term.
-20 5 goes into 23 4 times. 4×1 goes to the result so remove the 4×5 from here.
30 And carry the remaining 3 to the next term.
-30 5 goes into 30 6 times. 6×0.1 goes to the result so remove the 6×5 from here.
00 No more remainders to carry so we are done.
Result is 2×10 + 4×1 + 6×0.1 = 24.6
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